CorrectionKey=TX-A CorrectionKey=TX-B LESSON 9 . … Locker A2.6.G Analyze the effect on the graphs...

Resource Locker

A26G Analyze the effect on the graphs of f (x) = 1 __ x when f (x) is replaced by af (x) f (bx) f (x ndash c) and f (x) + d for specific positive and negative real values of a b c and d Also A22A A26K A27I

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Explore 1 Graphing and Analyzing f (x) = 1 __ x A rational function is a function of the form ƒ (x) = p (x)

___ q (x) where p(x) and q(x) are polynomials The most basic rational function with a variable expression in the denominator is ƒ (x) = 1 __ x

State the domain of ƒ (x) = 1 _ x

The function accepts all real numbers except because division by is undefined So the functionrsquos domain is as follows

bull As an inequality x lt or x gt

bull In set notation ⎧

⎨ ⎩ x| x ne

⎫

⎬ ⎭

bull In interval notation (where the symbol cup means union)

(-infin ) cup ( +infin) Determine the end behavior of ƒ (x) = 1 _ x

First complete the tables

Next summarize the results

bull As x rarr +infin ƒ (x) rarr

bull As x rarr -infin ƒ (x) rarr

x Increases without Bound

x f (x) = 1 _ x

100

1000

10000

x Decreases without Bound

x f (x) = 1 _ x

-100

-1000

-10000

0

0

0 0

0

001 -001

0001 -0001

00001 -00001

0 0

0

0

Module 9 469 Lesson 2

9 2 Graphing Simple Rational Functions

Essential Question How are the graphs of f (x) = a ( 1 ____ x - h ) + k and f (x) = 1 _______ 1 __ b (x - h)

+ k

related to the graph of f (x) = 1 __ x

DO NOT EDIT--Changes must be made through File infoCorrectionKey=TX-A

A2_MTXESE353947_U4M09L2indd 469 22114 113 AM

Texas Math StandardsThe student is expected to

A26G

Analyze the effect on the graphs of f (x) = 1 _ x when f(x) is replaced by af(x) f(bx) f(x ndash c) and f(x) + d for specific positive and negative real values of a b c and d Also A22A A26K A27I

Mathematical Processes

A21E

The student is expected to create and use representations to organize record and communicate mathematical ideas

Language Objective

1A 2i3 2I4 3E 3H3 4G

Explain to a partner the characteristics of the graphs of rational functions

HARDCOVER PAGES 333348

Turn to these pages to find this lesson in the hardcover student edition

Graphing Simple Rational Functions

ENGAGE Essential Question How are the

graphs of f (x) = a ( 1 ____

x - h ) + k and

f (x) = 1 _______

1 __ b

(x - h) + k related to

the graph of f (x) = 1 __ x Possible answer Both graphs involve

transformations of the graph of f (x) = 1 __ x The first

equation involves vertically stretching or

compressing the graph of f (x) = 1 __ x by a factor of a

reflecting it across the x-axis if a lt 0 and translating

it h units horizontally and k units vertically The

second equation involves horizontally stretching or

compressing the graph of f (x) = 1 __ x by a factor of b

reflecting it across the y-axis if b lt 0 and translating

it h units horizontally and k units vertically

PREVIEW LESSON PERFORMANCE TASKView the Engage section online Discuss the photo and how a one-time fee can be converted to a monthly fee Then preview the Lesson Performance Task

469

HARDCOVER PAGES

Turn to these pages to find this lesson in the hardcover student edition

Resource

Locker

A26G Analyze the effect on the graphs of f (x) = 1 __ x when f (x) is replaced by af (x) f (bx) f (x ndash c) and

f (x) + d for specific positive and negative real values of a b c and d Also A22A A26K A27I

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Explore 1 Graphing and Analyzing f (x) = 1 __ x

A rational function is a function of the form ƒ (x) = p (x)

___ q (x)

where p(x) and q(x) are polynomials

The most basic rational function with a variable expression in the denominator is ƒ (x) = 1 __ x

State the domain of ƒ (x) = 1 _ x

The function accepts all real numbers except because division by is undefined So

the functionrsquos domain is as follows

bull As an inequality x lt or x gt

bull In set notation ⎧ ⎨ ⎩ x| x ne

⎫ ⎬ ⎭

bull In interval notation (where the symbol cup means union)

(-infin ) cup ( +infin)

Determine the end behavior of ƒ (x) = 1 _ x




bull As x rarr -infin ƒ (x) rarr

x Increases without Bound

xf (x) =

1 _ x

100

1000

10000

x Decreases without Bound

xf (x) =

1 _ x

-100

-1000

-10000

0

0

0 0

0

001

-001

0001

-0001

00001

-00001

00

0

0

Module 9

469

Lesson 2

9 2 Graphing Simple Rational

Functions

Essential Question How are the graphs of f (x) = a ( 1 ____ x - h

) + k and f (x) = 1 _______ 1 __ b

(x - h) + k

related to the graph of f (x) = 1 __ x

DO NOT EDIT--Changes must be made through File info

CorrectionKey=TX-A

A2_MTXESE353947_U4M09L2indd 469

22114 113 AM

469 Lesson 9 2

L E S S O N 9 2

DO NOT EDIT--Changes must be made through File infoCorrectionKey=TX-B

2

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C Be more precise about the end behavior of ƒ (x) = 1 _ x and determine what this means for the graph of the function

You can be more precise about the end behavior by using the notation ƒ (x) rarr 0 + which means that the value of ƒ (x) approaches 0 from the positive direction (that is the value of ƒ (x) is positive as it approaches 0) and the notation ƒ (x) rarr 0 - which means that the value of ƒ (x) approaches 0 from the negative direction So the end behavior of the function is more precisely summarized as follows


bull As x rarr -infin ƒ (x) rarr The end behavior indicates that the graph of ƒ (x) approaches but does not cross the

[x-axisy-axis] so that axis is an asymptote for the graph

D Examine the behavior of ƒ (x) = 1 _ x near x = 0 and determine what this means for the graph of the function



bull As x rarr 0 + ƒ (x) rarr bull As x rarr 0 - ƒ (x) rarr

The behavior of ƒ (x) = 1 _ x near x = 0 indicates that the graph of ƒ (x) approaches but does not cross the [x-axisy-axis] so that axis is also an asymptote for the graph

E Graph ƒ (x) = 1 _ x

First determine the sign of ƒ (x) on the two parts of its domain

bull When x is a negative number ƒ (x) is a [positivenegative] numberbull When x is a positive number ƒ (x) is a [positivenegative] number

Next complete the tables

Finally use the information from this step and all previous steps to draw the graph Draw asymptotes as dashed lines

x Approaches 0 from the Positive Direction

x f (x) = 1 _ x

001

0001

00001

Negative Values of x

x f (x) = 1 _ x

-2

-1

-05

Positive Values of x

x f (x) = 1 _ x

05

1

2

x Approaches 0 from the Negative Direction

x f (x) = 1 _ x

-001

-0001

-00001

0 +

0 -

+infin

100 -100

-1000

-100001000

10000

-infin

-05 2

1

05

-1

-2


DO NOT EDIT--Changes must be made through File info CorrectionKey=TX-B


A2_MTXESE353947_U4M09L2 470 11315 854 AM

PROFESSIONAL DEVELOPMENT

Learning ProgressionsPreviously students looked at polynomial functions and their transformations inverse variation applications and the graphs of rational functions of the form f (x) = a __ x in the first quadrant Students now extend that knowledge by graphing and transforming the graph of the parent function f (x) = 1 __ x They learn how the values of h and k are related to the graph of f (x) = a ____ x - h + k and graph rational functions in the general form ax + b

_____ cx + d where ax + b and cx + d are both linear functions In the next lesson students examine the quotients of linear and quadratic polynomials and discontinuities

EXPLORE 1 Graphing and Analyzing f(x) = 1 __ X

INTEGRATE TECHNOLOGYStudents may complete the Explore activity either in the book or online

QUESTIONING STRATEGIESWhy are there two parts to the graph of the rational function There are positive values

and negative values for x but no value when x is 0

Why isnrsquot the graph symmetric across the y-axis Positive values of x result in positive

values of y but negative values of x result in

negative values of y

Graphing Simple Rational Functions 470


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FState the range of ƒ (x) = 1 _ x

The function takes on all real numbers except so the functionrsquos range is as follows

bull As an inequality y lt or y gt


⎨ ⎩ y|y ne ⎫

⎬ ⎭

bull In interval notation (where the symbol cup means union) (-infin ) cup ( +infin) GIdentify the intervals where the function is increasing and where it is decreasing

H Determine whether ƒ (x) = 1 _ x is an even function an odd function or neither

Reflect

1 How does the graph of ƒ (x) = 1 _ x show that the function has no zeros

2 Discussion A graph is said to be symmetric about the origin (and the origin is called the graphrsquos point of symmetry) if for every point (x y) on the graph the point (-x -y) is also on the graph Is the graph of ƒ (x) = 1 __ x symmetric about the origin Explain

3 Give any line(s) of symmetry for the graph of ƒ (x) = 1 _ x

Explore 2 Predicting Transformations of the Graph of f (x) = 1 __ x In an earlier lesson you learned how to transform the graph of a function using reflections across the x- and y-axes vertical and horizontal stretches and compressions and translations Here you will apply those transformations to the graph of the function ƒ (x) = 1 _ x

A Predict the effect of the parameter h on the graph of g (x) = 1 _ x - h

for each function

a The graph of g (x) = 1 _____ x - 3 is a of the graph of ƒ (x) [rightleftupdown] 3 units

b The graph of g (x) = 1 _____ x + 3 is a of the graph of ƒ (x) [rightleftupdown] 3 units Check your predictions using a graphing calculator

0

0

0

0 0

0

The function is never increasing

The function is decreasing on the intervals (-infin 0) and (0 +infin)

f (-x) = 1 _ -x = - 1 _ x = -f (x) so f (x) = 1 _ x is an odd function

The graph of the function never crosses the x-axis

The lines y = x and y = -x

Yes because for every point (x y) = (x 1 _ x ) on the graph of the function the point

(-x 1 _ -x ) = (-x - 1 _ x ) = (-x -y) is also on the graph

translation

translation



A2_MTXESE353947_U4M09L2 471 11115 957 AM

COLLABORATIVE LEARNING

Peer-to-Peer ActivityHave students work in pairs Give each pair graphs of eight transformed rational functions labeled A through H and cards with the eight functions on them Set aside one additional graph and function that do not match Have each pair match the correct functions and graphs and then decide why the remaining graph and function do not match

EXPLORE 2 Predicting Transformations of the Graph of f(x) = 1 __ x

QUESTIONING STRATEGIESHow is the process of graphing a rational function of the form f (x) = 1 ____ x - h + k related

to the process of graphing a polynomial function of the form f (x) = (x ndash h) + k For both start with the

graph of the parent function and translate the

graph h units horizontally and k units vertically

MULTIPLE REPRESENTATIONSHave students consider a graph of a rational function with a horizontal asymptote at y = 3 and a vertical asymptote at x = minus2 and have them state what form the function representing the graph would take Students should be able to conclude that h is -2 and k = 3 in a function of the form f (x) = a _____ x - h + k

471 Lesson 9 2


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B Predict the effect of the parameter k on the graph of g (x) = 1 _ x + k for each function

a The graph of g (x) = 1 _ x + 3 is a of the graph of ƒ (x) [rightleftupdown] 3 units

b The graph of g (x) = 1 _ x - 3 is a of the graph of ƒ (x) [rightleftupdown] 3 units

Check your predictions using a graphing calculator

C Predict the effect of the parameter a on the graph of g (x) = a ( 1 _ x ) for each function

a The graph of g (x) = 2 ( 1 __ x ) is a of the graph of ƒ (x) by a factor of

b The graph of g (x) = 05 ( 1 __ x ) is a of the graph of ƒ (x) by a factor of

c The graph of g (x) = minus05 ( 1 __ x ) is a of the graph of ƒ (x) by a factor of as

well as a across the

d The graph of g (x) = minus2 ( 1 __ x ) is a of the graph of ƒ (x) by a factor

of as well as a across the


D Predict the effect of the parameter b on the graph of g (x) = 1 ___ 1 __ b x

for each function

a The graph of g (x) = 1 ____ 05x is a of the graph of ƒ (x) by a

factor of

b The graph of g (x) = 1 __ 2x is a of the graph of ƒ (x) by a factor of

c The graph of g (x) = 1 _____ -05x is a of the graph of ƒ (x) by a factor of

as well as a across the

d The graph of g (x) = 1 ___ -2x is a of the graph of ƒ (x) by a factor of


translation

translation

horizontal stretch

2

1 __ 2 horizontal compression

horizontal stretch

reflection y-axis

1 __ 2

2

horizontal compression

reflection y-axis

vertical stretch

1 __ 2 vertical compression

vertical compression

2

1 __ 2

reflection x-axis

vertical stretch

2 reflection x-axis




A2_MTXESE353947_U4M09L2 472 11115 1002 AM

DIFFERENTIATE INSTRUCTION

Multiple RepresentationsInterpersonal learners may benefit from working in pairs with one student graphing a quotient of linear functions by rewriting the function in graphing form and the other graphing the function by making a table of values and plotting points Students should compare the strategies by identifying at least one advantage and one disadvantage for each method



