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Example 1 GRAPHING FUNCTIONS OF THE FORM (x) = ax n Solution Choose several values for x, and find the corresponding values of (x), or y. a. Graph the function. x (x) – 2– 2– 8– 8 – 1– 1– 1–

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If the zero has even multiplicity, the graph is tangent to the x-axis at the corresponding x-intercept (that is, it touches but does not cross the x-axis there).

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If the zero has odd multiplicity greater than one, the graph crosses the x-axis and is tangent to the x-axis at the corresponding x-intercept. This causes a change in concavity, or shape, at the x-intercept and the graph wiggles there.

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Turning Points and End Behavior The previous graphs show that polynomial functions often have turning points where the function changes from increasing to decreasing or from decreasing to increasing.

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Turning Points A polynomial function of degree n has at most n – 1 turning points, with at least one turning point between each pair of successive zeros.

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End Behavior The end behavior of a polynomial graph is determined by the dominating term, that is, the term of greatest degree. A polynomial of the form has the same end behavior as.

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End Behavior of Polynomials Suppose that ax n is the dominating term of a polynomial function of odd degree. 1.If a > 0, then as and as Therefore, the end behavior of the graph is of the type that looks like the figure shown here. We symbolize it as.

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End Behavior of Polynomials Suppose that ax n is the dominating term of a polynomial function of odd degree. 2. If a < 0, then as and as Therefore, the end behavior of the graph looks like the graph shown here. We symbolize it as.

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End Behavior of Polynomials Suppose that ax n is the dominating term of a polynomial function of even degree. 1.If a > 0, then as Therefore, the end behavior of the graph looks like the graph shown here. We symbolize it as.

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End Behavior of Polynomials Suppose that is the dominating term of a polynomial function of even degree. 2. If a < 0, then as Therefore, the end behavior of the graph looks like the graph shown here. We symbolize it as.

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Example 3 DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL Match each function with its graph. Solution Because is of even degree with positive leading coefficient, its graph is C. A.B.C. D.

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Example 3 DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL Match each function with its graph. Solution Because g is of even degree with negative leading coefficient, its graph is A. A.B.C. D.

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Example 3 DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL Match each function with its graph. Solution Because function h has odd degree and the dominating term is positive, its graph is in B. A.B.C. D.

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Example 3 DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL Match each function with its graph. Solution Because function k has odd degree and a negative dominating term, its graph is in D. A.B.C. D.

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Graphing Techniques We have discussed several characteristics of the graphs of polynomial functions that are useful for graphing the function by hand. A comprehensive graph of a polynomial function will show the following characteristics: 1. all x-intercepts (zeros) 2. the y-intercept 3. the sign of (x) within the intervals formed by the x- intercepts, and all turning points 4. enough of the domain to show the end behavior. In Example 4, we sketch the graph of a polynomial function by hand. While there are several ways to approach this, here are some guidelines.

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