1. Polynomial Functions &
Algebra II Mr. Gilbert
Chapter 7.1 & 7.2
Polynomial Functions &
Graphing Polynomials
Standard & Honors
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2. Agenda
Warm up
Home Work
Lesson
Practice
Homework
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3. Homework Review
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4. Communicate Effectively
Polynomial Function in one variable:
standard form: f(x)=a0xn+a1xn-1+…+an-1x+an
| a00 and a0, a1, … an are Real
Degree of the polynomial: the largest exponent.
Leading Coefficient: coefficient of the highest degree.
Relative Minimum: No other nearby points are smaller.
Relative Maximum: No other nearby points are larger.
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5. Example 1 Find Degrees and Leading Coefficients (5)
Example 2 Evaluate a Polynomial Function (5)
Example 3 Functional Values of Variables (5)
Example 4 Graphs of Polynomial Functions (5)
Example 1 Graph a Polynomial Function
Example 2 Locate Zeros of a Function
Example 3 Maximum and Minimum Points
Example 4 Graph a Polynomial Model
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Lesson 1 Contents

6. State the degree and leading coefficient of. in one variable
State the degree and leading coefficient of in one variable. 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.
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Example 1-1a

7. State the degree and leading coefficient of. in one variable
State the degree and leading coefficient of in one variable. 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.
State the degree and leading coefficient of in one variable. If it is not a polynomial in one variable, explain why.
Answer: This is not a polynomial in one variable. The term 2c–1 is not of the form ancn, where n is a nonnegative integer.
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Example 1-1b

8. Rewrite the expression so the powers of y are in decreasing order.
State the degree and leading coefficient of in one variable. If it is not a polynomial in one variable, explain why.
Rewrite the expression so the powers of y are in decreasing order.
Answer: This is a polynomial in one variable with degree of 4 and leading coefficient 1.
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Example 1-1d

9. Answer: degree 3, leading coefficient 3
State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. a. b.
Answer: degree 3, leading coefficient 3
Answer: This is not a polynomial in one variable. It contains two variables, x and y.
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Example 1-1e

10. Answer: degree 3, leading coefficient 1
Answer: This is not a polynomial in one variable. The term 3a–1 is not of the form ancn, where n is nonnegative.
Answer: degree 3, leading coefficient 1
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Example 1-1f

11. Find the values of f (4), f (5), and f (6).
Nature Refer to Example 2 on page 347 of your textbook. A sketch of the arrangement of hexagons shows a fourth ring of 18 hexagons, a fifth ring of 24 hexagons, and a sixth ring of 30 hexagons.
Show that the polynomial function gives the total number of hexagons when
Find the values of f (4), f (5), and f (6).
Original function
Replace r with 4.
Simplify.
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Example 1-2a

12. Original function Replace r with 5. Simplify. Original function
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Example 1-2b

13. Answer: These match the function values for respectively.
From the information given in Example 2 of your textbook, you know that the total number of hexagons for three rings is 19. So, the total number of hexagons for four rings is or 37, five rings is or 61, and six rings is or 91.
Answer: These match the function values for respectively.
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Example 1-2c

14. Find the total number of hexagons in a honeycomb with 20 rings.
Nature Refer to Example 2 on page 347 of your textbook. A sketch of the arrangement of hexagons shows a fourth ring of 18 hexagons, a fifth ring of 24 hexagons, and a sixth ring of 30 hexagons.
Find the total number of hexagons in a honeycomb with 20 rings.
Original function
Replace r with 20.
Answer:
Simplify.
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Example 1-2d

15. Nature A sketch of the arrangement of hexagons shows a seventh ring of 36 hexagons, an eighth ring of 42 hexagons, and a ninth ring of 48 hexagons.
a. Show that the polynomial function gives the total number of hexagons when Recall that the total number of hexagons in six rings is 91.
Answer: f (7) = 127; f (8) = 169; f (9) = 217; the total number of hexagons for seven rings is or 127, eight rings is or 169, and nine rings is or 217. These match the functional values for r = 7, 8, and 9, respectively.
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Example 1-2e

16. b. Find the total number of hexagons in a honeycomb with 30 rings.
Answer: 2611
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Example 1-2f

17. Find Original function Replace x with y 3. Answer: Property of powers
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Example 1-3a

18. To evaluate b(2x – 1), replace m in b(m) with 2x – 1.
Find
To evaluate b(2x – 1), replace m in b(m) with 2x – 1.
Original function
Replace m with 2x – 1.
Evaluate 2(2x – 1)2.
Simplify.
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Example 1-3b

19. Distributive Property
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
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Example 1-3c

20. Now evaluate b(2x – 1) – 3b(x).
Replace b(2x – 1) and 3b(x) with evaluated expressions.
Simplify.
Answer:
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Example 1-3d

21. a. Find b. Find
Answer:
Answer:
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Example 1-3e

22.  describe the end behavior,
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: 
 It is an even-degree polynomial function.
 The graph does not intersect the x-axis, so the function has no real zeros. .
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Example 1-4a

23.  describe the end behavior,
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: 
 It is an odd-degree polynomial function.
 The graph intersects the x-axis at one point, so the function has one real zero. .
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Example 1-4b