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Reflect

4 For each function from Step D there is a function from Step C that has an identical graph even though the graphs of the two functions involve different transformations of the graph of ƒ (x) Using the letters and numbers in the lists below match each function from Step D with a function from Step C and show why the functions are equivalent

Functions from Step D Functions from Step C

1 g (x) = 1 _ 05x a g (x) = 2 ( 1 _ x ) 2 g (x) = 1 _ 2x b g (x) = 05 ( 1 _ x ) 3 g (x) = 1 _ -05x c g (x) = -05 ( 1 _ x ) 4 g (x) = 1 _ -2x d g (x) = -2 ( 1 _ x )

Explain 1 Graphing Simple Rational FunctionsWhen graphing transformations of ƒ (x) = 1 _ x it helps to consider the effect of the transformations on the following features of the graph of ƒ (x) the vertical asymptote x = 0 the horizontal asymptote y = 0 and two reference points (minus1 minus1) and (1 1) The table lists these features of the graph of

ƒ (x) and the corresponding features of the graph of g (x) = a ( 1 _______ 1 __ b (x - h)

) + k Note that the asymptotes

are affected only by the parameters h and k while the reference points are affected by all four parameters

Feature f (x) = 1 _ x g (x) = a ( 1 _

1 _ b

(x - h) ) + k

Vertical asymptote x = 0 x = h

Horizontal asymptote y = 0 y = k

Reference point (minus1 minus1) (minusb + h minusa + k) Reference point (1 1) (b + h a + k)

Example 1 Identify the transformations of the graph of ƒ (x) = 1 __ x that produce the graph of the given function g (x) Then graph g (x) on the same coordinate plane as the graph of ƒ (x) by applying the transformations to the asymptotes x = 0 and y = 0 to the reference points (minus1 minus1) and (1 1) Also state the domain and range of g (x) using inequalities set notation and interval notation

Match (1) with (a) because 1 ____ 05x = 1 ___ 1 __ 2 x

= 2 __ x = 2 ( 1 __ x )

Match (2) with (b) because 1 ___ 2x = 1 __ 2 ( 1 __ x ) = 05 ( 1 __ x )

Match (3) with (d) because 1 _____ minus05x = 1 ____ minus 1 __

2 x

= minus2 ___ x = minus2 ( 1 __ x )

Match (4) with (c) because 1 ____ minus2x = minus 1 __ 2

( 1 __ x ) = minus05 ( 1 __ x )



A2_MTXESE353947_U4M09L2 473 11115 1004 AM

EXPLAIN 1 Graphing Simple Rational Functions

QUESTIONING STRATEGIES What is the midpoint of the x-coordinates of the two reference points (h k)

because (-b + h) + (b + h)

______________ 2 = 2h ___ 2 = h

and (-a + k) + (a + k)

_____________ 2 = 2k __ 2 = k

473 Lesson 9 2


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A g (x) = 3 ( 1 _ x minus 1 ) + 2

The transformations of the graph of ƒ (x) that produce the graph of g (x) arebull a vertical stretch by a factor of 3bull a translation of 1 unit to the right and 2 units up

Note that the translation of 1 unit to the right affects only the x-coordinates while the vertical stretch by a factor of 3 and the translation of 2 units up affect only the y-coordinates

Feature f (x) = 1 _ x g (x) = 3 ( 1 _ x - 1

) + 2

Vertical asymptote x = 0 x = 1

Horizontal asymptote y = 0 y = 2

Reference point (minus1 minus1) (minus1 + 1 3 (minus1) + 2) = (0 minus1)

Reference point (1 1) (1 + 1 3 (1) + 2) = (2 5)

Domain of g (x)

Inequality x lt 1 or x gt 1

Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫

⎬ ⎭

Interval notation (minusinfin 1) ⋃ (1 +infin)

Range of g (x)

Inequality y lt 2 or y gt 2

Set notation ⎧

⎨ ⎩ y| y ne 2 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 474 11315 810 AM

AVOID COMMON ERRORSStudents may make errors when finding a reference point Have students check them by confirming that the midpoint of the segment joining them is (h k)



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B g (x) = 1 _ 2 (x + 3)

- 1

The transformations of the graph of ƒ (x) that produce the graph of g (x) arebull a horizontal compression by a factor of 1 __ 2 bull a translation of 3 units to the left and 1 unit down

Note that the horizontal compression by a factor of 1 __ 2 and the translation of 3 units to the left affect only the x-coordinates of points on the graph of ƒ (x) while the translation of 1 unit down affects only the y-coordinates

Feature f (x) = 1 _ x g (x) = 1 _ 2 (x + 3)

minus 1

Vertical asymptote x = 0 x =

Horizontal asymptote y = 0 y =

Reference point (minus1 minus1) ( 1 _ 2

( ) minus3 minus1) = ( )

Reference point (1 1) ( 1 _ 2


Domain of g (x)

Inequality x lt or x gt

Set notation ⎧

⎨ ⎩ x|x ne ⎫

⎬ ⎭

Interval notation (-infin ) cup ( +infin) Range of g (x)

Inequality y lt or y gt

Set notation ⎧

⎨ ⎩ y|y ne ⎫

⎬ ⎭

Interval notation (-infin ) cup ( +infin)

minus3

minus1

minus1 minus3 1 _ 2

minus1 minus2

1 1 minus2 1 _ 2

0

minus3

minus3 minus3

minus3

minus3

minus1

minus1

minus1minus1

minus1


DO NOT EDIT--Changes must be made through File info CorrectionKey=TX-A


CONNECT VOCABULARY Connect the term rational function to rational numbers including fractions Remind students that a simple rational function has polynomials in the numerator and denominator

475 Lesson 9 2


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Your Turn

Identify the transformations of the graph of ƒ (x) = 1 __ x that produce the graph of the given function g (x) Then graph g (x) on the same coordinate plane as the graph of ƒ (x) by applying the transformations to the asymptotes x = 0 and y = 0 to the reference points (-1 -1) and (1 1) Also state the domain and range of g (x) using inequalities set notation and interval notation

5 g (x) = minus05 ( 1 _ x + 1 ) minus 3

6 g (x) = 1 __ minus05 (x minus 2)

+ 1

The transformations of the graph of f (x) that produce the graph of g (x) are

bulla vertical compression by a factor of 1 _ 2

bulla reflection across the x-axis

bulla translation of 1 units to the left and 3 units down

Vert Asymp x = 0 becomes x = -1

Horiz Asymp y = 0 becomes y = -3

(-1 -1) becomes (-2 - 5 _ 2

)

(1 1) becomes (0 - 7 _ 2

)

Domain of g (x) Range of g (x)

Inequality x lt -1 or x gt minus1 Inequality y lt minus3 or y gt minus3

Set notation ⎧

⎨ ⎩ x| x ne minus1 ⎫

⎬ ⎭ Set notation ⎧

⎨ ⎩ y| y ne minus3 ⎫

⎬ ⎭

Interval notation (minusinfin minus1) ⋃ (minus1 +infin) Interval notation (minus infin minus3) ⋃ (minus3 +infin)


bulla horizontal stretch by a factor of 2

bulla reflection across the y-axis

bulla translation of 2 units to the right and 1 unit up

Vert Asymp x = 0 becomes x = 2

Horiz Asymp y = 0 becomes y = 1

(ndash1 ndash1) becomes (4 0)

(1 1) becomes (0 2)


Inequality x lt 2 or x gt 2 Inequality y lt 1 or y gt 1

Set notation ⎧

⎨ ⎩ x|x ne 2 ⎫


⎨ ⎩ y|y ne 1 ⎫

⎬ ⎭

Interval notation (minus infin 2) ⋃ (2 +infin) Interval notation (-infin 1) ⋃ (1 +infin)




A2_MTXESE353947_U4M09L2 476 11115 1007 AM



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Explain 2 Rewriting Simple Rational Functions in Order to Graph Them

When given a rational function of the form g (x) = mx + n______px + q where m ne 0 and p ne 0 you can carry out the division of

the numerator by the denominator to write the function in the form g (x) = a ( 1____x - h) + k or g (x) = 1______

1 __ b (x - h) + k

in order to graph it

Example 2 Rewrite the function in the form g (x) = a ( 1 ____ x - h ) + k or g (x) = 1 ______ 1 __ b (x - h)

+ k

and graph it Also state the domain and range using inequalities set

notation and interval notation

g (x) = 3x - 4 _ x - 1

Use long division

3 x - 1 ⟌ ndashndashndashndashndashndashndash 3x - 4

3x - 3 _ -1

So the quotient is 3 and the remainder is -1 Using the fact that dividend = quotient + remainder ________ divisor you have g (x) = 3 + -1 ____ x - 1 or g (x) = - 1 ____ x - 1 + 3

The graph of g (x) has vertical asymptote x = 1 horizontal asymptote y = 3

and reference points (-1 + 1 - (-1) + 3) = (0 4) and ( 1 + 1 - ( 1 ) + 3 ) = ( 2 2 )

Domain of g (x) Domain of g (x) Inequality x lt 1 or x gt 1 Inequality y lt 3 or y gt 3

Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭

Interval notation (-infin 1) ⋃ (1 +infin) Interval notation (-infin 3) ⋃ (3 +infin)

g (x) = 4x - 7 _ -2x + 4

Use long division

-2 -2x + 4 ⟌ ndashndashndashndashndashndashndash 4x - 7

4x - 8 _

So the quotient is -2 and the remainder is Using the fact that dividend = quotient + remainder _ divisor

you have

g (x) = -2 + _ -2x + 4 or g (x) = __

-2 (x - ) - 2

1

1

1

2

1



A2_MTXESE353947_U4M09L2 477 11115 1013 AM

EXPLAIN 2 Rewriting Simple Rational Functions in Order to Graph Them

INTEGRATE MATHEMATICAL PROCESSESFocus on Math ConnectionsRemind students that division by a linear factor in the form x - h can be done by synthetic division However for simple quotients it is quicker to use long division

AVOID COMMON ERRORSStudents may not get the correct sign of the remainder when performing long division Remind them to consider the operation signs For example when subtracting 3x - 3 from 3x - 4 the result is -1

477 Lesson 9 2


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-2

2

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The graph of g (x) has vertical asymptote x = horizontal asymptote y = -2 and reference points

( - 1 _ 2 (-1) + -1 - 2 ) = ( -3) and (- 1 _ 2 (1) + 1 - 2) = ( - 1) Domain of g (x)


Set notation ⎧

⎨

⎩ x|x ne

⎫

⎬

⎭

Interval notation (-infin ) ⋃ ( + infin) Range of g (x)


Set notation ⎧

⎨

⎩ y|y ne

⎫

⎬

⎭

Interval notation (-infin ) ⋃ ( +infin) Reflect

7 In Part A the graph of g (x) is the result of what transformations of the graph of f (x) = 1 __ x

8 In Part B the graph of g (x) is the result of what transformations of the graph of f (x) = 1 __ x

2

2

2

-2

2

-2

2

-2

2

-2

2

-2

22 1 __ 2 1 1 __ 2

The graph of g (x) is a reflection of the graph of f (x) across the x-axis and a translation

1 unit to the right and 3 units up

The graph of g (x) is a horizontal compression of the graph of f (x) by a factor of 1 __ 2 a

reflection in the y-axis and a translation 2 units to the right and 2 units down




A2_MTXESE353947_U4M09L2 478 11115 1023 AM

QUESTIONING STRATEGIESWhy are the domain and range of the graphs of rational functions disjunctions or

inequalities and not conjunctions The domain

does not include the x-asymptote but it does

contain the values of x on each side of the

asymptote This is also true of the range and the

y-asymptote



6

2

2

y

x0

4

-4-6 -2

6

2

8

4 62

y

x0

(2 2)

(4 6)

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pan

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Your Turn

Rewrite the function in the form g (x) = a ( 1 _____ x - h ) + k or g (x) = 1 ________ 1 __ b (x - h)

+ k and

graph it Also state the domain and range using inequalities set notation and interval notation

