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2. Lesson 7-1 Polynomial Functions
Lesson 7-2 Graphing Polynomial Functions
Lesson 7-3 Solving Equations Using Quadratic Techniques
Lesson 7-4 The Remainder and Factor Theorems
Lesson 7-5 Roots and Zeros
Lesson 7-6 Rational Zero Theorem
Lesson 7-7 Operations on Functions
Lesson 7-8 Inverse Functions and Relations
Lesson 7-9 Square Root Functions and Inequalities
Contents

3. Example 1 Find Degrees and Leading Coefficients
Example 2 Evaluate a Polynomial Function
Example 3 Functional Values of Variables
Example 4 Graphs of Polynomial Functions
Lesson 1 Contents

4. Vocabulary
The leading coefficient of an expression is the coefficient (number) in front of the term with the highest degree.
The degree of a polynomial is the largest sum of exponents in a given term.

5. 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.
Example 1-1a

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 not a polynomial in one variable. It contains two variables, a and b.
Example 1-1b

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. The term 2c–1 is not of the form ancn, where n is a nonnegative integer.
Example 1-1c

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.
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.
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
Example 1-1f

11. Find the values of f (4), f (5), and f (6).
Original function
Replace r with 4.
Simplify.
Example 1-2a

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

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

14. 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
Example 1-3c

15. a. Find
Answer:
Example 1-3e

16. Odd/Even Degree Graph the equation y = x squared What is the degree?
Is the degree an odd or even number?
What does the graph do as x approaches infinity?
What does the graph do as x approaches negative infinity?
Does the graph do the same thing on both sides?

17. Even Degree
* When a function has an even degree the function does the same thing as x approaches infinity and negative infinity.
Odd Degree
* When the function does the opposite on one side as it does on the other.

18.  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. .
Example 1-4a

19.  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. .
Example 1-4b

20.  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. .
Example 1-4c

21.  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. .
Example 1-4d

22.  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. .
Example 1-4e

23.  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. .
Example 1-4f

24. End of Lesson 1

25. 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
Lesson 2 Contents

26. Sign Changes
If the value of the function goes from a negative value to a positive value, or the other way around that means it must hav e crossed the x-axis showing a zero exists between those two x-values.

27. 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:
Example 2-1a

28. 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:
Example 2-1b

29. 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:
Example 2-1c

30. 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
Example 2-2a

31. 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.
Example 2-2b

32. 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.
Example 2-2c

33. 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
Example 2-3a

34. 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
Example 2-3b

35. 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.
Example 2-3c

36. 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.
Example 2-4a

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

38. 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.
Example 2-4c

39. 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.
Example 2-4d

40. 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:
Example 2-4e

41. 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.
Example 2-4f

42. End of Lesson 2

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47. End of Slide Show