Lesson 1: 2.1 Symmetry (3-1) Lesson 2: 2.2 Graph Families (3-2, 3-3) Lesson 3: 2.3 Inverses (3-4) Lesson 4: 2.4 Continuity (3-5) Lesson 5: 2.5 Extrema (3-6) Lesson 6: 2.6 Rational Functions (3-7)

In this unit we will learn…STANDARD 2.1: use algebraic tests to determine

symmetry in graphs, including even-odd tests (3-1)STANDARD 2.2: graph parent functions and perform

transformations to them (3-2, 3-3)STANDARD 2.3: determine and graph inverses of

functions (3-4)STANDARD 2.4: determine the continuity and end

behavior of functions (3-5)STANDARD 2.5: use appropriate mathematical

terminology to describe the behavior of graphs (3-6)STANDARD 2.6: graph rational functions (3-7)

In this lesson we will…

Discuss what symmetry is and the different types that exist.

Learn to determine symmetry in graphs.

Classify functions as even or odd.

Point Symmetry: Symmetry about one point

Figure will spin about the point and land on itself in less than 360º.

Two distinct points P and P are symmetric to M

iff M is the midpoint of the segment PP .

M

P’

P

This is the main point we look at for symmetry.

Let’s build some symmetry!

The graph of the relation S is symmetric

with respect to the origin iff :

, implies ,a b S a b S

Easier: iff ( ) ( )f x f x

If the function is odd

(all odd exponents) then the graph will be

symmetric to the origin.

* plain numbers have exp = 0 *

Easier yet!

Two distinct points and ' are

symmetric with respect to a line

iff is the perpendicular bisector of '.

A point is symmetric to itself

with respect to line iff is on .

P P

PP

P

P

l

l

l l

U D

x-axis

y-axis

y = x

y = -x

, implies ,a b S a b S

, implies ,a b S a b S

Easier: iff ( ) ( )f x f x

If the function is even

(all even exponents) then the graph will be

symmetric to the axis.

* plain numbers have exp = 0 *

Easier yet!

y

, implies ,a b S b a S

, implies ,a b S b a S

HW 2.1: P 134 #15 – 35 odd

Get a piece of graph paper and a calculator.

Graph the following on separate axii:

2 3

0

y y x

y x y x

y x y x

y x

1

yx

In this section we will…

Identify the graphs of some simple functions. Recognize and perform transformations of

simple graphs. Sketch graphs of related functions.

Any function based on a simple function will have the basic “look” of that family.

Multiplying, dividing, adding or subtracting from the function may move it, shrink it or stretch it but won’t change its basic shape.

5 4

How are these two f unctions the same?

How are they diff erent?

y x y x

Reflections

Vertical Translations

Horizontal Translations

Vertical Dilations

Horizontal Dilations

Send One person from your group to get a white board with a graph on it, a pen and an eraser.

In this section we will…

Use function families to graph inequalities.

3 2y x

HW1 2.2: P 143 #13-29 odd, 33 HW2 2.2: P 150 #21-31 odd

In this section we will…

Determine inverses of relations and functions.

Graph functions and their inverses.

An inverse of function will take the answers (range) from the function and give back the original domain.

Easy!!! Just switch the domain and range!

Are they both functions?

1

( ) (1,4),(2,6),(3,8),(4,4)

( )

f x

f x

If f(x) and f –1(x) are inverse functions, then

In other words…◦ Two relations are inverse relations iff one relation

contains the element (b,a) whenever the other relation contains (a,b).

◦ Does this remind you of something?

1( ) if and only if ( )f x y f y x

1

( ) (1,4),(2,6),(3,8),(4,4)

( ) (4,1),(6,2),(8,3),(4,4)

f x

f x

Are reflections of each other over the line y = x.

Graph the inverse of 2 3.y x

Is the inverse a function?

If the original function passes the HORIZONTAL line test then the inverse will be a function.

Let’s check our parent graphs.

Is the inverse a function?

3

3

Use the parent f unction

to graph 3 2.

y x

y x

Is this a function?

2Use the parent f unction

to graph 1 2

y x

y x

If two functions are actually inverses then both the composites of the functions will equal x.

You must prove BOTH true.

( ) ( )f g x g f x x

2( ) 3 and ( ) 3f x x f x x

Replace f(x) with y (it is just easier to look at this way).

Switch the x and y in the equation. Resolve the equation for y. The result is the inverse. Now check!