9 g (x) = 3x + 8 _ x + 2

Explain 3 Writing Simple Rational FunctionsWhen given the graph of a simple rational function you can write its equation using one of the general forms g (x) = a ( 1 ____ x - h ) + k and g (x) = 1 ______

1 __ b (x - h) + k after identifying the values of the parameters using information obtained

from the graph

Example 3

Write the equation of the function whose graph is shown Use the form g (x) = a ( 1 _____ x - h

) + k

Since the graphrsquos vertical asymptote is x = 3 the value of the parameter h is 3 Since the graphrsquos horizontal asymptote is y = 4 the value of the parameter k is 4

Substitute these values into the general form of the function

g (x) = a ( 1 _ x - 3 ) + 4

Now use one of the points such as (4 6) to find the value of the parameter a

g (x) = a ( 1 _ x - 3 ) + 4

6 = a ( 1 _ 4 - 3 ) + 4

6 = a + 4 2 = a

So g (x) = 2 ( 1 _____ x - 3 ) + 4

3x + 2 ⟌ ndashndashndashndashndashndashndash 3x + 8

3x + 6

_ 2

So g (x) = 3 + 2 _ x + 2

or g (x) = 2 ( 1 _ x + 2

) + 3

The graph of g (x) has vertical asymptote x = -2 horizontal asymptote y = 3 and reference points (-1 - 2 2 (-1) + 3) = (-3 1) and (1 - 2 2 (1) + 3) = (-1 5)

Domain of g (x) Domain of g (x)

Inequality x lt -2 or x gt -2 Inequality y lt 3 or y gt 3

Set notation ⎧

⎨ ⎩ x|x ne -2 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭

Interval notation (-infin -2) ⋃ (-2 +infin) Interval notation (-infin 3) ⋃ (3 +infin)



A2_MTXESE353947_U4M09L2 479 11115 1026 AM

EXPLAIN 3 Writing Simple Rational Functions

QUESTIONING STRATEGIESHow do you know that the graph can be modeled by a rational function There is a

vertical asymptote there is a horizontal asymptote

as x decreases without bound f (x) approaches the

horizontal asymptote as x increases without bound

f (x) approaches the horizontal asymptote as x

approaches the vertical asymptote from the left

f (x) rarr -infin as x approaches the vertical asymptote

from the right f (x) rarr infin

479 Lesson 9 2


-4

2

-4 -2-6

y

x0

(-5 0)

(-1 -2)

2

4

-4 2-6

y

x0

(-3 5)

(-1 -3)

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pany

B Write the equation of the function whose graph is shown Use the form g (x) = 1 _______ 1 __ b (x - h)

+ k

Since the graphrsquos vertical asymptote is x = -3 the value of the

parameter h is -3 Since the graphrsquos horizontal asymptote is y =

the value of the parameter k is


g (x) = 1 _ 1 _ b

(x + 3) +

Now use one of the points such as (-5 0) to find the value of the parameter a

g (x) = 1 _ 1 _ b

(x + 3) +

0 = 1 _ 1 _ b

(-5 + 3) +

= 1 _ 1 _ b

(-2)

1 _ b

(-2) = 1

1 _ b

=

b =

So g (x) = 1 __ 1 _ (x + 3)

+ or g (x) = 1 __ -05 (x + 3)

-1

Reflect

10 Discussion In Parts A and B the coordinates of a second point on the graph of g (x) are given In what way can those coordinates be useful

Your Turn

11 Write the function whose graph is shown Use the form g (x) = a ( 1 _____ x - h ) + k

-1

-1

-1

-1

-1

1

1

-1 ___ 2

-2

-1

-2

You can use those coordinates as a check on the correctness of the equation by

substituting the coordinates into the equation and seeing if you get a true statement

h = -2

k = 1

Use the point (-3 5)

g (x) = a ( 1 _ x + 2

) + 1

5 = a ( 1 _ -3 + 2 ) + 1

5 = -a + 1 4 = -a -4 = a

So g (x) = -4 ( 1 _ x + 2

) + 1




A2_MTXESE353947_U4M09L2 480 11115 1028 AM



-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

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Com

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mag

e C

red

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copyZu

zan

a D

ole

zalo

vaA

lam

y

12 Write the equation of the function whose graph is shown Use the form g (x) = 1 _______ 1 _ b

(x - h) + k

Explain 4 Modeling with Simple Rational FunctionsIn a real-world situation where there is a shared cost and a per-person or per-item cost you can model the situation using a rational function that has the general form f (x) = a ____ x - h + k where f (x) is the total cost for each person or item

Example 4

Mary and some of her friends are thinking about renting a car while staying at a beach resort for a vacation The cost per person for staying at the beach resort is $300 and the cost of the car rental is $220 If the friends agree to share the cost of the car rental what is the minimum number of people who must go on the trip so that the total cost for each person is no more than $350

Analyze Information

Identify the important informationbull The cost per person for the resort is

bull The cost of the car rental is

bull The most that each person will spend is

Formulate a Plan

Create a rational function that gives the total cost for each person Graph the function and use the graph to answer the question

h = 4

k = -3

Use the point (45 -2)

g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM

EXPLAIN 4 Modeling with Simple Rational Functions

QUESTIONING STRATEGIESWhat indicates that a rational function is a good model for a situation A constant

quantity divided among varying numbers of

another quantity is one possibility

481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

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pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve

Let p be the number of people who agree to go on the trip Let C (p) be the cost (in dollars) for each person

C (p) = _ p +

Graph the function recognizing that the graph involves two transformations of the graph of the parent rational function

bull a vertical stretch by a factor of

bull a vertical translation of units up

Also draw the line C (p) = 350

The graphs intersect between p = and p = so the minimum number of people who must go on the trip in order for the total cost for each person to be no more than $350

is

Justify and Evaluate

Check the solution by evaluating the function C (p) Since C (4) = gt 350 and C (5) = lt 350 the minimum number of people who must go on the trip is

Your Turn

13 Justin has purchased a basic silk screening kit for applying designs to fabric The kit costs $200 He plans to buy T-shirts for $10 each apply a design that he creates to them and then sell them Model this situation with a rational function that gives the average cost of a silk-screened T-shirt when the cost of the kit is included in the calculation Use the graph of the function to determine the minimum number of T-shirts that brings the average cost below $1750

220300

220

300

4 5

5

5355

344

Let s be the number of T-shirts Let C (s) be the average cost

(in dollars) of the silk-screened T-shirts when the cost of the

kit is included C (s) = 200 _ s + 10

The graphs appear to intersect between s = 26 and s = 27

Since C (26) = 200 ____ 26 + 10 asymp 1769 and C (26) = 200 ____ 27 + 10 asymp 1741

the minimum number of T-shirts that brings the average cost

below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM

INTEGRATE MATHEMATICAL PROCESSESFocus on TechnologyStudents can graph a rational function using a graphing calculator and use the TRACE function to explore the relationship between the quantities After TRACE is pressed the left and right arrow keys move the cursor along the graph As the cursor moves its x- and y-coordinates are updated at the bottom of the viewing window



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Elaborate

14 Compare and contrast the attributes of ƒ (x) = 1 __ x and the attributes of g (x) = - 1 __ x

15 State the domain and range of ƒ (x) = a ( 1 ____ x - h ) + k using inequalities set notation and interval notation

16 Given that the model C (p) = 100 ___ p + 50 represents the total cost C (in dollars) for each person in a group of p people when there is a shared expense and an

individual expense describe what the expressions 100 ___ p and 50 represent

17 Essential Question Check-In Describe the transformations you must perform on the graph of f (x) = 1 __ x to obtain the graph of ƒ (x) = a ( 1 ____ x - h ) + k

Both functions have a domain that excludes 0 and a range that excludes 0 The graphs of

both functions have the x-axis and the y-axis as asymptotes However the graphs approach

the asymptotes from different directions (For instance while the graph of f (x) approaches

the x-axis from the positive direction as x increases without bound the graph of g (x)

approaches the x-axis from the negative direction as x increases without bound) The graph

of f (x) lies in Quadrants I and III whereas the graph of g (x) lies in Quadrants II and IV While

f (x) decreases on both parts of its domain g (x) increases on both parts of its domain

The expression 100 ____ p represents each personrsquos share of a $100 expense for the group The

expression 50 represents an individual expense of $50

If a lt 0 reflect the parent graph across the x-axis Then either stretch the graph vertically

by a factor of a if |a| gt 1 or compress the graph vertically by a factor of a if |a| lt 1 Finally

translate the graph h units horizontally and k units vertically

Domain

Inequality x lt h or x gt h

Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭

Interval notation (-infin h) ⋃ (h + infin) Range

Inequality y lt k or y gt k

Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭

Interval notation (-infin k) ⋃ (k + infin)



A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT

Communicate MathHave students work in pairs to look at graphs of rational functions and their equations The first student describes the graph to the second student identifying the asymptotes and the domain range and end behavior of the function Students switch roles and repeat the process with a different rational function and its graph

ELABORATE INTEGRATE MATHEMATICAL PROCESSESFocus on Critical ThinkingAsk how could you graph C (a) = a + 5

_____ a + 20 in the first quadrant without using a graphing calculator Elicit that you know that the graph begins at (0 025) and that you can determine that the graph has C(a) = 1 as its horizontal asymptote (which is the functionrsquos end behavior as a increases without bound) You can then make a table of values for several positive values of a plot the points and draw a smooth curve that approaches the line C(a) = 1 for greater values of a

QUESTIONING STRATEGIESFor the function f (x) = 2x - 5 _____ x - 4 what attributes of the graph can be directly read from the

coefficients y = 2 is the horizontal asymptote

x = 4 is the vertical asymptote

SUMMARIZE THE LESSONHow do you transform the graph of f (x) = 1 __ x f (x) = a ( 1

_____ 1 __ b

x - h ) + k vertically stretches or

compresses the graph of f (x) = 1 __ x by a factor of a and reflects it across the x-axis if a lt 0 horizontally stretches or compresses the graph by a factor of b and reflects it across the y-axis if b lt 0 and translates it h units horizontally and k units vertically

483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

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arcourt Publishin

g Com

pany

bull Online Homeworkbull Hints and Helpbull Extra Practice

Evaluate Homework and Practice

Describe how the graph of g (x) is related to the graph of ƒ (x) = 1 __ x

1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x

Identify the transformations of the graph of f (x) that produce the graph of the given function g (x) Then graph g (x) on the same coordinate plane as the graph of f (x) by applying the transformations to the asymptotes x = 0 and y = 0 and to the reference points (-1 -1) and (1 1) Also state the domain and range of g (x) using inequalities set notation and interval notation

9 g (x) = 3 ( 1 _ x + 1 ) - 2

The graph of g (x) is a translation of the graph of f (x) up 4 units

The graph of g (x) is a vertical stretch of the graph of f (x) by a factor of 5

The graph of g (x) is a translation of the graph of f (x) left 3 units

The graph of g (x) is a horizontal stretch of the graph of f (x) by a factor of 10

The graph of g (x) is a translation of the graph of f (x) down 7 units

The graph of g (x) is a vertical compression of the graph of f (x) by a factor of 01 as well as a reflection of the graph of f (x) across the x-axis

The graph of g (x) is a translation of the graph of f (x) right 8 units

The graph of g (x) is a horizontal

compression of the graph of f (x) by a

factor of 1 _ 3 as well as a reflection of the

graph of f (x) across the y-axis

The transformations are

bull a vertical stretch by a factor of 3

bull a translation of 1 unit to the left and 2 units down


Inequality x lt -1 or x gt -1 Inequality y lt -2 or y gt -2

Set notation x∣x ne -1 Set notation y∣y ne -2

Interval notation (-infin -1) ⋃ (-1 +infin) Interval notation (-infin -2) ⋃ (-2 +infin)




A2_MTXESE353947_U4M09L2indd 484 22114 106 AMExercise Depth of Knowledge (DOK) Mathematical Processes

1ndash8 1 Recall of Information 1F Analyze relationships

9ndash11 2 SkillsConcepts 1F Analyze relationships


21 1 Recall of Information 1A Everyday life

22 2 SkillsConcepts 1A Everyday life

23 2 SkillsConcepts 1C Select tools

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

Explore 1Graphing and Analyzing f (x) = 1 __ x

Explore 2Predicting Transformations of the Graph of f (x) = 1 __ x

Exercises 1ndash8

Example 1Graphing Simple Rational Functions

Exercises 9ndash12

Example 2Rewriting Simple Rational Functions in Order to Graph Them

Exercises 13ndash16

Example 3Writing Simple Rational Functions

Exercises 17ndash20

Example 4Modeling with Simple Rational Functions

Exercises 21ndash22

QUESTIONING STRATEGIESIf the coordinates of (h k) are (-6 15) and a and b are each equal to 2 what are the

coordinates of the reference points (-2 + (-6)

