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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

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

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.

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

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

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

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

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

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

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

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

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

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

a. Find Answer: Example 1-3e

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?

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.

 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

 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

 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

 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

 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

 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

End of Lesson 1

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

End of Lesson 2

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