24.  describe the end behavior,
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: 
 It is an even-degree polynomial function.
 The graph intersects the x-axis at two points, so the function has two real zeros. .
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Example 1-4c

25.  describe the end behavior,
For each graph, a.
 describe the end behavior,
 determine whether it represents an odd-degree or an even-degree function, and
 state the number of real zeros.
Answer: 
 It is an even-degree polynomial function.
 The graph intersects the x-axis at two points, so the function has two real zeros. .
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Example 1-4d

26.  describe the end behavior,
For each graph, b.
 describe the end behavior,
 determine whether it represents an odd-degree or an even-degree function, and
 state the number of real zeros.
Answer: 
 It is an odd-degree polynomial function.
 The graph intersects the x-axis at three points, so the function has three real zeros. .
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Example 1-4e

27.  describe the end behavior,
For each graph, c.
 describe the end behavior,
 determine whether it represents an odd-degree or an even-degree function, and
 state the number of real zeros.
Answer: 
 It is an even-degree polynomial function.
 The graph intersects the x-axis at one point, so the function has one real zero. .
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Example 1-4f

28. Graphing
Answer:
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End of Lesson 1

29. making a table of values. –4 5
Graph by
making a table of values. x
f(x) –4 5 –3 –2 –1 2 1
–19
Answer:
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Example 2-1a

30. making a table of values.
Graph by
making a table of values.
This is an odd degree polynomial with a negative leading coefficient, so f (x)  + as x  – and f (x)  – as x  +. Notice that the graph intersects the x-axis at 3 points indicating that there are 3 real zeros.
Answer:
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Example 2-1b

31. making a table of values. –3 –8
Graph by
making a table of values. x
f (x) –3 –8 –2 1 –1 2 4 17
Answer:
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Example 2-1c

32. Determine consecutive values of x between which each real zero of the function is located. Then draw the graph.
Make a table of values. Since f (x) is a 4th degree polynomial function, it will have between 0 and 4 zeros, inclusive. x
f (x) –2 9 –1 1 –3 2 –7 3 19
change in signs
change in signs
change in signs
change in signs
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Example 2-2a

33. Look at the value of f (x) to locate the zeros
Look at the value of f (x) to locate the zeros. Then use the points to sketch the graph of the function.
Answer:
There are zeros between x = –2 and –1, x = –1 and 0, x = 0 and 1, and x = 2 and 3.
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Example 2-2b

34. There are zeros between x = –1 and 0, x = 0 and 1, and x = 3 and 4.
Determine consecutive values of x between which each real zero of the function is located. Then draw the graph.
Answer:
There are zeros between x = –1 and 0, x = 0 and 1, and x = 3 and 4.
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Example 2-2c

35. Make a table of values and graph the function.
Graph Estimate the x-coordinates at which the relative maximum and relative minimum occur.
Make a table of values and graph the function. x
f (x) –2
–19 –1 5 1 2 –3 3 –4 4 30
zero at x = –1
indicates a relative maximum
zero between x = 1 and x = 2
indicates a relative minimum
zero between x = 3 and x = 4
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Example 2-3a

36. Answer: The value of f (x) at x = 0 is greater than the surrounding points, so it is a relative maximum. The value of f (x) at x = 3 is less than the surrounding points, so it is a relative minimum. x
f (x) –2
–19 –1 5 1 2 –3 3 –4 4 30
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Example 2-3b

37. Graph Estimate the x-coordinates at which the relative maximum and relative minimum occur.
Answer: The value of f (x) at x = 0 is less than the surrounding points, so it is a relative minimum. The value of f (x) at x = –2 is greater than the surrounding points, so it is a relative maximum.
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Example 2-3c

38. Health The weight w, in pounds, of a patient during a 7-week illness is modeled by the cubic equation where n is the number of weeks since the patient became ill.
Graph the equation.
Make a table of values for weeks 0 through 7. Plot the points and connect with a smooth curve.
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Example 2-4a

39. n
w(n)
110 1
109.5 2
108.4 3
107.3 4
106.8 5
107.5 6 7
114.9
Answer:
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Example 2-4b

40. Describe the turning points of the graph and its end behavior.
Answer: There is a relative minimum at week 4. For the end behavior, w (n) increases as n increases.
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Example 2-4c

41. What trends in the patient’s weight does the graph suggest?
Answer: The patient lost weight for each of 4 weeks after becoming ill. After 4 weeks, the patient started to gain weight and continues to gain weight.
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Example 2-4d

42. Weather The rainfall r, in inches per month, in a Midwestern town during a 7-month period is modeled by the cubic equation where m is the number of months after March 1.
a. Graph the equation.
Answer:
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Example 2-4e

43. b. Describe the turning. points of the graph. and its end behavior. c
b. Describe the turning points of the graph and its end behavior c. What trends in the amount of rainfall received by the town does the graph suggest?
Answer: There is a relative maximum at Month 2, or May. For the end behavior, r (m) decreases as m increases.
Answer: The rainfall increased for two months following March. After two months, the amount of rainfall decreased for the next five months and continues to decrease.
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Example 2-4f

44. Homework
See Syllabus 7.1 & 7.2
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