(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

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10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

The transformations arebulla horizontal stretch by a factor of 2





Set notation x∣x ne 3 Set notation y∣y ne 1Interval notation (-infin 3) ⋃ (3 +infin) Interval notation (-infin 1) ⋃ (1 +infin)

The transformations arebulla vertical compression by a factor of 1 _ 2


bulla translation of 1 unit to the right and 2 units down


Inequality x lt 1 or x gt 1 Inequality y lt -2 or y gt -2

Set notation x∣x ne 1 Set notation y∣y ne -2

Interval notation (-infin 1) ⋃ (1 +infin) Interval notation (-infin -2) ⋃ (-2 +infin)

The transformations arebulla horizontal compression by a factor of 1 _ 2

bulla translation of 2 units to the left and 3 units up



Set notation x∣x ne -2 Set notation y∣y ne 3





24ndash25 3 Strategic Thinking 1G Explain and justify arguments

VISUAL CUESAfter horizontal and vertical asymptotes of a graph have been identified visual learners may benefit from drawing horizontal and vertical arrows to find the reference points

485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Rewrite the function in the form g (x) = a 1 ______ (x - h)

+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it

Also state the domain and range using inequalities set notation and interval notation

13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2

3x - 1 ⟌ ndashndashndashndashndashndashndash 3x - 5

――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3




205x + 2 ⟌ ndashndashndashndashndashndash x + 5

―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2

(ndash1 ndash1) becomes (ndash6 1)

(1 1) becomes (ndash2 3)



Set notation x∣x ne -4 Set notation y∣y ne 2Interval notation (-infin -4) ⋃ (-4 +infin) Interval notation (-infin 2) ⋃ (2 +infin)

-4x - 2 ⟌ ndashndashndashndashndashndashndashndashndashndash -4x + 11

―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4

(ndash1 ndash1) becomes (1 ndash7)

(1 1) becomes (3 ndash1) Domain of g (x) Range of g (x)


Set notation x∣x ne 2 Set notation y∣y ne -4Interval notation (-infin 2) ⋃ (2 +infin) Interval notation (-infin -4) ⋃ (-4 +infin)

Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)

Vert Asymp x = ndash4

Horiz Asymp y = 2

Vert Asymp x = 2

Horiz Asymp y = ndash4




A2_MTXESE353947_U4M09L2 486 230214 256 PM

CONNECT VOCABULARY Relate the term reference point to the noun reference meaning a source of information A reference on a job application is person who serves as a source for information about the applicant



-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

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16 g (x) = 4x + 13 _ -2x - 6

17 Write the function whose graph is shown Use the form g (x) = a ( 1 ____ x - h ) + k

18 Write the function whose graph is shown Use the form g (x) = 1 ______ 1 _ b

(x - h) + k

-2-2x - 6 ⟌ ndashndashndashndashndashndashndashndash 4x + 13

――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3


(ndash1 ndash1) becomes ( ndash5 __ 2 ndash3)

(1 1) becomes ( ndash7 __ 2 ndash1)



Set notation x∣x ne -3 Set notation y∣y ne -2Interval notation (-infin -3) ⋃ (-3 +infin) Interval notation (-infin -2) ⋃ (-2 +winfin)

h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2




AVOID COMMON ERRORSFor real-world problems students may not check to see whether the domain is continuous or discrete and if it is discrete how a solution should be rounded Stress the importance of checking whether solutions are defined and whether it is appropriate to round up or down

487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)

Number of suncatchers

C

s

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pany bull Im

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redits copy

Bernie Pearson

Alam

y


(x - h) + k


21 Maria has purchased a basic stained glass kit for $100 She plans to make stained glass suncatchers and sell them She estimates that the materials for making each suncatcher will cost $15 Model this situation with a rational function that gives the average cost of a stained glass suncatcher when the cost of the kit is included in the calculation Use the graph of the function to determine the minimum number of suncatchers that brings the average cost below $2250

h = 3

k = -1

Use the point (35 0)

0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4

Use the point (-1 -6)

-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4

Let s be the number of suncatchers that Maria makes Let C(s) be the average cost (in dollars) of a suncatcher when the cost of the kit is included

C (s) = 100 _ s + 15

The graphs appear to intersect between s = 13

and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14

+ 15 asymp 2214 the minimum number

of suncatchers that brings the average cost below

$2250 is 14





INTEGRATE MATHEMATICAL PROCESSESFocus on Math ConnectionsStudents may lightly connect the reference points to see that they are symmetric about (h k)



0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)

Number of invitations

C

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Com

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22 Amy has purchased a basic letterpress kit for $140 She plans to make wedding invitations She estimates that the cost of the paper and envelope for each invitation is $2 Model this situation with a rational function that gives the average cost of a wedding invitation when the cost of the kit is included in the calculation Use the graph of the function to determine the minimum number of invitations that brings the average cost below $5

23 Multiple Response Select the transformations of the graph of the parent rational function that result in the graph of g (x) = 1 ______

2 (x - 3) + 1

A Horizontal stretch by a factor of 2 E Translation 1 unit up

B Horizontal compression by a factor of 1 _ 2 F Translation 1 unit down

C Vertical stretch by a factor of 2 G Translation 3 units right

D Vertical compression by a factor of 1 __ 2 H Translation 3 units left

HOT Focus on Higher Order Thinking

24 Justify Reasoning Explain why for positive numbers a and b a vertical stretch or compression of the graph of ƒ (x) = 1 __ x by a factor of a and separately a horizontal stretch or compression of the graph of ƒ (x) by a factor of b result in the same graph when a and b are equal

25 Communicate Mathematical Ideas Determine the domain and range of the rational function g (x) = mx + n _____ px + q where p ne 0 Give your answer in set notation and explain your reasoning

Let w be the number of wedding invitations that Maria makes Let C (w) be the average cost (in dollars) of a wedding invitation when the cost of the kit is included

C (w) = 140 _ w + 2

The graphs appear to intersect between w = 46

and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47

+ 2 asymp 498 the minimum number of

invitations that brings the average below $5 is 47

A horizontal stretch or compression of the graph of f (x) by a factor of b is given by f ( 1 __ b

x)

which you can rewrite as follows f ( 1 __ b

x) = 1 ___ 1 __ b

x = 1 __

1 __ b

∙ 1 __ x = b ∙ 1 __ x = b ∙ f (x) Since a vertical

stretch or compression of the graph of f (x) by a factor of a is given by a ∙ f (x) you can

conclude that the two transformations are identical provided a = b

The domain consists of all real numbers except for the value of x that makes the denominator

equal to 0 Solving px + q = 0 for x gives x = - q __ p So the domain is ⎧ ⎨ ⎩ x | |

x ne - q __ p ⎫ ⎬ ⎭ When you

divide the numerator by the denominator the quotient determines the horizontal asymptote

of the graph and this y-value is the only real number that isnrsquot in the range of the function

Dividing mx + n by px + q gives a quotient of m __ p so the range is ⎧ ⎨ ⎩ y | |

y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM

JOURNALHave students make a table that shows the steps for graphing a function of the form ax + b ______

cx + d by two

methods (1) writing the function in graphing form before graphing and (2) graphing the function without writing the function in graphing form first

489 Lesson 9 2


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g Com

pany

Lesson Performance Task

Graham wants to take snowboarding lessons at a nearby ski resort that charges $40 per week for a class that meets once during the week for 1 hour and gives him a lift ticket valid for 4 hours The resort also charges a one-time equipment-rental fee of $99 for uninterrupted enrollment in classes The resort requires that learners pay for three weeks of classes at a time

a Write a model that gives Grahamrsquos average weekly enrollment cost (in dollars) as a function of the time (in weeks) that Graham takes classes

b How much would Grahamrsquos average weekly enrollment cost be if he took classes only for the minimum of three weeks

c For how many weeks would Graham need to take classes for his average weekly enrollment cost to be at most $60 Describe how you can use a graphing calculator to graph the function from part a in order to answer this question and then state the answer

a Let t be the time (in weeks) that Graham takes classes Let C (t) be Grahamrsquos average weekly enrollment cost (in dollars) when the rental fee is included The model is C (t) = 99 ___ t + 40

b If Graham takes classes for 3 weeks his average weekly enrollment cost is C (3) = 99 ___ 3 + 40 = $73

c To find when Grahamrsquos average weekly enrollment cost will be at most $60 graph y = 99 ___ x + 40 and y = 60 on a graphing calculator and find the x-coordinate of the point where the graphs intersect The graphing calculator gives 495 as that x-coordinate So since Graham must pay for three weeks of classes at a time he will need to take classes for 6 weeks in order to have an average weekly enrollment cost of at most $60




EXTENSION ACTIVITY

Ask students to graph the cumulative amount Graham has spent on his membership at the ski resort after x months Ask them to describ e the parent function and tell what translation they need to make Then have them discuss when Graham might want to use this function and when he would want to use the function in the Performance Task

INTEGRATE MATHEMATICAL PROCESSESFocus on CommunicationAsk students to describe the domain and range of the continuous function f (x) in the Performance Task Then ask them how the domain and range are different for the real-world situation Have them discuss how the constraints in the real-world situation affect the domain and range

QUESTIONING STRATEGIESWhat effect does the monthly fee have on f (x) It translates f (x) up or down the y-axis

As x increases why does f (x) approach the monthly fee The initial fee of $240 is spread

out over more months so its monthly contribution

becomes smaller and smaller

Scoring Rubric2 points Student correctly solves the problem and explains hisher reasoning1 point Student shows good understanding of the problem but does not fully solve or explain hisher reasoning0 points Student does not demonstrate understanding of the problem



2

-2

-4

4

-4 -2 42

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x0

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C Be more precise about the end behavior of ƒ (x) = 1 _ x and determine what this means for the graph of the function

You can be more precise about the end behavior by using the notation ƒ (x) rarr 0 + which means that the value of ƒ (x) approaches 0 from the positive direction (that is the value of ƒ (x) is positive as it approaches 0) and the notation ƒ (x) rarr 0 - which means that the value of ƒ (x) approaches 0 from the negative direction So the end behavior of the function is more precisely summarized as follows


bull As x rarr -infin ƒ (x) rarr The end behavior indicates that the graph of ƒ (x) approaches but does not cross the

[x-axisy-axis] so that axis is an asymptote for the graph

D Examine the behavior of ƒ (x) = 1 _ x near x = 0 and determine what this means for the graph of the function



bull As x rarr 0 + ƒ (x) rarr bull As x rarr 0 - ƒ (x) rarr

The behavior of ƒ (x) = 1 _ x near x = 0 indicates that the graph of ƒ (x) approaches but does not cross the [x-axisy-axis] so that axis is also an asymptote for the graph

E Graph ƒ (x) = 1 _ x

First determine the sign of ƒ (x) on the two parts of its domain

bull When x is a negative number ƒ (x) is a [positivenegative] numberbull When x is a positive number ƒ (x) is a [positivenegative] number

Next complete the tables

Finally use the information from this step and all previous steps to draw the graph Draw asymptotes as dashed lines

x Approaches 0 from the Positive Direction

x f (x) = 1 _ x

001

0001

00001

Negative Values of x

x f (x) = 1 _ x

-2

-1

-05

Positive Values of x

x f (x) = 1 _ x

05

1

2

x Approaches 0 from the Negative Direction

x f (x) = 1 _ x

-001

-0001

-00001

0 +

0 -

+infin

100 -100

-1000

-100001000

10000

-infin

-05 2

1

05

-1

-2




A2_MTXESE353947_U4M09L2 470 11315 854 AM

PROFESSIONAL DEVELOPMENT

Learning ProgressionsPreviously students looked at polynomial functions and their transformations inverse variation applications and the graphs of rational functions of the form f (x) = a __ x in the first quadrant Students now extend that knowledge by graphing and transforming the graph of the parent function f (x) = 1 __ x They learn how the values of h and k are related to the graph of f (x) = a ____ x - h + k and graph rational functions in the general form ax + b

_____ cx + d where ax + b and cx + d are both linear functions In the next lesson students examine the quotients of linear and quadratic polynomials and discontinuities

EXPLORE 1 Graphing and Analyzing f(x) = 1 __ X

INTEGRATE TECHNOLOGYStudents may complete the Explore activity either in the book or online

