1. Splash Screen

2. Example 1: Real-World Example: Estimate Function Values
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
Lesson Menu

3. 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.
5–Minute Check 1

4. 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.
5–Minute Check 2

5. Find f (–2) if f (x) = 6 – x 2.
A. 2
B. 4
C. 8
D. 10
5–Minute Check 3

6. Find
A. 3a – 2 B. C. D.
5–Minute Check 4

7. State the domain of A.
B. (–3, 3) C. D.
5–Minute Check 5

8. 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.
5–Minute Check 1

9. 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.
5–Minute Check 2

10. Find f (–2) if f (x) = 6 – x 2.
A. 2
B. 4
C. 8
D. 10
5–Minute Check 3

11. Find
A. 3a – 2 B. C. D.
5–Minute Check 4

12. State the domain of A.
B. (–3, 3) C. D.
5–Minute Check 5

13. 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.
Then/Now

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

15. Estimate Function Values
A. ADVERTISING The function f (x) = –5x x 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.
Example 1

16. The graphical estimate of about $1,000,000 is reasonable.
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

17. Estimate Function Values
B. ADVERTISING The function f (x) = –5x x 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.
Example 1

18. 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

19. 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

20. 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

21. 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

22. 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

23. Find Domain and Range
Answer: D: R:
Example 2

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

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

33. 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

34. 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

35. 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

36. Key Concept 1

37. 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

38. A table of values supports this conjecture.
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

39. 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

40. 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

41. A table of values supports this conjecture.
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

42. 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

43. 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

44. 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

45. Key Concept 2

46. 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

47. 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

48. 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

49. 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

50. 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

51. 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

52. 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

53. 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

54. That’s a wrap Practice 1-2 HW # 1- 41 (odd), 107-110
Get ready for the 1-1, 1-2 Quiz!