1. Splash Screen

2. Five-Minute Check (over Lesson 5–2) CCSS Then/Now New Vocabulary
Example 1: Degrees and Leading Coefficients
Example 2: Real-World Example: Evaluate a Polynomial Function
Example 3: Function Values of Variables
Key Concept: End Behavior of a Polynomial Function
Key Concept: Zeros of Even- and Odd-Degree Functions
Example 4: Graphs of Polynomial Functions
Lesson Menu

3. A. 6m2y3 – 3my B. 6my – 3y C. 3m2y3 – 3my D. 2m2y3 – my
5-Minute Check 1

4. A. 6m2y3 – 3my B. 6my – 3y C. 3m2y3 – 3my D. 2m2y3 – my
5-Minute Check 1

5. Simplify (m3 – 3m2 – 18m + 40) ÷ (m + 4).
A. m3 + 10
B. m2 + m + 6
C. m2 – 9m + 6
D. m2 – 7m + 10
5-Minute Check 2

6. Simplify (m3 – 3m2 – 18m + 40) ÷ (m + 4).
A. m3 + 10
B. m2 + m + 6
C. m2 – 9m + 6
D. m2 – 7m + 10
5-Minute Check 2

7. Simplify (p3 – 8) ÷ (p – 2). A. p3 + 4 B. p2 + 2p + 4 C. p2 + p + 4
D. p2 + 4
5-Minute Check 3

8. Simplify (p3 – 8) ÷ (p – 2). A. p3 + 4 B. p2 + 2p + 4 C. p2 + p + 4
D. p2 + 4
5-Minute Check 3

9. Simplify (4x4 – x3 – 19x2 + 11x – 3) ÷ (x – 2).
A. 4x4 – x3 – 5x2 + x – 1 B. C.
D. x3 – 4x2 – 5x + 1
5-Minute Check 4

10. Simplify (4x4 – x3 – 19x2 + 11x – 3) ÷ (x – 2).
A. 4x4 – x3 – 5x2 + x – 1 B. C.
D. x3 – 4x2 – 5x + 1
5-Minute Check 4

11. If the area of a parallelogram is given by x2 – 5x + 4 and the base is x – 1, what is the height of the figure?
A. x + 4
B. x – 4
C. x – 2
D. x + 2
5-Minute Check 5

12. If the area of a parallelogram is given by x2 – 5x + 4 and the base is x – 1, what is the height of the figure?
A. x + 4
B. x – 4
C. x – 2
D. x + 2
5-Minute Check 5

13. The volume of a box is given by the expression x3 + 3x2 – x – 3
The volume of a box is given by the expression x3 + 3x2 – x – 3. The height of the box is given by the expression x – 1. Find an expression for the area of the base of the box.
A. x2 + 4x + 3
B. x2 + 2x – 3
C. x2 + 2x –
D. x + 3
5-Minute Check 6

14. The volume of a box is given by the expression x3 + 3x2 – x – 3
The volume of a box is given by the expression x3 + 3x2 – x – 3. The height of the box is given by the expression x – 1. Find an expression for the area of the base of the box.
A. x2 + 4x + 3
B. x2 + 2x – 3
C. x2 + 2x –
D. x + 3
5-Minute Check 6

15. Mathematical Practices
Content Standards
F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
F.IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Mathematical Practices
1 Make sense of problems and persevere in solving them.
CCSS

16. You analyzed graphs of quadratic functions.
Evaluate polynomial functions.
Identify general shapes of graphs of polynomial functions.
Then/Now

17. polynomial in one variable leading coefficient polynomial function
power function
quartic function
quintic function
end behavior
Vocabulary

18. Degrees and Leading Coefficients
A. State the degree and leading coefficient of 7z3 – 4z2 + z. If it is not a polynomial in one variable, explain why.
Answer:
Example 1

19. Degrees and Leading Coefficients
A. State the degree and leading coefficient of 7z3 – 4z2 + z. If it is not a polynomial in one variable, explain why.
Answer: This is a polynomial in one variable. The degree is 3 and the leading coefficient is 7.
Example 1

20. Degrees and Leading Coefficients
B. State the degree and leading coefficient of 6a3 – 4a2 + ab2. If it is not a polynomial in one variable, explain why.
Answer:
Example 1

21. Degrees and Leading Coefficients
B. State the degree and leading coefficient of 6a3 – 4a2 + ab2. If it is not a polynomial in one variable, explain why.
Answer: This is not a polynomial in one variable. It contains two variables, a and b.
Example 1