QUESTIONING STRATEGIESWhy are there two parts to the graph of the rational function There are positive values

and negative values for x but no value when x is 0

Why isnrsquot the graph symmetric across the y-axis Positive values of x result in positive

values of y but negative values of x result in

negative values of y



copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y





⎨ ⎩ y|y ne ⎫

⎬ ⎭



Reflect






for each function



0

0

0

0 0

0








translation

translation



A2_MTXESE353947_U4M09L2 471 11115 957 AM









471 Lesson 9 2


copy H

oug

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arcourt Publishin

g Com

pany














for each function


factor of






translation

translation

horizontal stretch

2


horizontal stretch

reflection y-axis

1 __ 2

2


reflection y-axis

vertical stretch



2

1 __ 2

reflection x-axis

vertical stretch

2 reflection x-axis




A2_MTXESE353947_U4M09L2 472 11115 1002 AM





copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Reflect








Feature f (x) = 1 _ x g (x) = a ( 1 _

1 _ b

(x - h) ) + k






= 2 __ x = 2 ( 1 __ x )



2 x

= minus2 ___ x = minus2 ( 1 __ x )


( 1 __ x ) = minus05 ( 1 __ x )



A2_MTXESE353947_U4M09L2 473 11115 1004 AM




______________ 2 = 2h ___ 2 = h

and (-a + k) + (a + k)

_____________ 2 = 2k __ 2 = k

473 Lesson 9 2


4

6

42

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x0

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arcourt Publishin

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pany

A g (x) = 3 ( 1 _ x minus 1 ) + 2



Feature f (x) = 1 _ x g (x) = 3 ( 1 _ x - 1

) + 2




Reference point (1 1) (1 + 1 3 (1) + 2) = (2 5)

Domain of g (x)


Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫

⎬ ⎭


Range of g (x)


Set notation ⎧

⎨ ⎩ y| y ne 2 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 474 11315 810 AM




-4

2

-4-6

y

x0copy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any

B g (x) = 1 _ 2 (x + 3)

- 1



Feature f (x) = 1 _ x g (x) = 1 _ 2 (x + 3)

minus 1







Domain of g (x)


Set notation ⎧

⎨ ⎩ x|x ne ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne ⎫

⎬ ⎭


minus3

minus1

minus1 minus3 1 _ 2

minus1 minus2

1 1 minus2 1 _ 2

0

minus3

minus3 minus3

minus3

minus3

minus1

minus1

minus1minus1

minus1





475 Lesson 9 2


-4

-6

-4 2

yx

0

y

0-2

2

4

-24 62

x

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Your Turn


5 g (x) = minus05 ( 1 _ x + 1 ) minus 3

6 g (x) = 1 __ minus05 (x minus 2)

+ 1







(-1 -1) becomes (-2 - 5 _ 2

)

(1 1) becomes (0 - 7 _ 2

)



Set notation ⎧




⎬ ⎭









(1 1) becomes (0 2)



Set notation ⎧

⎨ ⎩ x|x ne 2 ⎫


⎨ ⎩ y|y ne 1 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 476 11115 1007 AM



y

0-2

4

2

6

42

xcopy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any




1 __ b (x - h) + k



+ k



g (x) = 3x - 4 _ x - 1

Use long division


3x - 3 _ -1





Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭


g (x) = 4x - 7 _ -2x + 4

Use long division


4x - 8 _


you have

g (x) = -2 + _ -2x + 4 or g (x) = __

-2 (x - ) - 2

1

1

1

2

1



A2_MTXESE353947_U4M09L2 477 11115 1013 AM




477 Lesson 9 2


-4

-6

-2

2

-2 2 4 6

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


( - 1 _ 2 (-1) + -1 - 2 ) = ( -3) and (- 1 _ 2 (1) + 1 - 2) = ( - 1) Domain of g (x)


Set notation ⎧

⎨

⎩ x|x ne

⎫

⎬

⎭



Set notation ⎧

⎨

⎩ y|y ne

⎫

⎬

⎭




2

2

2

-2

2

-2

2

-2

2

-2

2

-2

22 1 __ 2 1 1 __ 2








A2_MTXESE353947_U4M09L2 478 11115 1023 AM






y-asymptote



6

2

2

y

x0

4

-4-6 -2

6

2

8

4 62

y

x0

(2 2)

(4 6)

copy H

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hton

Mif

flin

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cour

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lishi

ng

Com

pan

y

Your Turn


+ k and


9 g (x) = 3x + 8 _ x + 2



from the graph

Example 3


) + k



g (x) = a ( 1 _ x - 3 ) + 4


g (x) = a ( 1 _ x - 3 ) + 4

6 = a ( 1 _ 4 - 3 ) + 4

6 = a + 4 2 = a

So g (x) = 2 ( 1 _____ x - 3 ) + 4


3x + 6

_ 2

So g (x) = 3 + 2 _ x + 2

or g (x) = 2 ( 1 _ x + 2

) + 3




Set notation ⎧

⎨ ⎩ x|x ne -2 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 479 11115 1026 AM










479 Lesson 9 2


-4

2

-4 -2-6

y

x0

(-5 0)

(-1 -2)

2

4

-4 2-6

y

x0

(-3 5)

(-1 -3)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k





g (x) = 1 _ 1 _ b

(x + 3) +


g (x) = 1 _ 1 _ b

(x + 3) +

0 = 1 _ 1 _ b

(-5 + 3) +

= 1 _ 1 _ b

(-2)

1 _ b

(-2) = 1

1 _ b

=

b =

So g (x) = 1 __ 1 _ (x + 3)

+ or g (x) = 1 __ -05 (x + 3)

-1

Reflect


Your Turn


-1

-1

-1

-1

-1

1

1

-1 ___ 2

-2

-1

-2



h = -2

k = 1


g (x) = a ( 1 _ x + 2

) + 1

5 = a ( 1 _ -3 + 2 ) + 1

5 = -a + 1 4 = -a -4 = a

So g (x) = -4 ( 1 _ x + 2

) + 1




A2_MTXESE353947_U4M09L2 480 11115 1028 AM



-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y bull I

mag

e C

red

its

copyZu

zan

a D

ole

zalo

vaA

lam

y


(x - h) + k


Example 4


Analyze Information




Formulate a Plan


h = 4

k = -3


g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM





481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

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oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

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flin

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pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

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lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

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lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y





⎨ ⎩ y|y ne ⎫

⎬ ⎭



Reflect






for each function



0

0

0

0 0

0








translation

translation



A2_MTXESE353947_U4M09L2 471 11115 957 AM









471 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany














for each function


factor of






translation

translation

horizontal stretch

2


horizontal stretch

reflection y-axis

1 __ 2

2


reflection y-axis

vertical stretch



2

1 __ 2

reflection x-axis

vertical stretch

2 reflection x-axis




A2_MTXESE353947_U4M09L2 472 11115 1002 AM





copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Reflect








Feature f (x) = 1 _ x g (x) = a ( 1 _

1 _ b

(x - h) ) + k






= 2 __ x = 2 ( 1 __ x )



2 x

= minus2 ___ x = minus2 ( 1 __ x )


( 1 __ x ) = minus05 ( 1 __ x )



A2_MTXESE353947_U4M09L2 473 11115 1004 AM




______________ 2 = 2h ___ 2 = h

and (-a + k) + (a + k)

_____________ 2 = 2k __ 2 = k

473 Lesson 9 2


4

6

42

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

A g (x) = 3 ( 1 _ x minus 1 ) + 2



Feature f (x) = 1 _ x g (x) = 3 ( 1 _ x - 1

) + 2




Reference point (1 1) (1 + 1 3 (1) + 2) = (2 5)

Domain of g (x)


Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫

⎬ ⎭


Range of g (x)


Set notation ⎧

⎨ ⎩ y| y ne 2 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 474 11315 810 AM




-4

2

-4-6

y

x0copy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any

B g (x) = 1 _ 2 (x + 3)

- 1



Feature f (x) = 1 _ x g (x) = 1 _ 2 (x + 3)

minus 1







Domain of g (x)


Set notation ⎧

⎨ ⎩ x|x ne ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne ⎫

⎬ ⎭


minus3

minus1

minus1 minus3 1 _ 2

minus1 minus2

1 1 minus2 1 _ 2

0

minus3

minus3 minus3

minus3

minus3

minus1

minus1

minus1minus1

minus1





475 Lesson 9 2


-4

-6

-4 2

yx

0

y

0-2

2

4

-24 62

x

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Your Turn


5 g (x) = minus05 ( 1 _ x + 1 ) minus 3

6 g (x) = 1 __ minus05 (x minus 2)

+ 1







(-1 -1) becomes (-2 - 5 _ 2

)

(1 1) becomes (0 - 7 _ 2

)



Set notation ⎧




⎬ ⎭









(1 1) becomes (0 2)



Set notation ⎧

⎨ ⎩ x|x ne 2 ⎫


⎨ ⎩ y|y ne 1 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 476 11115 1007 AM



y

0-2

4

2

6

42

xcopy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any




1 __ b (x - h) + k



+ k



g (x) = 3x - 4 _ x - 1

Use long division


3x - 3 _ -1





Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭


g (x) = 4x - 7 _ -2x + 4

Use long division


4x - 8 _


you have

g (x) = -2 + _ -2x + 4 or g (x) = __

-2 (x - ) - 2

1

1

1

2

1



A2_MTXESE353947_U4M09L2 477 11115 1013 AM




477 Lesson 9 2


-4

-6

-2

2

-2 2 4 6

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


( - 1 _ 2 (-1) + -1 - 2 ) = ( -3) and (- 1 _ 2 (1) + 1 - 2) = ( - 1) Domain of g (x)


Set notation ⎧

⎨

⎩ x|x ne

⎫

⎬

⎭



Set notation ⎧

⎨

⎩ y|y ne

⎫

⎬

⎭




2

2

2

-2

2

-2

2

-2

2

-2

2

-2

22 1 __ 2 1 1 __ 2








A2_MTXESE353947_U4M09L2 478 11115 1023 AM






y-asymptote



6

2

2

y

x0

4

-4-6 -2

6

2

8

4 62

y

x0

(2 2)

(4 6)

copy H

oug

hton

Mif

flin

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cour

t Pub

lishi

ng

Com

pan

y

Your Turn


+ k and


9 g (x) = 3x + 8 _ x + 2



from the graph

Example 3


) + k



g (x) = a ( 1 _ x - 3 ) + 4


g (x) = a ( 1 _ x - 3 ) + 4

6 = a ( 1 _ 4 - 3 ) + 4

6 = a + 4 2 = a

So g (x) = 2 ( 1 _____ x - 3 ) + 4


3x + 6

_ 2

So g (x) = 3 + 2 _ x + 2

or g (x) = 2 ( 1 _ x + 2

) + 3




Set notation ⎧

⎨ ⎩ x|x ne -2 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 479 11115 1026 AM










479 Lesson 9 2


-4

2

-4 -2-6

y

x0

(-5 0)

(-1 -2)

2

4

-4 2-6

y

x0

(-3 5)

(-1 -3)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k





g (x) = 1 _ 1 _ b

(x + 3) +


g (x) = 1 _ 1 _ b

(x + 3) +

0 = 1 _ 1 _ b

(-5 + 3) +

= 1 _ 1 _ b

(-2)

1 _ b

(-2) = 1

1 _ b

=

b =

So g (x) = 1 __ 1 _ (x + 3)

+ or g (x) = 1 __ -05 (x + 3)

-1

Reflect


Your Turn


-1

-1

-1

-1

-1

1

1

-1 ___ 2

-2

-1

-2



h = -2

k = 1


g (x) = a ( 1 _ x + 2

) + 1

5 = a ( 1 _ -3 + 2 ) + 1

5 = -a + 1 4 = -a -4 = a

So g (x) = -4 ( 1 _ x + 2

) + 1




A2_MTXESE353947_U4M09L2 480 11115 1028 AM



-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y bull I

mag

e C

red

its

copyZu

zan

a D

ole

zalo

vaA

lam

y


(x - h) + k


Example 4


Analyze Information




Formulate a Plan


h = 4

k = -3


g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM





481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

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hton

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Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

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cour

t Pub

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Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

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Har

cour

t Pub

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Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany














for each function


factor of






translation

translation

horizontal stretch

2


horizontal stretch

reflection y-axis

1 __ 2

2


reflection y-axis

vertical stretch



2

1 __ 2

reflection x-axis

vertical stretch

2 reflection x-axis




A2_MTXESE353947_U4M09L2 472 11115 1002 AM





copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Reflect








Feature f (x) = 1 _ x g (x) = a ( 1 _

1 _ b

(x - h) ) + k






= 2 __ x = 2 ( 1 __ x )



2 x

= minus2 ___ x = minus2 ( 1 __ x )


( 1 __ x ) = minus05 ( 1 __ x )



A2_MTXESE353947_U4M09L2 473 11115 1004 AM




______________ 2 = 2h ___ 2 = h

and (-a + k) + (a + k)