6( )

3x

f x

2( ) 2 1f x x x

Now check: Does ( ) ( ) ?f g x g f x x

The fixed costs for manufacturing a particular stereo system are $96,000, and the variable costs are $80 per unit.◦ A. Write an equation that expresses the total cost

C(x) as a function of x given that x units are manufactured.

B. Determine the equation for the inverse process and describe the real-world situation it models.

C. Determine the number of units that can be made for $144,000.

HW 2.3: P 156 #15 – 39 odd and 45

In this section we will…

Determine the continuity or discontinuity of a function.

Identify the end behavior of functions.

Determine whether a function is increasing or decreasing on an interval.

A continuous function’s graph can be drawn without ever lifting up your pencil.

It has no holes or gaps.

Anything which disrupts the flow of the graph.

What parent graphs do we have which demonstrate discontinuous functions?

Function is undefined at a value but, otherwise, the graph matches up.

Graph has a “hole”.

Graph stops at one y-value, then “jumps” to a different y-value for the same x-value.

Common in piece-wise functions.

A major disruption in the graph.

As graph approaches the domain restriction, the graph will shoot towards either positive or negative infinity.

A function is continuous at if it satisfies the following conditions:

1) the function is defined at ; in other words, ( ) exists. 2) the function approaches the same -value on the l

x c

c f cy

eft and on the right sides of ; and 3) the -value that the function approaches from each side is ( ).

x cy

f c

A function is continuous on an interval iff it is continuous at each number in the interval.

A f unction is said to be increasing on an interval, , iff

f or every and contained on , ( ) ( ), wherever .

A f unction is said to be decreasing on an interval, , iff

f or every and containe

I

a b I f a f b a b

I

a b

d on , ( ) ( ), wherever .

A f unction is said to be constant on an interval, , iff

f or every and contained on , ( ) ( ), wherever .

I f a f b a b

I

a b I f a f b a b

Increasing means uphill left to right.

Decreasing means downhill left to right.

Constant means a flat or horizontal line left to right.

P 166 #26, 28, 30

Determine the intervals where the functions are increasing or decreasing.

Write the intervals in interval notation and in in terms of x.

26.

28.

30.

What will the function be doing at the outermost reaches of its domain and range?

2 4 3

3 2

6 9 ( ) 3

3, 2( ) 7 2 ( )

3 2, 2

1( )

3

y x x f x x x

xf x x x f x

x x

f xx

HW 2.4: P 166 #13 – 31 odd, 39

You will need a graphing calculator.

2 4 3

3 2

6 9 ( ) 3

3, 2( ) 7 2 ( )

3 2, 2

1( )

3

y x x f x x x

xf x x x f x

x x

f xx

In this section we will…

Find the extrema of functions.

Learn the difference between Absolute Extrema and Relative Extrema.

Find the point of inflection of a functions (if it exists).

An Absolute Minimum of Maximum is the lowest or highest value the range of the function can have.

The slope of the line drawn tangent to the min or max will have a slope of zero.

That point is called a critical point for the graph.

2

2

3

( ) ( 2) 1

( ) 4 3

( )

f x x

f x x x

D t t t

These points are not the absolute highs or lows for the function but they are the high or low over a certain interval.

The slope of the line tangent to a relative min or max is still zero so the point is a critical point.

Minimums are said to be concave up and maximums are concave down.

3

4 2

( ) 7 1

( ) 4 2

f x x x

h x x x

A point of inflection occurs when a graph changes from one concavity to another.

The slope of the tangent line to this point is undefined ( a vertical line). This point is also considered a critical point.

You will learn to calculate the point of inflection in calculus.

2

4 2

5 3 2

6 1, 3

3 5, 0

2 2 , 0

y x x x

y x x x

y x x x x

HW 2.5: P 177 #13 – 29 every other odd and 34

You will need a graphing calculator.

Describe the end behavior of the graph.

In this section we will…

Graph Rational Functions

Determine vertical, horizontal and slant asymptotes

Have a variable in the denominator.

The denominator restriction will have a profound effect on the function’s graph.

Caused by values which make the denominator 0.

Also known as removable and non-removable discontinuities.

3 cases possible:

1. Degree of numerator < Degree of denominator H.A. at y = 0.

2. Degree of numerator = Degree of denominator H.A. is the ratio of the coefficients.

3. Degree of numerator > Degree of denominator Do long division to find the Slant Asymptote.

HW 2.6: P 186 #15 – 39 odd, 43