Splash Screen

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

http://connected.mcgraw-hill.com/connected/

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.


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.


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

A. 2

B. 4

C. 8

D. 10


A. 3a – 2

B.

C.

D.

Find the function value for


State the domain of .

A.

B. (–3, 3)

C.

D.


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.


• zeros

• roots

• line symmetry

• point symmetry

• even function

• odd function


http://www.glencoe.com/apps/eGlossary612/grade.php

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.


Answer: $1,050,000


$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.



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.


Answer: $50,000


$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.


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 < 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


Find Domain and Range

Use the graph of f to find the domain and range of the function.



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.

The domain of f is . In set-builder notation, the

domain is .

Range

The graph does not extend above y = 2, but f (x) decreases without bound for smaller and smaller values of x. So the range of f is .



Answer: D:

R:


Use the graph of f to find the domain and range of the function.

A. Domain:

Range:

B. Domain:

Range:

C. Domain:

Range:

D. Domain:

Range:


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.


Find y-Intercepts

Answer: 4

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.


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.


Find y-Intercepts

Answer: 1


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.


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


Find Zeros

Use the graph of f (x) = x 3 – x to approximate its

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


Answer: 1, 0, 1

Find Zeros


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.


A. –2.5

B. –1

C. 5

D. 9

Use the graph of to approximate

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



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.


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.


Answer: symmetric with respect to the y-axis

Test for Symmetry

Confirm Algebraically

Because x2 + 2 is equivalent to (x)2 + 2, the graph is symmetric with respect to the y-axis.


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.


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.


Answer: symmetric with respect to the origin

Test for Symmetry

Confirm Algebraically

Because (x)(y) = 6 is equivalent to (x)(y) = 6, the graph is symmetric with respect to the origin.


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

C. symmetric with respect to the origin

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



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.

Identify Even and Odd Functions


Answer: neither


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).


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.



Answer: even; symmetric with respect to the y-axis


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).


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.



Answer: neither


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).


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



Splash Screen

Documents

function of x

function f x

estimate function valuesa

estimate function valuesb

function valuesexample

graphical estimate

graph of f

polynomial p x