_____________ 2 = 2k __ 2 = k

473 Lesson 9 2


4

6

42

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

A g (x) = 3 ( 1 _ x minus 1 ) + 2



Feature f (x) = 1 _ x g (x) = 3 ( 1 _ x - 1

) + 2




Reference point (1 1) (1 + 1 3 (1) + 2) = (2 5)

Domain of g (x)


Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫

⎬ ⎭


Range of g (x)


Set notation ⎧

⎨ ⎩ y| y ne 2 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 474 11315 810 AM




-4

2

-4-6

y

x0copy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any

B g (x) = 1 _ 2 (x + 3)

- 1



Feature f (x) = 1 _ x g (x) = 1 _ 2 (x + 3)

minus 1







Domain of g (x)


Set notation ⎧

⎨ ⎩ x|x ne ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne ⎫

⎬ ⎭


minus3

minus1

minus1 minus3 1 _ 2

minus1 minus2

1 1 minus2 1 _ 2

0

minus3

minus3 minus3

minus3

minus3

minus1

minus1

minus1minus1

minus1





475 Lesson 9 2


-4

-6

-4 2

yx

0

y

0-2

2

4

-24 62

x

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Your Turn


5 g (x) = minus05 ( 1 _ x + 1 ) minus 3

6 g (x) = 1 __ minus05 (x minus 2)

+ 1







(-1 -1) becomes (-2 - 5 _ 2

)

(1 1) becomes (0 - 7 _ 2

)



Set notation ⎧




⎬ ⎭









(1 1) becomes (0 2)



Set notation ⎧

⎨ ⎩ x|x ne 2 ⎫


⎨ ⎩ y|y ne 1 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 476 11115 1007 AM



y

0-2

4

2

6

42

xcopy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any




1 __ b (x - h) + k



+ k



g (x) = 3x - 4 _ x - 1

Use long division


3x - 3 _ -1





Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭


g (x) = 4x - 7 _ -2x + 4

Use long division


4x - 8 _


you have

g (x) = -2 + _ -2x + 4 or g (x) = __

-2 (x - ) - 2

1

1

1

2

1



A2_MTXESE353947_U4M09L2 477 11115 1013 AM




477 Lesson 9 2


-4

-6

-2

2

-2 2 4 6

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


( - 1 _ 2 (-1) + -1 - 2 ) = ( -3) and (- 1 _ 2 (1) + 1 - 2) = ( - 1) Domain of g (x)


Set notation ⎧

⎨

⎩ x|x ne

⎫

⎬

⎭



Set notation ⎧

⎨

⎩ y|y ne

⎫

⎬

⎭




2

2

2

-2

2

-2

2

-2

2

-2

2

-2

22 1 __ 2 1 1 __ 2








A2_MTXESE353947_U4M09L2 478 11115 1023 AM






y-asymptote



6

2

2

y

x0

4

-4-6 -2

6

2

8

4 62

y

x0

(2 2)

(4 6)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Your Turn


+ k and


9 g (x) = 3x + 8 _ x + 2



from the graph

Example 3


) + k



g (x) = a ( 1 _ x - 3 ) + 4


g (x) = a ( 1 _ x - 3 ) + 4

6 = a ( 1 _ 4 - 3 ) + 4

6 = a + 4 2 = a

So g (x) = 2 ( 1 _____ x - 3 ) + 4


3x + 6

_ 2

So g (x) = 3 + 2 _ x + 2

or g (x) = 2 ( 1 _ x + 2

) + 3




Set notation ⎧

⎨ ⎩ x|x ne -2 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 479 11115 1026 AM










479 Lesson 9 2


-4

2

-4 -2-6

y

x0

(-5 0)

(-1 -2)

2

4

-4 2-6

y

x0

(-3 5)

(-1 -3)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k





g (x) = 1 _ 1 _ b

(x + 3) +


g (x) = 1 _ 1 _ b

(x + 3) +

0 = 1 _ 1 _ b

(-5 + 3) +

= 1 _ 1 _ b

(-2)

1 _ b

(-2) = 1

1 _ b

=

b =

So g (x) = 1 __ 1 _ (x + 3)

+ or g (x) = 1 __ -05 (x + 3)

-1

Reflect


Your Turn


-1

-1

-1

-1

-1

1

1

-1 ___ 2

-2

-1

-2



h = -2

k = 1


g (x) = a ( 1 _ x + 2

) + 1

5 = a ( 1 _ -3 + 2 ) + 1

5 = -a + 1 4 = -a -4 = a

So g (x) = -4 ( 1 _ x + 2

) + 1




A2_MTXESE353947_U4M09L2 480 11115 1028 AM



-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y bull I

mag

e C

red

its

copyZu

zan

a D

ole

zalo

vaA

lam

y


(x - h) + k


Example 4


Analyze Information




Formulate a Plan


h = 4

k = -3


g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM





481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Reflect








Feature f (x) = 1 _ x g (x) = a ( 1 _

1 _ b

(x - h) ) + k






= 2 __ x = 2 ( 1 __ x )



2 x

= minus2 ___ x = minus2 ( 1 __ x )


( 1 __ x ) = minus05 ( 1 __ x )



A2_MTXESE353947_U4M09L2 473 11115 1004 AM




______________ 2 = 2h ___ 2 = h

and (-a + k) + (a + k)

_____________ 2 = 2k __ 2 = k

473 Lesson 9 2


4

6

42

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

A g (x) = 3 ( 1 _ x minus 1 ) + 2



Feature f (x) = 1 _ x g (x) = 3 ( 1 _ x - 1

) + 2




Reference point (1 1) (1 + 1 3 (1) + 2) = (2 5)

Domain of g (x)


Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫

⎬ ⎭


Range of g (x)


Set notation ⎧

⎨ ⎩ y| y ne 2 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 474 11315 810 AM




-4

2

-4-6

y

x0copy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any

B g (x) = 1 _ 2 (x + 3)

- 1



Feature f (x) = 1 _ x g (x) = 1 _ 2 (x + 3)

minus 1







Domain of g (x)


Set notation ⎧

⎨ ⎩ x|x ne ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne ⎫

⎬ ⎭


minus3

minus1

minus1 minus3 1 _ 2

minus1 minus2

1 1 minus2 1 _ 2

0

minus3

minus3 minus3

minus3

minus3

minus1

minus1

minus1minus1

minus1





475 Lesson 9 2


-4

-6

-4 2

yx

0

y

0-2

2

4

-24 62

x

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Your Turn


5 g (x) = minus05 ( 1 _ x + 1 ) minus 3

6 g (x) = 1 __ minus05 (x minus 2)

+ 1







(-1 -1) becomes (-2 - 5 _ 2

)

(1 1) becomes (0 - 7 _ 2

)



Set notation ⎧




⎬ ⎭









(1 1) becomes (0 2)



Set notation ⎧

⎨ ⎩ x|x ne 2 ⎫


⎨ ⎩ y|y ne 1 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 476 11115 1007 AM



y

0-2

4

2

6

42

xcopy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any




1 __ b (x - h) + k



+ k



g (x) = 3x - 4 _ x - 1

Use long division


3x - 3 _ -1





Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭


g (x) = 4x - 7 _ -2x + 4

Use long division


4x - 8 _


you have

g (x) = -2 + _ -2x + 4 or g (x) = __

-2 (x - ) - 2

1

1

1

2

1



A2_MTXESE353947_U4M09L2 477 11115 1013 AM




477 Lesson 9 2


-4

-6

-2

2

-2 2 4 6

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


( - 1 _ 2 (-1) + -1 - 2 ) = ( -3) and (- 1 _ 2 (1) + 1 - 2) = ( - 1) Domain of g (x)


Set notation ⎧

⎨

⎩ x|x ne

⎫

⎬

⎭



Set notation ⎧

⎨

⎩ y|y ne

⎫

⎬

⎭




2

2

2

-2

2

-2

2

-2

2

-2

2

-2

22 1 __ 2 1 1 __ 2








A2_MTXESE353947_U4M09L2 478 11115 1023 AM






y-asymptote



6

2

2

y

x0

4

-4-6 -2

6

2

8

4 62

y

x0

(2 2)

(4 6)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Your Turn


+ k and


9 g (x) = 3x + 8 _ x + 2



from the graph

Example 3


) + k



g (x) = a ( 1 _ x - 3 ) + 4


g (x) = a ( 1 _ x - 3 ) + 4

6 = a ( 1 _ 4 - 3 ) + 4

6 = a + 4 2 = a

So g (x) = 2 ( 1 _____ x - 3 ) + 4


3x + 6

_ 2

So g (x) = 3 + 2 _ x + 2

or g (x) = 2 ( 1 _ x + 2

) + 3




Set notation ⎧

⎨ ⎩ x|x ne -2 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 479 11115 1026 AM










479 Lesson 9 2


-4

2

-4 -2-6

y

x0

(-5 0)

(-1 -2)

2

4

-4 2-6

y

x0

(-3 5)

(-1 -3)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k





g (x) = 1 _ 1 _ b

(x + 3) +


g (x) = 1 _ 1 _ b

(x + 3) +

0 = 1 _ 1 _ b

(-5 + 3) +

= 1 _ 1 _ b

(-2)

1 _ b

(-2) = 1

1 _ b

=

b =

So g (x) = 1 __ 1 _ (x + 3)

+ or g (x) = 1 __ -05 (x + 3)

-1

Reflect


Your Turn


-1

-1

-1

-1

-1

1

1

-1 ___ 2

-2

-1

-2



h = -2

k = 1


g (x) = a ( 1 _ x + 2

) + 1

5 = a ( 1 _ -3 + 2 ) + 1

5 = -a + 1 4 = -a -4 = a

So g (x) = -4 ( 1 _ x + 2

) + 1




A2_MTXESE353947_U4M09L2 480 11115 1028 AM



-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y bull I

mag

e C

red

its

copyZu

zan

a D

ole

zalo

vaA

lam

y


(x - h) + k


Example 4


Analyze Information




Formulate a Plan


h = 4

k = -3


g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM





481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










4

6

42

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

A g (x) = 3 ( 1 _ x minus 1 ) + 2



Feature f (x) = 1 _ x g (x) = 3 ( 1 _ x - 1

) + 2




Reference point (1 1) (1 + 1 3 (1) + 2) = (2 5)

Domain of g (x)


Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫

⎬ ⎭


Range of g (x)


Set notation ⎧

⎨ ⎩ y| y ne 2 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 474 11315 810 AM




-4

2

-4-6

y

x0copy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any

B g (x) = 1 _ 2 (x + 3)

- 1



Feature f (x) = 1 _ x g (x) = 1 _ 2 (x + 3)

minus 1







Domain of g (x)


Set notation ⎧

⎨ ⎩ x|x ne ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne ⎫

⎬ ⎭


minus3

minus1

minus1 minus3 1 _ 2

minus1 minus2

1 1 minus2 1 _ 2

0

minus3

minus3 minus3

minus3

minus3

minus1

minus1

minus1minus1

minus1





475 Lesson 9 2


-4

-6

-4 2

yx

0

y

0-2

2

4

-24 62

x

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Your Turn


5 g (x) = minus05 ( 1 _ x + 1 ) minus 3

6 g (x) = 1 __ minus05 (x minus 2)

+ 1







(-1 -1) becomes (-2 - 5 _ 2

)

(1 1) becomes (0 - 7 _ 2

)



Set notation ⎧




⎬ ⎭









(1 1) becomes (0 2)



Set notation ⎧

⎨ ⎩ x|x ne 2 ⎫


⎨ ⎩ y|y ne 1 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 476 11115 1007 AM



y

0-2

4

2

6

42

xcopy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any




1 __ b (x - h) + k



+ k



g (x) = 3x - 4 _ x - 1

Use long division


3x - 3 _ -1





Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭


g (x) = 4x - 7 _ -2x + 4

Use long division


4x - 8 _


you have

g (x) = -2 + _ -2x + 4 or g (x) = __

-2 (x - ) - 2

1

1

1

2

1



A2_MTXESE353947_U4M09L2 477 11115 1013 AM




477 Lesson 9 2


-4

-6

-2

2

-2 2 4 6

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


( - 1 _ 2 (-1) + -1 - 2 ) = ( -3) and (- 1 _ 2 (1) + 1 - 2) = ( - 1) Domain of g (x)


Set notation ⎧

⎨

⎩ x|x ne

⎫

⎬

⎭



Set notation ⎧

⎨

⎩ y|y ne

⎫

⎬

⎭




2

2

2

-2

2

-2

2

-2

2

-2

2

-2

22 1 __ 2 1 1 __ 2








A2_MTXESE353947_U4M09L2 478 11115 1023 AM






y-asymptote



6

2

2

y

x0

4

-4-6 -2

6

2

8

4 62

y

x0

(2 2)