22. Degrees and Leading Coefficients
C. State the degree and leading coefficient of 3x5 + 2x2 – 4 – 8x6. If it is not a polynomial in one variable, explain why.
Answer:
Example 1

23. Degrees and Leading Coefficients
C. State the degree and leading coefficient of 3x5 + 2x2 – 4 – 8x6. If it is not a polynomial in one variable, explain why.
Answer: This is a polynomial in one variable. The greatest exponent is 6, so the degree is 6 and the leading coefficient is –8.
Example 1

24. A. degree: 3 leading coefficient: 2
A. Determine whether 3x3 + 2x2 – 3 is a polynomial in one variable. If so, state the degree and leading coefficient.
A. degree: 3 leading coefficient: 2
B. degree: 3 leading coefficient: 3
C. degree: 2 leading coefficient: –3
D. not a polynomial in one variable
Example 1

25. A. degree: 3 leading coefficient: 2
A. Determine whether 3x3 + 2x2 – 3 is a polynomial in one variable. If so, state the degree and leading coefficient.
A. degree: 3 leading coefficient: 2
B. degree: 3 leading coefficient: 3
C. degree: 2 leading coefficient: –3
D. not a polynomial in one variable
Example 1

26. A. degree: 2 leading coefficient: 3
B. Determine whether 3x2 + 2xy – 5 is a polynomial in one variable. If so, state the degree and leading coefficient.
A. degree: 2 leading coefficient: 3
B. degree: 2 leading coefficient: 2
C. degree: 1 leading coefficient: –5
D. not a polynomial in one variable
Example 1

27. A. degree: 2 leading coefficient: 3
B. Determine whether 3x2 + 2xy – 5 is a polynomial in one variable. If so, state the degree and leading coefficient.
A. degree: 2 leading coefficient: 3
B. degree: 2 leading coefficient: 2
C. degree: 1 leading coefficient: –5
D. not a polynomial in one variable
Example 1

28. A. degree: 6 leading coefficient: 4
C. Determine whether 9y3 + 4y6 – 45 – 8y2 – 5y7 is a polynomial in one variable. If so, state the degree and leading coefficient.
A. degree: 6 leading coefficient: 4
B. degree: 7 leading coefficient: –5
C. degree: 7 leading coefficient: 5
D. not a polynomial in one variable
Example 1

29. A. degree: 6 leading coefficient: 4
C. Determine whether 9y3 + 4y6 – 45 – 8y2 – 5y7 is a polynomial in one variable. If so, state the degree and leading coefficient.
A. degree: 6 leading coefficient: 4
B. degree: 7 leading coefficient: –5
C. degree: 7 leading coefficient: 5
D. not a polynomial in one variable
Example 1

30. Evaluate a Polynomial Function
RESPIRATION The volume of air in the lungs during a 5-second respiratory cycle can be modeled by v(t) = –0.037t t t, where v is the volume in liters and t is the time in seconds. This model is an example of a polynomial function. Find the volume of air in the lungs 1.5 seconds into the respiratory cycle.
By substituting 1.5 into the function we can find v(1.5), the volume of air in the lungs 1.5 seconds into the respiration cycle.
Example 2

31. v(t) = –0.037t 3 + 0.152t 2 + 0.173t Original function
Evaluate a Polynomial Function
v(t) = –0.037t t t Original function
v(1.5) = –0.037(1.5) (1.5) (1.5) Replace t with 1.5.
≈ – Simplify.
≈ Add.
Answer:
Example 2

32. v(t) = –0.037t 3 + 0.152t 2 + 0.173t Original function
Evaluate a Polynomial Function
v(t) = –0.037t t t Original function
v(1.5) = –0.037(1.5) (1.5) (1.5) Replace t with 1.5.
≈ – Simplify.
≈ Add.
Answer: L
Example 2

33. The height of a toy rocket during a 2
The height of a toy rocket during a 2.35 second flight is predicted by the function h(t) = –4t 3 + 6t 2 + 8t, where h is the height in meters and t is the time in seconds. This model is an example of a polynomial function. Find the height of the toy rocket seconds into the flight.
A meters
B meters
C meters
D meters
Example 2

34. The height of a toy rocket during a 2
The height of a toy rocket during a 2.35 second flight is predicted by the function h(t) = –4t 3 + 6t 2 + 8t, where h is the height in meters and t is the time in seconds. This model is an example of a polynomial function. Find the height of the toy rocket seconds into the flight.
A meters
B meters
C meters
D meters
Example 2

