Explanation Given: The polynomial function f ( x ) = 2 ( x + 2 ) 2 ( x − 3 ) . Procedure used: Graphing a Polynomial Function: Step 1: Determine the graph’s end behavior using the Leading Coefficient Test as follows. As x increases or decreases without bound, the graph of the polynomial function eventually rises or falls. In particular, (a) For n odd: If the leading coefficient is positive, the graph falls to the left and rises to the right. If the leading coefficient is negative, the graph rises to the left and falls to the right (b) For n even: If the leading coefficient is positive, the graph rises to the left and rises to the right. If the leading coefficient is negative, the graph falls to the left and falls to the right. Step 2: Find x -intercepts by setting f ( x ) = 0 and solving the resulting polynomial equation. If there is an x -intercept at r as a result of ( x − r ) k in the complete factorization of f ( x ), then (a). If k is even, the graph touches the x -axis at r and turns around. (b). If k is odd, the graph crosses the x -axis at r . (c). If k > 1 , the graph flattens out near ( r , 0 ) . Step 3: Find the y -intercept by computing f ( 0 ) . Step 4: Use symmetry, if applicable, to help draw the graph: (a). y -axis symmetry: f ( − x ) = f ( x ) . (b). Origin symmetry: f ( − x ) = − f ( x ) . Step 5: Use the fact that the maximum number of turning points of the graph is n − 1 , where n is the degree of the polynomial function, to check whether it is drawn correctly. Step 1: Obtain the sum of the degree of factors in order to determine the degree of the function. The degree of the factors are observed to be 2 and 1. The sum of the above degree is computed as, 2 + 1 = 3 Thus, the degree of the given function is 3, which is odd. The graph of odd polynomial function have opposite behavior at each end. The number a n , which is the coefficient of the variable with the highest power is called the leading coefficient. The coefficient of x 3 is noticed to be 2. That is, the leading coefficient is positive. By above procedure, the graph of f ( x ) = 2 ( x + 2 ) 2 ( x − 3 ) falls to the left end and rises to the right end. Step 2: The x -intercept is obtained as follows. Substitute f ( x ) = 0 in f ( x ) = 2 ( x + 2 ) 2 ( x − 3 ) . 2 ( x + 2 ) 2 ( x − 3 ) = 0 ( x + 2 ) 2 ( x − 3 ) = 0 ( x + 2 ) 2 = 0 or x − 3 = 0 x = − 2 or x = 3 Thus, the x -intercept of the given function are x = − 2 and x = 3 . It is observed that the multiplicity of x = − 2 and x = 3 are observed to be 2 and 1, respectively. Since the multiplicity of x = − 2 is even, the graph touches the x -axis at −2 and turns around. Similarly, the graph will cross the x -axis at 3 as the multiplicity of x = 3 is odd. Step 3: Obtain the y -intercept as follows. Substitute x = 0 in f ( x ) = 2 ( x + 2 ) 2 ( x − 3 )

College Algebra Essentials (5th Edition)

5th Edition

ISBN: 9780134469294

Author: Blitzer, Robert F.

Publisher: PEARSON

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Chapter 3.2, Problem 10CP

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To sketch: The graph of the polynomial function f(x)=2(x+2)2(x−3) using five-step strategy.

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Chapter 3.2, Problem 9CP

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College Algebra Essentials (5th Edition)

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The graph of the polynomial function f ( x ) = 2 ( x + 2 ) 2 ( x − 3 ) using five-step strategy.

Solution Summary: The author illustrates the graph of the polynomial function using the Leading Coefficient Test.

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To sketch: The graph of the polynomial function f(x)=2(x+2)2(x−3) using five-step strategy.

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