Homework, Page 147

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 1

Homework, 47

Describe how the graph of y = x2 can be transformed to the graph of the given equation.

1.

The graph of y = x2 can be transformed to the graph of y = x2 – 3 by a vertical translation down three units.

2 3y x


Homework, Page 147


5.

The graph of y = x2 can be transformed to the graph of y = (100 – x)2 by a horizontal translation right 100 units.

2100y x

2 2100 100y x x


Homework, Page 147

Show that the graph of can be transformed to the graph of the given equation.

9.

The graph of can be transformed to the graph of by reflecting about the x–axis.

y x

y x

y x

y x


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13.

The graph of y = x3 can be transformed to the graph of y = 2x3 by a vertical stretch of 2.

32y x


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Describe how to transform the graph of f into the graph of g.

17.

The graph of f can be transformed into the graph of g by a horizontal translation six units right.

2 and 4f x x g x x


Homework, Page 147

Sketch the graphs of f, g, and h by hand. Support your answers with a grapher.

21. 2 222 ; 3 2; 2 3f x x g x x h x x

x

y

x

y

x

y


Homework, Page 147

The graph is that of a function y = f(x) that can

be obtained by transforming the graph of

Write a formula for the function f.

25.

.y x

5y x


Homework, Page 147

Find the equation of the reflection of f across (a) the x–axis and (b) across the y–axis.

29. 3 25 3 2f x x x x

3

3 2

3

2

2

5 3 2

5 3 2

5 3 2

f x x

f x x

x x

f x x x x

x x

x

y


Homework, Page 147

33. Prove that the graph of an odd function is the same when reflected across the x–axis as it is when reflected across the y–axis.

An odd function is defined as f(–x) = –f(x). Reflecting y = f(x) across the x–axis yields

y = –f(x). Reflecting y = f(x) across the y–axis yields y = f(–x). Since f(–x) = –f(x), we have the same reflection across both axes.


Homework, Page 147

Refer to the graph of y = f (x) shown. Sketch the graph of the new function.

37. y f x

x

y


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Transform the given function by (a) vertical stretch by a factor of 2 and (b) a horizontal shrink by a factor of 1/3.

41. 2 2f x x x

2 2

2

2 2

2 2 2 2 4

2 2 4

( ) 3 3 2 9 3 2

a f x x x f x x x

f x x x

b f x x x f x x x


Homework, Page 147

Describe a basic graph and a sequence of transformations that can be used to produce a graph of the given function.

45.

If the function f (x) = x2 is given a horizontal shrink of 1/3, we get f (x) = (3x)2. If we then apply a vertical translation of –4, we get

f (x) = (3x)2 – 4.

23 4f x x


Homework, Page 147

A graph G is obtained from a graph of y by the sequence of transformations indicated. Write an equation whose graph is G.

49. ; a shift left two units, then a vertical

stretch by a factor of two, and finally a shift

down 4 units

y x

2 2 2

4 2 2 2 2 4

y x y x y x

y x G x x


Homework, 7

Refer to the function f (x) whose graph is shown.

53. Sketch the graph of 2y f x

x

y


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57. The graph shows the temperature (ºC) for one day. Describe the transformation to convert this graph to one showing degrees Fahrenheit.

The graph would have a vertical stretch of 9/5 and vertical translation of positive 32.


Homework, Page 147

61. Given a function f, which of the following represents a vertical stretch by a factor of three.

a. b.

c. d.

e.

3y f x

3y f x

3y f x 3

f xy

3xy f


Homework, Page 147

65. The table shows the price of Dell stock for the first eight months of 2004.

Month Price ($) Month Price ($)

1 33.44 5 35.24

2 32.65 6 35.82

3 33.62 7 35.47

4 34.78 8 34.84


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65. a. Graph price (y) as a function of month (x) as a line graph, connecting the points to make a continuous graph.

x

y

Months

Price ($10)


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65. b. Explain what transformation you would apply to this graph to produce a graph showing the price of the stock in Japanese yen.

The graph would need to be stretched vertically by a factor of about 120, the approximate number of yen in one dollar.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

1.7

Modeling with Functions


Quick Review

2

3

Solve the given formula for the given variable.

11. Solve for :

21

2. Solve for : 3

43. Solve for :

34.

b A bh

h V r h

r V r

Area of a Triangle

Volume of a Right Circular Cylinder

Volume of a Sphere

Surface 2 Solve for : 4

5. Solve for :

r A r

P I Prt

Area of a Sphere

Simple Interest


Quick Review Solutions

2

2

Solve the given formula for the given variable.

11. Solve for :

21

2. Solve for : 3

3.

2

3

Solve for

Ab

hb A bh

hV

hV r hr

Area of a Triangle

Volume of a Right Circular Cylinder

Volume of a Sphere 33

2

4:

3

4. Solve for :

3

4

4 4

5. Solve for :

r V r

r A r

P I Prt

Vr

Ar

IP

rt

Surface Area of a Sphere

Simple Interest


What you’ll learn about

Functions from Formulas Functions from Graphs Functions from Verbal Descriptions Functions from Data

… and whyUsing a function to model a variable under observation in terms of another variable often allows one to make predictionsin practical situations, such as predicting the future growth of a business based on data.


Example A Maximum Value Problem

A square of side inches is cut out of each corner of an 8 in. by 15 in. piece

of cardboard and the sides are folded up to form an open-topped box.

(a) Write the volume as a function of .

(b) Find th

x

V x

e domain of as a function of .

(c) Graph as a function of over the domain found in part (b) and use

the maximum finder on your grapher to determine the maximum volume

such a box can hold.

(d) How b

V x

V x

ig should the cut-out squares be in order to produce the box of

maximum volume?





(a) Write the volume as a function of .

x

V x





(b) Find the domain of as a function of .

x

V x





(c) Graph as a function of over the domain found

x

V x in part (b) and use

the maximum finder on your grapher to determine the maximum volume

such a box can hold. Use a window of [0, 5] by [0, 100].





(d) How big should the cut-out squares be in order to

x

produce the box of

maximum volume?


Example Finding the Model and Solving

Grain is leaking through a hole in a storage bin at a constant rate of 5 cubic inches per minute. The grain forms a cone-shaped pile on the ground below. As it grows, the height of the cone always remains equal to its radius. If the cone is one foot tall now, how tall will it be in one hour?


Constructing a Function from Data

Given a set of data points of the form (x, y), to construct

a formula that approximates y as a function of x:

1. Make a scatter plot of the data points. The points do not need to pass the vertical line test.

2. Determine from the shape of the plot whether the points seem to follow the graph of a familiar type of function (line, parabola, cubic, sine curve, etc.).

3. Transform a basic function of that type to fit the points as closely as possible.


Models from DataThe table shows low and high temperatures in 20 cities. Find a function that relates temperatures and predict the high for a low

of 46. City Low High City Low High

1 70 86 11 76 92

2 62 80 12 70 85

3 52 72 13 50 70

4 70 94 14 67 89

5 68 86 15 64 88

6 61 86 16 57 79

7 82 106 17 62 90

8 64 90 18 60 86

9 65 87 19 52 68

10 54 76 20 66 84


Models from DataFind a function that relates temperatures and predict the high for a low of 46.


Functions


Functions (cont’d)


Homework

Review Section 1.7 Page 160, Exercises: 1 – 49 (EOO) Quiz next time

Homework, Page 147

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