Graphs of Functions

Lesson 3

Warm Up – Perform the Operations and Simplify

2

412.

x

a

5420

3.

22

xx

x

xxb

352

10

5

6112.

2

23

xx

x

x

xxxc

92

7

92

7.

x

x

x

xd

2

412.

x

a

Solution

2

20122

424122

4

2

2122

412

x

xx

xxx

xx

Solution

5420

3.

22

xx

x

xxb

145

3

145

433

145

413

1545

3

2

2

xxx

xx

xxx

xxx

xxx

xxx

xx

x

xx

352

10

5

6112.

2

23

xx

x

x

xxxc

Solution

35

106

3125

10612

312

10

5

6112

352

10

5

6112

2

2

23

x

xx

xxx

xxxx

xx

x

x

xxx

xx

x

x

xxx

92

7

92

7.

x

x

x

xd

Solution

7

7

7

92

92

7

92

7

92

7

x

x

x

x

x

x

x

x

x

x

Domain & Range of a Function

What is the domain ofthe graph of the functionf?

4,1: A

Domain & Range of a Function

What is the range ofthe graph of the functionf?

4,5

Domain & Range of a Function .21 fandfFind

51 f

42 f

Let’s look at domain and range of a function using an algebraic approach.

Then, let’s check it with a graphical approach.

Find the domain and range of

Algebraic Approach

.4 xxf

The expression under the radical can not be negative.Therefore, Domain .04 x

,4

4:

or

xA Since the domain is never negative the range is the set of all nonnegative real numbers.

,0

0:

or

yARange

Find the domain and range of

Graphical Approach

.4 xxf

Increasing and Decreasing Functions

The more you know about the graph of a function, the more you know about the function itself.

Consider the graph on the next slide.

Falls from x = -2 to x = 0.

Is constant from x = 0 to x = 2.

Rises from x = 2 to x = 4.

Ex: Find the open intervals on which the function is increasing, decreasing, or constant.

Increases over the entire real line.

Ex: Find the open intervals on which the function is increasing, decreasing, or constant.

,11,

:

and

INCREASING

1,1

:

DECREASING

Ex: Find the open intervals on which the function is increasing, decreasing, or constant.

0,

:

INCREASING

2,0

:CONSTANT

.2

:DECREASING

Relative Minimum and Maximum Values

The point at which a function changes its increasing, decreasing, or constant behavior are helpful in determining the relative maximum or relative minimum values of a function.

General Points – We’ll find EXACT points later……

Approximating a Relative Minimum

Example: Use a GDC to approximate the relative minimum of the function given by

.243 2 xxxf

Put the function into the “y = “ the press zoom 6 to look at the graph.

Press trace to follow the line to the lowest point.

.243 2 xxxf

Example

Use a GDC to approximate the relative minimum and relative maximum of the function given by

.3 xxxf

Solution

Relative Minimum

(-0.58, -0.38)

Solution

Relative Maximum

(0.58, 0.38)

Step Functions and Piecewise-Defined Functions

Because of the vertical jumps, the greatest integer function is an example

of a step function.

Let’s graph a Piecewise-Defined Function

Sketch the graph of

1,4

1,32

xx

xxxf

Notice when open dots and closeddots are used. Why?

Even and Odd Functions

Graphically

Algebraically

Let’s look at the graphs again and see if this applies.

Graphically

☺ ☺

Example

Determine whether each function is even, odd, or neither.

AlgebraicGraphical – Symmetric to Origin

Algebraic

Graphical – Symmetric to y-axis

Algebraic

Graphical – NOT Symmetric to origin OR y-axis.

You Try

Is the function

Even, Odd, of Neither?

xxf

Solution xxf

Symmetric about the y-axis.