1. Splash Screen

2. Lesson Menu 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

3. 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.

4. Vocabulary zeros roots line symmetry point symmetry even function odd function

5. Example 1 Estimate Function Values A. ADVERTISING The function f (x) = –5x 2 + 50x approximates the profit at a toy company, where x represents marketing costs and f (x) represents 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.

6. Example 1 Estimate Function Values B. ADVERTISING The function f (x) = –5x 2 + 50x approximates the profit at a toy company, where x represents marketing costs and f (x) represents 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.

7. 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 function p (x) = –0.325x 3 + 1.5x 2 + 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

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

9. Example 2 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:

10. 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.

11. 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.

12. 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

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

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

15. Key Concept 1

16. 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.

17. 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.

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

19. Key Concept 2

20. Example 6 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

21. Example 6 Answer: neither 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 + 4Simplify. Since –f (x) =  x 2 + 4x  4, the function is neither even nor odd because f (  x) ≠ f (x) or –f (x).

22. Example 6 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. Identify Even and Odd Functions

23. Example 6 Answer: even; symmetric with respect to the y-axis 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).

24. Example 6 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. Identify Even and Odd Functions

25. Example 6 Answer: neither 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).

26. 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 nor odd

27. End of the Lesson