Explanation 1) Approach: For draw the graph of the polynomial, Step 1: Find the x -and y -intercepts of the graph of the polynomial. Step 2: Determine the end behavior. Step 3: Make a sign chart to determine where the graph is above and below the x -axis. Step 4: Test for symmetry about y -axis, and the origin Step 5: Find an additional point and table the co-ordinates. Step 6: Draw the graph of the polynomial function. 2) Calculation: Given, f ( x ) = − x 4 + 2 x 3 + 8 x 2 Step 1: Find the x - and y -intercepts of the graph. To find the x -intercepts, we substitute 0 for f ( x ) and solve for x . 0 = − x 4 + 2 x 3 + 8 x 2 (i.e.) − x 4 + 2 x 3 + 8 x 2 = 0 Factorize x 4 + 2 x 3 + 8 x 2 as − x 2 ( x 2 − 2 x − 8 ) By the zero factor property, − x 2 = 0 and x 2 − 2 x − 8 = 0 Taking, − x 2 = 0 Multiply − 1 on both sides, x 2 = 0 Taking, x 2 − 2 x − 8 = 0 Factorize x 2 − 2 x − 8 = 0 ( x − 4 ) ( x + 2 ) = 0 By the zero factor property, x − 4 = 0 , x + 2 = 0 Taking x − 4 = 0 Add 4 on both sides, x − 4 + 4 = 0 + 4 x = 4 Taking x + 2 = 0 Add − 2 on both sides, x + 2 − 2 = 0 − 2 x = − 2 Thus, the x -intercepts are ( − 2 , 0 ) ( 0 , 0 ) and ( 4 , 0 ) To find the y -intercept, we substitute 0 for x and solve for y . Here, f ( x ) = y y = − x 4 + 2 x 3 + 8 x 2 y = − ( 0 ) 4 + 2 ( 0 ) 3 + 8 ( 0 ) 2 y = 0 + 0 + 0 y = 0 Thus, the y -intercept is ( 0 , 0 ) . Step 2: Determine the end behavior. The degree of the function f ( x ) = − x 4 + 2 x 3 + 8 x 2 is 4 which is even, and the leading co-efficient is − 1 which is negative. The graph falls on the left and falls on the right. Step 3: Make a sign chart to determine where the graph is above and below the x -axis. We will make a sign chart to determine the behavior of the function between the x -intercepts. The zero of the polynomial function divides the x -axis into four intervals. ( − ∞ , − 2 ) , ( − 2 , 0 ) , ( 0 , 4 ) and ( 4 , ∞ ) Table the test value of x from each interval and determine the sign. f ( x ) = − x 4 + 2 x 3 + 8 x 2 Interval Test point of f ( x ) Sign of f ( x ) Graph f ( x ) ( − ∞ , − 2 ) f ( − 3 ) = − ( − 3 ) 4 + 2 ( − 3 ) 3 + 8 ( − 3 ) 2 = − 81 + 2 ( − 27 ) + 8 ( 9 ) = − 81 − 54 + 72 f ( − 3 ) = − <

WebAssign with Corequisite Support for Gustafson/Hughes' College Algebra, 12th Edition [Instant Access], Single-Term

12th Edition

ISBN: 9780357422533

Author: Gustafson; R. David; Hughes; Jeff

Publisher: Cengage Learning US

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Chapter 4.2, Problem 64E

To determine

To graph:

The polynomial function

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Chapter 4.2, Problem 63E

Chapter 4.2, Problem 65E

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Ch. 4.1 - Find the vertex of the graph of the quadratic...Ch. 4.1 - Prob. 2SC Ch. 4.1 - Prob. 3SC Ch. 4.1 - Prob. 4SC Ch. 4.1 - Find the largest area possible if the association...Ch. 4.1 - A company that makes and sells baseball caps has...Ch. 4.1 - Fill in the blanks. A quadratic function is...Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E

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To graph: The polynomial function

Solution Summary: The author explains how the polynomial f(x) is graphed.