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Five-Minute Check Then/Now New Vocabulary Example

1 Real-World Example Estimate Function

Values Example 2 Find Domain and Range Example

3 Find y-Intercepts Example 4 Find Zeros Key

Concept Tests for Symmetry Example 5 Test for

Symmetry Key Concept Even and Odd

Functions Example 6 Identify Even and Odd

Functions

5Minute Check 1

Determine whether 2y 5x 7 represents y as a

function of x.

A. y is a function of x. B. y is not a function

of x.

5Minute Check 2

Determine whether the graph represents y as a

function of x.

A. y is a function of x. B. y is not a function

of x.

5Minute Check 3

Find the function value for f (2) if f (x) 6

x 2.

A. 2 B. 4 C. 8 D. 10

5Minute Check 4

5Minute Check 5

Then/Now

You identified functions. (Lesson 1-1)

- Use graphs of functions to estimate function

values and find domains, ranges, y-intercepts,

and zeros of functions. - Explore symmetries of graphs, and identify even

and odd functions.

Vocabulary

- zeros
- roots
- line symmetry
- point symmetry
- even function
- odd function

Example 1

Estimate Function Values

A. ADVERTISING The function f (x) 5x 2 50x

approximates the profit at a toy company, where

x is marketing costs and f (x) is profit. Both

costs and profits are measured in tens of

thousands of dollars. Use the graph to estimate

the profit when marketing costs are 30,000.

Confirm your estimate algebraically.

Example 1

Estimate Function Values

30,000 is three ten thousands. The function

value at x 3 appears to be about 100 ten

thousands, so the total profit was about

1,000,000. To confirm this estimate

algebraically, find f(3). f(3) -5(3)2 50(3)

105, or about 1,050,000. The graphical

estimate of about 1,000,000 is reasonable.

Answer 1,050,000

Example 1

Estimate Function Values

B. ADVERTISING The function f (x) 5x 2 50x

approximates the profit at a toy company, where

x is marketing costs and f (x) is profit. Both

costs and profits are measured in tens of

thousands of dollars. Use the graph to estimate

marketing costs when the profit is 1,250,000.

Confirm your estimate algebraically.

Example 1

Estimate Function Values

1,250,000 is 125 ten thousands. The value of the

function appears to reach 125 ten thousands for

an x-value of about 5. This represents an

estimate of 5 ? 10,000 or 50,000. To confirm

algebraically, find f(5). f(5) -5(5)2 50(5)

125, or about 1,250,000. The graphical estimate

that marketing costs are 50,000 when the profit

is 1,250,000 is reasonable.

Answer 50,000

Example 1

PROFIT A-Z Toy Boat Company found the average

price of its boats over a six month period. The

average price for each boat can be represented by

the polynomial p (x) 0.325x3 1.5x2 22,

where x is the month, and 0 lt x 6. Use the

graph to estimate the average price of a boat in

the fourth month. Confirm you estimate

algebraically.

A. 25 B. 23 C. 22 D. 20

Example 2

Find Domain and Range

Use the graph of f to find the domain and range

of the function.

Example 2

Find Domain and Range

- Domain
- The dot at (3, -3) indicates that the domain of

f ends at 3 and includes 3. - The arrow on the left side indicates that the

graph will continue without bound.

Example 2

Find Domain and Range

Example 2

Use the graph of f to find the domain and range

of the function.

Example 3

Find y-Intercepts

A. Use the graph of the function f (x) x 2 4x

4 to approximate its y-intercept. Then find the

y-intercept algebraically.

Example 3

Find y-Intercepts

Estimate Graphically It appears that f (x)

intersects the y-axis at approximately (0, 4), so

the y-intercept is about 4. Solve

Algebraically Find f (0). f (0) (0)2 4(0) 4

4. The y-intercept is 4.

Answer 4

Example 3

Find y-Intercepts

B. Use the graph of the function g (x) x 2

3 to approximate its y-intercept. Then find the

y-intercept algebraically.

Example 3

Find y-Intercepts

Estimate Graphically g (x) intersects the y-axis

at approximately (0, -1), so the y-intercept is

about -1. Solve Algebraically Find g (0). g (0)

0 2 3 or 1 The y-intercept is 1.

Answer -1

Example 3

Use the graph of the function to approximate its

y-intercept. Then find the y-intercept

algebraically.

A. 1 f (0) 1 B. 0 f (0) 0 C. 1 f (0)

1 D. 2 f (0) 2

Example 4

Find Zeros

Use the graph of f (x) x 3 x to approximate

its zero(s). Then find its zero(s) algebraically.

