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 function p (x) = –0.325x x2 + 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
Example 1

2. Use the graph of f to find the domain and range of the function.
Find Domain and Range
Use the graph of f to find the domain and range of the function.
Example 2

3. The domain of f is . In set-builder notation, the domain is .
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.
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
Example 2

4. 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:
Example 2

5. 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

6. 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

7. 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

8. 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

9. Use the graph of the function to approximate its y-intercept
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 3

10. Find Zeros
Use the graph of f (x) = x 3 – x to approximate its zero(s). Then find its zero(s) algebraically.
Example 4

11. The x-intercepts appear to be at about -1, 0, and 1.
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 = x = -1 Solve for x.
The zeros of f are 0, 1, and -1.
Answer: -1, 0, 1
Example 4

12. Use the graph of to approximate its zero(s)
Use the graph of to approximate its zero(s). Then find its zero(s) algebraically.
A. –2.5
B. –1
C. 5
D. 9
Example 4

13. Key Concept 1

14. Test for Symmetry
A. Use the graph of the equation y = x 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

15. Confirm Algebraically
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

16. 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

17. Confirm Algebraically
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

18. A. symmetric with respect to the x-axis
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
Example 5

19. Key Concept 2

20. 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

21. f (-x) = (-x) 2 – 4(-x) + 4 Substitute -x for x.
Identify Even and Odd Functions
It appears that the graph of the function is neither symmetric with respect to the y-axis nor to the origin. Test this conjecture algebraically.
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

22. 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

23. f (-x) = (-x)2 – 4 Substitute -x for x. = x 2 - 4 Simplify.
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 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

24. 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

25. f (–x) = (–x) 3 – 3(–x)2 – (–x) + 3 Substitute –x for x.
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

26. A. odd; symmetric with respect to the origin
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 nor odd
Example 6