Explanation Procedure used: The graph of a polynomial, follow the below steps. 1. Use the leading term test to determine the end behavior. 2. Find the zeros of the function by solving f ( x ) = 0 . Any real zeros are the first coordinates of the x -intercepts. 3. Use the x -intercepts (zeros) to divide the x -axis into the intervals, and choose a test point in each interval to determine the sign of all function values in that interval. 4. Find f ( 0 ) . This gives the y -intercept of the function. 5. If necessary, find additional function values to determine the general shape of the graph and then draw the graph. 6. As a partial check, use the facts that the graph has at most n x -intercepts and at most n − 1 turning points. Multiplicity of zeros can also be considered in order to check where the graph crosses or is tangent to the x -axis. Calculation: The given polynomial function is f ( x ) = − x 3 − 2 x 2 . The leading term in the given polynomial is − x 3 . So, the degree of the polynomial is 3, which is an odd number. The coefficient of the leading term is − 1 , which is a negative number. So, it is observed that the value of f ( x ) → ∞ as x → − ∞ and f ( x ) → − ∞ as x → ∞ . Obtain the zeros of the function by solving f ( x ) = 0 . Any real zeros are the first coordinates of the x -intercepts. f ( x ) = 0 − x 3 − 2 x 2 = 0 − x 2 ( x + 2 ) = 0 − x 2 = 0 or x + 2 = 0 x = 0 or x = − 2 That is, the zeros of the function are 0 and − 2 . Thus, the x -intercepts will be ( 0 , 0 ) and ( − 2 , 0 ) . Use the x -intercepts to divide the x axis into 3 intervals, ( − ∞ , − 2 ) , ( − 2 , 0 ) and ( 0 , ∞ ) . Now, choose some points for x from each of these intervals to obtain the sign of the function. Tabulate the values and the signs as shown below in Table 1. Interval ( − ∞ , − 2 ) ( − 2 , 0 ) ( 0 , ∞ ) Test value, x − 4 − 1 1 Function value, f ( x ) f ( − 4 ) = [ − ( − 4 ) 3 − 2 ( − 4 ) 2 ] = 64 − 32 = 32 f ( − 1 ) = [ − ( − 1 ) 3

College Algebra: Graphs and Models (6th Edition)

6th Edition

ISBN: 9780134179032

Author: Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna

Publisher: PEARSON

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Chapter 4.2, Problem 13E

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To graph: The polynomial function f(x)=−x3−2x2.

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Chapter 4.2, Problem 12E

Chapter 4.2, Problem 14E

Chapter 4 Solutions

College Algebra: Graphs and Models (6th Edition)

Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 2E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 10E

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To graph: The polynomial function f ( x ) = − x 3 − 2 x 2 .

Solution Summary: The author explains how to graph a polynomial by using the leading term test to determine the end behavior and finding the zeros of the function.

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To graph: The polynomial function f(x)=−x3−2x2.

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