35. Find b(2x – 1) – 3b(x) if b(m) = 2m2 + m – 1.
Function Values of Variables
Find b(2x – 1) – 3b(x) if b(m) = 2m2 + m – 1.
To evaluate b(2x – 1), replace the m in b(m) with 2x – 1.
Original function
Replace m with 2x – 1.
Evaluate 2(2x – 1)2.
Simplify.
Example 3

36. Distributive Property
Function Values of Variables
To evaluate 3b(x), replace m with x in b(m), then multiply the expression by 3.
Original function
Replace m with x.
Distributive Property
Example 3

37. Now evaluate b(2x – 1) – 3b(x).
Function Values of Variables
Now evaluate b(2x – 1) – 3b(x).
b(2x – 1) – 3b(x)
Replace b(2x – 1) and 3b(x) with evaluated expressions.
Distribute.
= 2x2 – 9x + 3 Simplify.
Answer:
Example 3

38. Now evaluate b(2x – 1) – 3b(x).
Function Values of Variables
Now evaluate b(2x – 1) – 3b(x).
b(2x – 1) – 3b(x)
Replace b(2x – 1) and 3b(x) with evaluated expressions.
Distribute.
= 2x2 – 9x + 3 Simplify.
Answer: 2x2 – 9x + 3
Example 3

39. Find g(2x + 1) – 2g(x) if g(b) = b2 + 3.
A. 1
B. 2x 2 + 4x – 2
C. 2x 2 + 4x + 10
D. 2x 2 – 2
Example 3

40. Find g(2x + 1) – 2g(x) if g(b) = b2 + 3.
A. 1
B. 2x 2 + 4x – 2
C. 2x 2 + 4x + 10
D. 2x 2 – 2
Example 3

41. Concept

42. Concept

43. describe the end behavior,
Graphs of Polynomial Functions
A. For the graph,
describe the end behavior,
determine whether it represents an odd-degree or an even- degree function, and
state the number of real zeros.
Answer:
Example 4

44. describe the end behavior,
Graphs of Polynomial Functions
A. For the graph,
describe the end behavior,
determine whether it represents an odd-degree or an even- degree function, and
state the number of real zeros.
Answer:
f(x) → –∞ as x → +∞
f(x) → –∞ as x → –∞
It is an even-degree polynomial function.
The graph does not intersect the x-axis, so the function has no real zeros.
Example 4

45. describe the end behavior,
Graphs of Polynomial Functions
B. For the graph,
describe the end behavior,
determine whether it represents an odd-degree or an even-degree function, and
state the number of real zeros.
Answer:
Example 4

46. describe the end behavior,
Graphs of Polynomial Functions
B. For the graph,
describe the end behavior,
determine whether it represents an odd-degree or an even-degree function, and
state the number of real zeros.
Answer:
f(x) → +∞ as x → +∞
f(x) → –∞ as x → –∞
It is an odd-degree polynomial function.
The graph intersects the x-axis at one point, so the function has one real zero.
Example 4

47. A. For the graph, determine whether it represents an odd-degree or an even-degree function, and state the number of real zeros.
A. It is an even-degree polynomial function and has no real zeros.
B. It is an even-degree polynomial function and has two real zeros.
C. It is an odd-degree polynomial function and has two real zeros.
D. It is an odd-degree polynomial function and has no real zeros.
Example 4

48. A. For the graph, determine whether it represents an odd-degree or an even-degree function, and state the number of real zeros.
A. It is an even-degree polynomial function and has no real zeros.
B. It is an even-degree polynomial function and has two real zeros.
C. It is an odd-degree polynomial function and has two real zeros.
D. It is an odd-degree polynomial function and has no real zeros.
Example 4

49. B. For the graph, determine whether it represents an odd-degree or an even-degree function, and state the number of real zeros.
A. It is an even-degree polynomial function and has three real zeros.
B. It is an odd-degree polynomial function and has no real zeros.
C. It is an odd-degree polynomial function and has three real zeros.
D. It is an even-degree polynomial function and has no real zeros.
Example 4

50. B. For the graph, determine whether it represents an odd-degree or an even-degree function, and state the number of real zeros.
A. It is an even-degree polynomial function and has three real zeros.
B. It is an odd-degree polynomial function and has no real zeros.
C. It is an odd-degree polynomial function and has three real zeros.
D. It is an even-degree polynomial function and has no real zeros.
Example 4

51. End of the Lesson