(4 6)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Your Turn


+ k and


9 g (x) = 3x + 8 _ x + 2



from the graph

Example 3


) + k



g (x) = a ( 1 _ x - 3 ) + 4


g (x) = a ( 1 _ x - 3 ) + 4

6 = a ( 1 _ 4 - 3 ) + 4

6 = a + 4 2 = a

So g (x) = 2 ( 1 _____ x - 3 ) + 4


3x + 6

_ 2

So g (x) = 3 + 2 _ x + 2

or g (x) = 2 ( 1 _ x + 2

) + 3




Set notation ⎧

⎨ ⎩ x|x ne -2 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 479 11115 1026 AM










479 Lesson 9 2


-4

2

-4 -2-6

y

x0

(-5 0)

(-1 -2)

2

4

-4 2-6

y

x0

(-3 5)

(-1 -3)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k





g (x) = 1 _ 1 _ b

(x + 3) +


g (x) = 1 _ 1 _ b

(x + 3) +

0 = 1 _ 1 _ b

(-5 + 3) +

= 1 _ 1 _ b

(-2)

1 _ b

(-2) = 1

1 _ b

=

b =

So g (x) = 1 __ 1 _ (x + 3)

+ or g (x) = 1 __ -05 (x + 3)

-1

Reflect


Your Turn


-1

-1

-1

-1

-1

1

1

-1 ___ 2

-2

-1

-2



h = -2

k = 1


g (x) = a ( 1 _ x + 2

) + 1

5 = a ( 1 _ -3 + 2 ) + 1

5 = -a + 1 4 = -a -4 = a

So g (x) = -4 ( 1 _ x + 2

) + 1




A2_MTXESE353947_U4M09L2 480 11115 1028 AM



-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y bull I

mag

e C

red

its

copyZu

zan

a D

ole

zalo

vaA

lam

y


(x - h) + k


Example 4


Analyze Information




Formulate a Plan


h = 4

k = -3


g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM





481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










-4

2

-4-6

y

x0copy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any

B g (x) = 1 _ 2 (x + 3)

- 1



Feature f (x) = 1 _ x g (x) = 1 _ 2 (x + 3)

minus 1







Domain of g (x)


Set notation ⎧

⎨ ⎩ x|x ne ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne ⎫

⎬ ⎭


minus3

minus1

minus1 minus3 1 _ 2

minus1 minus2

1 1 minus2 1 _ 2

0

minus3

minus3 minus3

minus3

minus3

minus1

minus1

minus1minus1

minus1





475 Lesson 9 2


-4

-6

-4 2

yx

0

y

0-2

2

4

-24 62

x

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Your Turn


5 g (x) = minus05 ( 1 _ x + 1 ) minus 3

6 g (x) = 1 __ minus05 (x minus 2)

+ 1







(-1 -1) becomes (-2 - 5 _ 2

)

(1 1) becomes (0 - 7 _ 2

)



Set notation ⎧




⎬ ⎭









(1 1) becomes (0 2)



Set notation ⎧

⎨ ⎩ x|x ne 2 ⎫


⎨ ⎩ y|y ne 1 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 476 11115 1007 AM



y

0-2

4

2

6

42

xcopy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any




1 __ b (x - h) + k



+ k



g (x) = 3x - 4 _ x - 1

Use long division


3x - 3 _ -1





Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭


g (x) = 4x - 7 _ -2x + 4

Use long division


4x - 8 _


you have

g (x) = -2 + _ -2x + 4 or g (x) = __

-2 (x - ) - 2

1

1

1

2

1



A2_MTXESE353947_U4M09L2 477 11115 1013 AM




477 Lesson 9 2


-4

-6

-2

2

-2 2 4 6

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


( - 1 _ 2 (-1) + -1 - 2 ) = ( -3) and (- 1 _ 2 (1) + 1 - 2) = ( - 1) Domain of g (x)


Set notation ⎧

⎨

⎩ x|x ne

⎫

⎬

⎭



Set notation ⎧

⎨

⎩ y|y ne

⎫

⎬

⎭




2

2

2

-2

2

-2

2

-2

2

-2

2

-2

22 1 __ 2 1 1 __ 2








A2_MTXESE353947_U4M09L2 478 11115 1023 AM






y-asymptote



6

2

2

y

x0

4

-4-6 -2

6

2

8

4 62

y

x0

(2 2)

(4 6)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Your Turn


+ k and


9 g (x) = 3x + 8 _ x + 2



from the graph

Example 3


) + k



g (x) = a ( 1 _ x - 3 ) + 4


g (x) = a ( 1 _ x - 3 ) + 4

6 = a ( 1 _ 4 - 3 ) + 4

6 = a + 4 2 = a

So g (x) = 2 ( 1 _____ x - 3 ) + 4


3x + 6

_ 2

So g (x) = 3 + 2 _ x + 2

or g (x) = 2 ( 1 _ x + 2

) + 3




Set notation ⎧

⎨ ⎩ x|x ne -2 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 479 11115 1026 AM










479 Lesson 9 2


-4

2

-4 -2-6

y

x0

(-5 0)

(-1 -2)

2

4

-4 2-6

y

x0

(-3 5)

(-1 -3)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k





g (x) = 1 _ 1 _ b

(x + 3) +


g (x) = 1 _ 1 _ b

(x + 3) +

0 = 1 _ 1 _ b

(-5 + 3) +

= 1 _ 1 _ b

(-2)

1 _ b

(-2) = 1

1 _ b

=

b =

So g (x) = 1 __ 1 _ (x + 3)

+ or g (x) = 1 __ -05 (x + 3)

-1

Reflect


Your Turn


-1

-1

-1

-1

-1

1

1

-1 ___ 2

-2

-1

-2



h = -2

k = 1


g (x) = a ( 1 _ x + 2

) + 1

5 = a ( 1 _ -3 + 2 ) + 1

5 = -a + 1 4 = -a -4 = a

So g (x) = -4 ( 1 _ x + 2

) + 1




A2_MTXESE353947_U4M09L2 480 11115 1028 AM



-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y bull I

mag

e C

red

its

copyZu

zan

a D

ole

zalo

vaA

lam

y


(x - h) + k


Example 4


Analyze Information




Formulate a Plan


h = 4

k = -3


g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM





481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










-4

-6

-4 2

yx

0

y

0-2

2

4

-24 62

x

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Your Turn


5 g (x) = minus05 ( 1 _ x + 1 ) minus 3

6 g (x) = 1 __ minus05 (x minus 2)

+ 1







(-1 -1) becomes (-2 - 5 _ 2

)

(1 1) becomes (0 - 7 _ 2

)



Set notation ⎧




⎬ ⎭









(1 1) becomes (0 2)



Set notation ⎧

⎨ ⎩ x|x ne 2 ⎫


⎨ ⎩ y|y ne 1 ⎫

⎬ ⎭





A2_MTXESE353947_U4M09L2 476 11115 1007 AM



y

0-2

4

2

6

42

xcopy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any




1 __ b (x - h) + k



+ k



g (x) = 3x - 4 _ x - 1

Use long division


3x - 3 _ -1





Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭


g (x) = 4x - 7 _ -2x + 4

Use long division


4x - 8 _


you have

g (x) = -2 + _ -2x + 4 or g (x) = __

-2 (x - ) - 2

1

1

1

2

1



A2_MTXESE353947_U4M09L2 477 11115 1013 AM




477 Lesson 9 2


-4

-6

-2

2

-2 2 4 6

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


( - 1 _ 2 (-1) + -1 - 2 ) = ( -3) and (- 1 _ 2 (1) + 1 - 2) = ( - 1) Domain of g (x)


Set notation ⎧

⎨

⎩ x|x ne

⎫

⎬

⎭



Set notation ⎧

⎨

⎩ y|y ne

⎫

⎬

⎭




2

2

2

-2

2

-2

2

-2

2

-2

2

-2

22 1 __ 2 1 1 __ 2








A2_MTXESE353947_U4M09L2 478 11115 1023 AM






y-asymptote



6

2

2

y

x0

4

-4-6 -2

6

2

8

4 62

y

x0

(2 2)

(4 6)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Your Turn


+ k and


9 g (x) = 3x + 8 _ x + 2



from the graph

Example 3


) + k



g (x) = a ( 1 _ x - 3 ) + 4


g (x) = a ( 1 _ x - 3 ) + 4

6 = a ( 1 _ 4 - 3 ) + 4

6 = a + 4 2 = a

So g (x) = 2 ( 1 _____ x - 3 ) + 4


3x + 6

_ 2

So g (x) = 3 + 2 _ x + 2

or g (x) = 2 ( 1 _ x + 2

) + 3




Set notation ⎧

⎨ ⎩ x|x ne -2 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 479 11115 1026 AM










479 Lesson 9 2


-4

2

-4 -2-6

y

x0

(-5 0)

(-1 -2)

2

4

-4 2-6

y

x0

(-3 5)

(-1 -3)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k





g (x) = 1 _ 1 _ b

(x + 3) +


g (x) = 1 _ 1 _ b

(x + 3) +

0 = 1 _ 1 _ b

(-5 + 3) +

= 1 _ 1 _ b

(-2)

1 _ b

(-2) = 1

1 _ b

=

b =

So g (x) = 1 __ 1 _ (x + 3)

+ or g (x) = 1 __ -05 (x + 3)

-1

Reflect


Your Turn


-1

-1

-1

-1

-1

1

1

-1 ___ 2

-2

-1

-2



h = -2

k = 1


g (x) = a ( 1 _ x + 2

) + 1

5 = a ( 1 _ -3 + 2 ) + 1

5 = -a + 1 4 = -a -4 = a

So g (x) = -4 ( 1 _ x + 2

) + 1




A2_MTXESE353947_U4M09L2 480 11115 1028 AM



-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y bull I

mag

e C

red

its

copyZu

zan

a D

ole

zalo

vaA

lam

y


(x - h) + k


Example 4


Analyze Information




Formulate a Plan


h = 4

k = -3


g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM





481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

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cour

t Pub

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Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










y

0-2

4

2

6

42

xcopy

Hou

ght

on M

iffl

in H

arco

urt P

ublis

hin

g C

omp

any




1 __ b (x - h) + k



+ k



g (x) = 3x - 4 _ x - 1

Use long division


3x - 3 _ -1





Set notation ⎧

⎨ ⎩ x|x ne 1 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭


g (x) = 4x - 7 _ -2x + 4

Use long division


4x - 8 _


you have

g (x) = -2 + _ -2x + 4 or g (x) = __

-2 (x - ) - 2

1

1

1

2

1



A2_MTXESE353947_U4M09L2 477 11115 1013 AM




477 Lesson 9 2


-4

-6

-2

2

-2 2 4 6

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


( - 1 _ 2 (-1) + -1 - 2 ) = ( -3) and (- 1 _ 2 (1) + 1 - 2) = ( - 1) Domain of g (x)


Set notation ⎧

⎨

⎩ x|x ne

⎫

⎬

⎭



Set notation ⎧

⎨

⎩ y|y ne

⎫

⎬

⎭




2

2

2

-2

2

-2

2

-2

2

-2

2

-2

22 1 __ 2 1 1 __ 2








A2_MTXESE353947_U4M09L2 478 11115 1023 AM






y-asymptote



6

2

2

y

x0

4

-4-6 -2

6

2

8

4 62

y

x0

(2 2)

(4 6)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Your Turn


+ k and


9 g (x) = 3x + 8 _ x + 2



from the graph

Example 3


) + k



g (x) = a ( 1 _ x - 3 ) + 4


g (x) = a ( 1 _ x - 3 ) + 4

6 = a ( 1 _ 4 - 3 ) + 4

6 = a + 4 2 = a

So g (x) = 2 ( 1 _____ x - 3 ) + 4


3x + 6

_ 2

So g (x) = 3 + 2 _ x + 2

or g (x) = 2 ( 1 _ x + 2

) + 3




Set notation ⎧

⎨ ⎩ x|x ne -2 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 479 11115 1026 AM










479 Lesson 9 2


-4

2

-4 -2-6

y

x0

(-5 0)

(-1 -2)

2

4

-4 2-6

y

x0

(-3 5)

(-1 -3)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k





g (x) = 1 _ 1 _ b

(x + 3) +


g (x) = 1 _ 1 _ b

(x + 3) +

0 = 1 _ 1 _ b

(-5 + 3) +

= 1 _ 1 _ b

(-2)

1 _ b

(-2) = 1

1 _ b

=

b =

So g (x) = 1 __ 1 _ (x + 3)

+ or g (x) = 1 __ -05 (x + 3)