Example 4

Find Zeros

Estimate Graphically The x-intercepts appear to

be at about -1, 0, and 1. Solve

Algebraically x 3 x 0 Let f (x)

0. x(x 2 1) 0 Factor. x(x 1)(x 1)

0 Factor. x 0 or x 1 0 or x 1

0 Zero Product Property x 0 x 1

x -1 Solve for x. The zeros of f are 0, 1, and

-1.

Answer -1, 0, 1

Example 4

A. 2.5 B. 1 C. 5 D. 9

Key Concept 1

Example 5

Test for Symmetry

A. Use the graph of the equation y x 2 2 to

test for symmetry with respect to the x-axis, the

y-axis, and the origin. Support the answer

numerically. Then confirm algebraically.

Example 5

Test for Symmetry

Analyze Graphically The graph appears to be

symmetric with respect to the y-axis because for

every point (x, y) on the graph, there is a point

(-x, y). Support Numerically A table of values

supports this conjecture.

Example 5

Test for Symmetry

Confirm Algebraically Because x2 2 is

equivalent to (-x)2 2, the graph is symmetric

with respect to the y-axis.

Answer symmetric with respect to the y-axis

Example 5

Test for Symmetry

B. Use the graph of the equation xy 6 to test

for symmetry with respect to the x-axis, the

y-axis, and the origin. Support the answer

numerically. Then confirm algebraically.

Example 5

Test for Symmetry

Analyze Graphically The graph appears to be

symmetric with respect to the origin because for

every point (x, y) on the graph, there is a point

(-x, -y). Support Numerically A table of values

supports this conjecture.

Example 5

Test for Symmetry

Confirm Algebraically Because (-x)( -y) -6 is

equivalent to (x)(y) -6, the graph is symmetric

with respect to the origin.

Answer symmetric with respect to the origin

Example 5

Use the graph of the equation y x 3 to test

for symmetry with respect to the x-axis, the

y-axis, and the origin. Support the answer

numerically. Then confirm algebraically.

- A. symmetric with respect to the x-axis
- B. symmetric with respect to the y-axis
- symmetric with respect to the origin
- D. not symmetric with respect to the x-axis,

y-axis, or the origin

Key Concept 2

Example 6

Identify Even and Odd Functions

A. Graph the function f (x) x 2 4x 4 using

a graphing calculator. Analyze the graph to

determine whether the function is even, odd, or

neither. Confirm algebraically. If even or odd,

describe the symmetry of the graph of the

function.

Example 6

Identify Even and Odd Functions

It appears that the graph of the function is

neither symmetric with respect to the y-axis or

to the origin. Test this conjecture. f (-x)

(-x) 2 4(-x) 4 Substitute -x for x. x 2

4x 4 Simplify.

Since f (x) -x 2 4x - 4, the function is

neither even nor odd because f (-x) ? f (x) or f

(x).

Answer neither

Example 6

Identify Even and Odd Functions

B. Graph the function f (x) x 2 4 using a

graphing calculator. Analyze the graph to

determine whether the function is even, odd, or

neither. Confirm algebraically. If even or odd,

describe the symmetry of the graph of the

function.

Example 6

Identify Even and Odd Functions

From the graph, it appears that the function is

symmetric with respect to the y-axis. Test this

conjecture algebraically. f (-x) (-x)2 4

Substitute -x for x. x 2 - 4 Simplify. f

(x) Original function f (x) x 2 4

The function is even because f (-x) f (x).

Answer even symmetric with respect to the

y-axis

Example 6

Identify Even and Odd Functions

C. Graph the function f (x) x 3 3x 2 x 3

using a graphing calculator. Analyze the graph to

determine whether the function is even, odd, or

neither. Confirm algebraically. If even or odd,

describe the symmetry of the graph of the

function.

Example 6

Identify Even and Odd Functions

From the graph, it appears that the function is

neither symmetric with respect to the y-axis nor

to the origin. Test this conjecture

algebraically. f (x) (x) 3 3(x)2 (x)

3 Substitute x for x. x 3 3x 2 x 3

Simplify.

Because f (x) x 3 3x 2 x 3, the

function is neither even nor odd because f (x) ?

f (x) or f (x).

Answer neither

Example 6

Graph the function f (x) x 4 8 using a

graphing calculator. Analyze the graph to

determine whether the graph is even, odd, or

neither. Confirm algebraically. If even or odd,

describe the symmetry of the graph of the

function.

A. odd symmetric with respect to the

origin B. even symmetric with respect to the

y-axis C. neither even or odd

End of the Lesson