1. Name:__________ warm-up 5-4 State the degree and leading coefficient of –4x 5 + 2x 3 + 6. Find p(3) and p(–5) for p(x) = x 3 – 10x + 40. Determine whether the statement is sometimes, always, or never true. The graph of a polynomial of degree three will intersect the x-axis three times. Describe the end behavior of the graph of function f(x) = –x 2 + 4

3. Details of the Day EQ: How do polynomials functions model real world problems and their solutions? I will be able to… Graph polynomial functions and locate their zeros. Find the relative maxima and minima of polynomial functions. Activities: Warm-up Review homework Notes: Class work/ HW MP Exam – Thursday, November 13 All make-up work due Monday, November 1 no exceptions Vocabulary: Location Principle relative maximum relative minimum extrema turning points.

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5. A Quick Review State the degree and leading coefficient of –4x 5 + 2x 3 + 6. Find p(3) and p(–5) for p(x) = x 3 – 10x + 40. Determine whether the statement is sometimes, always, or never true. The graph of a polynomial of degree three will intersect the x-axis three times. Describe the end behavior of the graph of function f(x) = –x 2 + 4

6. Notes and examples State the degree and leading coefficient of 7z 3 – 4z 2 + z. If it is not a polynomial in one variable, explain why. State the degree and leading coefficient of 6a 3 – 4a 2 + ab 2. If it is not a polynomial in one variable, explain why State the degree and leading coefficient of 3x 5 + 2x 2 – 4 – 8x 6. If it is not a polynomial in one variable, explain why. Determine whether 3x 3 + 2x 2 – 3 is a polynomial in one variable. If so, state the degree and leading coefficient.

7. Notes and examples Graph f(x) = –x 3 – 4x 2 + 5 by making a table of values. Which graph is the graph of f(x) = x 3 + 2x 2 + 1?

8. Notes and examples

9. Determine consecutive values of x between which each real zero of the function f(x) = x 4 – x 3 – 4x 2 + 1 is located. Then draw the graph.

10. Notes and examples Look at the values of f(x) to locate the zeros. Then use the points to sketch the graph of the function. There are zeros between x = –2 and –1, x = –1 and 0, x = 0 and 1, and x = 2 and 3.

11. Notes and examples Graph f(x) = x 3 – 3x 2 + 5. Estimate the x- coordinates at which the relative maxima and relative minima occur.

12. Notes and examples You can use a graphing calculator to find the relative maximum and relative minimum of a function and confirm your estimate. Enter y = x 3 – 3x 2 + 5 in the Y= list and graph the function. Use the CALC menu to find each maximum and minimum. When selecting the left bound, move the cursor to the left of the maximum or minimum. When selecting the right bound, move the cursor to the right of the maximum or minimum. The estimates for a relative maximum near x = 0 and a relative minimum near x = 2 are accurate.

13. Notes and examples Consider the graph of f(x) = x 3 + 3x 2 + 2. Estimate the x-coordinates at which the relative maximum and relative minimum occur. A. relative minimum: x = 0 relative maximum: x = –2 B.relative minimum: x = –2 relative maximum: x = 0 C.relative minimum: x = –3 relative maximum: x = 1 D.relative minimum: x = 0 relative maximum: x = 2

14. Notes and examples A. HEALTH The weight w, in pounds, of a patient during a 7-week illness is modeled by the function w(n) = 0.1n 3 – 0.6n 2 + 110, where n is the number of weeks since the patient became ill. Graph the equation.

15. Notes and examples

16. Describe the turning points of the graph and its end behavior.

17. Notes and examples What trends in the patient’s weight does the graph suggest? Is it reasonable to assume the trend will continue indefinitely?

18. Notes and examples. WEATHER The rainfall r, in inches per month, in a Midwestern town during a 7- month period is modeled by the function r(m) = 0.01m 3 – 0.18m 2 + 0.67m + 3.23, where m is the number of months after March 1. Graph the equation. Describe the turning points of the graph and its end behavior WEATHER What trends in the amount of rainfall received by the town does the graph suggest? Is it reasonable to assume the trend will continue indefinitely?