-1

Reflect


Your Turn


-1

-1

-1

-1

-1

1

1

-1 ___ 2

-2

-1

-2



h = -2

k = 1


g (x) = a ( 1 _ x + 2

) + 1

5 = a ( 1 _ -3 + 2 ) + 1

5 = -a + 1 4 = -a -4 = a

So g (x) = -4 ( 1 _ x + 2

) + 1




A2_MTXESE353947_U4M09L2 480 11115 1028 AM



-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y bull I

mag

e C

red

its

copyZu

zan

a D

ole

zalo

vaA

lam

y


(x - h) + k


Example 4


Analyze Information




Formulate a Plan


h = 4

k = -3


g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM





481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










-4

-6

-2

2

-2 2 4 6

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


( - 1 _ 2 (-1) + -1 - 2 ) = ( -3) and (- 1 _ 2 (1) + 1 - 2) = ( - 1) Domain of g (x)


Set notation ⎧

⎨

⎩ x|x ne

⎫

⎬

⎭



Set notation ⎧

⎨

⎩ y|y ne

⎫

⎬

⎭




2

2

2

-2

2

-2

2

-2

2

-2

2

-2

22 1 __ 2 1 1 __ 2








A2_MTXESE353947_U4M09L2 478 11115 1023 AM






y-asymptote



6

2

2

y

x0

4

-4-6 -2

6

2

8

4 62

y

x0

(2 2)

(4 6)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Your Turn


+ k and


9 g (x) = 3x + 8 _ x + 2



from the graph

Example 3


) + k



g (x) = a ( 1 _ x - 3 ) + 4


g (x) = a ( 1 _ x - 3 ) + 4

6 = a ( 1 _ 4 - 3 ) + 4

6 = a + 4 2 = a

So g (x) = 2 ( 1 _____ x - 3 ) + 4


3x + 6

_ 2

So g (x) = 3 + 2 _ x + 2

or g (x) = 2 ( 1 _ x + 2

) + 3




Set notation ⎧

⎨ ⎩ x|x ne -2 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 479 11115 1026 AM










479 Lesson 9 2


-4

2

-4 -2-6

y

x0

(-5 0)

(-1 -2)

2

4

-4 2-6

y

x0

(-3 5)

(-1 -3)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k





g (x) = 1 _ 1 _ b

(x + 3) +


g (x) = 1 _ 1 _ b

(x + 3) +

0 = 1 _ 1 _ b

(-5 + 3) +

= 1 _ 1 _ b

(-2)

1 _ b

(-2) = 1

1 _ b

=

b =

So g (x) = 1 __ 1 _ (x + 3)

+ or g (x) = 1 __ -05 (x + 3)

-1

Reflect


Your Turn


-1

-1

-1

-1

-1

1

1

-1 ___ 2

-2

-1

-2



h = -2

k = 1


g (x) = a ( 1 _ x + 2

) + 1

5 = a ( 1 _ -3 + 2 ) + 1

5 = -a + 1 4 = -a -4 = a

So g (x) = -4 ( 1 _ x + 2

) + 1




A2_MTXESE353947_U4M09L2 480 11115 1028 AM



-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y bull I

mag

e C

red

its

copyZu

zan

a D

ole

zalo

vaA

lam

y


(x - h) + k


Example 4


Analyze Information




Formulate a Plan


h = 4

k = -3


g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM





481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










6

2

2

y

x0

4

-4-6 -2

6

2

8

4 62

y

x0

(2 2)

(4 6)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Your Turn


+ k and


9 g (x) = 3x + 8 _ x + 2



from the graph

Example 3


) + k



g (x) = a ( 1 _ x - 3 ) + 4


g (x) = a ( 1 _ x - 3 ) + 4

6 = a ( 1 _ 4 - 3 ) + 4

6 = a + 4 2 = a

So g (x) = 2 ( 1 _____ x - 3 ) + 4


3x + 6

_ 2

So g (x) = 3 + 2 _ x + 2

or g (x) = 2 ( 1 _ x + 2

) + 3




Set notation ⎧

⎨ ⎩ x|x ne -2 ⎫


⎨ ⎩ y|y ne 3 ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 479 11115 1026 AM










479 Lesson 9 2


-4

2

-4 -2-6

y

x0

(-5 0)

(-1 -2)

2

4

-4 2-6

y

x0

(-3 5)

(-1 -3)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k





g (x) = 1 _ 1 _ b

(x + 3) +


g (x) = 1 _ 1 _ b

(x + 3) +

0 = 1 _ 1 _ b

(-5 + 3) +

= 1 _ 1 _ b

(-2)

1 _ b

(-2) = 1

1 _ b

=

b =

So g (x) = 1 __ 1 _ (x + 3)

+ or g (x) = 1 __ -05 (x + 3)

-1

Reflect


Your Turn


-1

-1

-1

-1

-1

1

1

-1 ___ 2

-2

-1

-2



h = -2

k = 1


g (x) = a ( 1 _ x + 2

) + 1

5 = a ( 1 _ -3 + 2 ) + 1

5 = -a + 1 4 = -a -4 = a

So g (x) = -4 ( 1 _ x + 2

) + 1




A2_MTXESE353947_U4M09L2 480 11115 1028 AM



-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y bull I

mag

e C

red

its

copyZu

zan

a D

ole

zalo

vaA

lam

y


(x - h) + k


Example 4


Analyze Information




Formulate a Plan


h = 4

k = -3


g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM





481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










-4

2

-4 -2-6

y

x0

(-5 0)

(-1 -2)

2

4

-4 2-6

y

x0

(-3 5)

(-1 -3)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k





g (x) = 1 _ 1 _ b

(x + 3) +


g (x) = 1 _ 1 _ b

(x + 3) +

0 = 1 _ 1 _ b

(-5 + 3) +

= 1 _ 1 _ b

(-2)

1 _ b

(-2) = 1

1 _ b

=

b =

So g (x) = 1 __ 1 _ (x + 3)

+ or g (x) = 1 __ -05 (x + 3)

-1

Reflect


Your Turn


-1

-1

-1

-1

-1

1

1

-1 ___ 2

-2

-1

-2



h = -2

k = 1


g (x) = a ( 1 _ x + 2

) + 1

5 = a ( 1 _ -3 + 2 ) + 1

5 = -a + 1 4 = -a -4 = a

So g (x) = -4 ( 1 _ x + 2

) + 1




A2_MTXESE353947_U4M09L2 480 11115 1028 AM



-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y bull I

mag

e C

red

its

copyZu

zan

a D

ole

zalo

vaA

lam

y


(x - h) + k


Example 4


Analyze Information




Formulate a Plan


h = 4

k = -3


g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM





481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










-2

-4

-6

yx

0 2 86

(35 -4)

(45 -2)

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y bull I

mag

e C

red

its

copyZu

zan

a D

ole

zalo

vaA

lam

y


(x - h) + k


Example 4


Analyze Information




Formulate a Plan


h = 4

k = -3


g (x) = 1 _ 1 _ b

(x - 4) + (-3)

-2 = 1 __ 1 _ b

(45 - 4) -3

-2 = 1 _ 1 _ b

(05) -3

1 = 1 _ 1 _ b

(05)

1 _ b

(05) = 1

1 _ b

= 2

b = 1 _ 2

So g (x) = 1 _ 2 (x - 4)

- 3

$300

$220

$350



A2_MTXESE353947_U4M09L2 481 11115 1033 AM





481 Lesson 9 2


2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










2

100200300400500600700800

4 6 8

C

p0

Number of people

Each

per

sonrsquo

s co

st (d

olla

rs)

51015

2520

303540

10 20 30 40

C

s0

Number of T-shirts

Ave

rage

cos

t (do

llars

)

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Hero

Imag

esCorb

is

Solve


C (p) = _ p +






is



Your Turn


220300

220

300

4 5

5

5355

344







below $1750 is 27




A2_MTXESE353947_U4M09L2 482 11115 1035 AM




copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate


















Domain


Set notation ⎧

⎨ ⎩ x|x ne h ⎫

⎬ ⎭



Set notation ⎧

⎨ ⎩ y|y ne k ⎫

⎬ ⎭




A2_MTXESE353947_U4M09L2 483 11315 813 AM

LANGUAGE SUPPORT








_____ 1 __ b



483 Lesson 9 2


-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










-4

-6

2

-4 2

y

x0

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




1 g (x) = 1 _ x + 4 2 g (x) = 5 ( 1 _ x )

3 g (x) = 1 _ x + 3 4 g (x) = 1 _ 01x

5 g (x) = 1 _ x - 7 6 g (x) = 1 _ x - 8

7 g (x) = -01 ( 1 _ x ) 8 g (x) = 1 _ -3x


9 g (x) = 3 ( 1 _ x + 1 ) - 2





























EVALUATE

ASSIGNMENT GUIDE




Exercises 1ndash8


Exercises 9ndash12


Exercises 13ndash16


Exercises 17ndash20


Exercises 21ndash22



(-2 + 15) = (-8 13) and (2 + (-6)

(2 + 15) = (-4 17)



y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










y

0

2

4

-24 62

x

y

0

2

-2

-4

-6

2-2 4

x

2

4

6

y

x0 2-4-6

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

10 g (x) = 1 __ -05 (x - 3)

+ 1

11 g (x) = -05 ( 1 _ x - 1 ) - 2

12 g (x) = 1 _ 2 (x + 2)

+ 3

























485 Lesson 9 2


2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










2

4

6

y

x0 2 4-2

-2

2

4

6y

x0-2-4-6-8

-6

-4

-2

-8

642

y x0-2

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany


+ k or g (x) = 1 ______ 1 __ b (x - h)

+ k and graph it


13 g (x) = 3x - 5 _ x - 1

14 g (x) = x + 5 _ 05x + 2

15 g (x) = -4x + 11 _ x - 2


――― 3x - 3

- 2

g (x) = -2 ( 1 ____ x - 1 ) + 3





―― x + 4

1

g (x) = 1 _______ 05 (x + 4)

+ 2







―――― -4x + 8

3

g (x) = 3 ( 1 ____ x - 2 ) - 4





Vert Asymp x = 1

Horiz Asymp y = 3


(1 1) becomes (2 1)


Horiz Asymp y = 2

Vert Asymp x = 2





A2_MTXESE353947_U4M09L2 486 230214 256 PM




-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

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)


C

s

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Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

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2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










-4

-2

-6

2y

x0-6 -4 -2

6

2

4

(2 1)

(4 7)8

642

y

x0

-2

4

2

6y

x0

(-6 3)

(-2 1)

-2-4-6-8

copy H

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lishi

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Com

pan

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16 g (x) = 4x + 13 _ -2x - 6



(x - h) + k


――― 4x + 12

1

g (x) = 1 _______ -2 (x + 3)

- 2

Vert Asymp x = -3







h = 3

k = 4

Use the point (4 7)

7= a ( 1 _ 4 - 3

) + 4

3 = a

g (x) = 3 ( 1 _ x - 3

) + 4

h = -4

k = 2


1 = 1 __ 1 _ b

(-2 + 4) + 2

b = -2

g (x) = 1 __ - 1 _

2 (x + 4)

+ 2





487 Lesson 9 2


-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










-4

-2

2

2 4 6

y

x0

(35 0)

(25 -2)

-2

-8

2

y x0-4-6

(-3 -2)

(-1 -6)

0

510152025303540

10 20 30 40

Ave

rage

cos

t (do

llars

)


C

s

copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany bull Im

age C

redits copy

Bernie Pearson

Alam

y


(x - h) + k



h = 3

k = -1


0 = 1 __ 1 _ b

(35 - 3) - 1

b = 1 _ 2

g (x) = 1 _ 2 (x - 3)

- 1

h = -2

k = -4


-6 = a ( 1 _ -1 + 2 ) - 4

-2 = a

g (x) = -2 ( 1 _ x + 2

) - 4


C (s) = 100 _ s + 15


and s = 14 Since C (13) = 100 _ 13

+ 15 asymp 2269 and

C (14) = 100 _ 14



$2250 is 14








0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










0

12345678

20 40 60 80

wAve

rage

cos

t (do

llars

)


C

copy H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y



2 (x - 3) + 1









C (w) = 140 _ w + 2


and w = 47 Since C (46) = 140 _ 46

+ 2 asymp 504 and

C (47) = 140 _ 47




x)


x) = 1 ___ 1 __ b

x = 1 __

1 __ b










y ne m __ p ⎫ ⎬ ⎭



A2_MTXESE353947_U4M09L2 489 11315 826 AM


cx + d by two


489 Lesson 9 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany












EXTENSION ACTIVITY










CorrectionKey=TX-A CorrectionKey=TX-B LESSON 9 . … Locker A2.6.G Analyze the effect on the graphs...

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Transcript of CorrectionKey=TX-A CorrectionKey=TX-B LESSON 9 . … Locker A2.6.G Analyze the effect on the graphs...