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 ) = − 4 x 3 − 4 x 2 − x + 1 Step 1: Find the x - and y -intercepts of the graph. To find the x -intercept, we substitute 0 for f ( x ) and solve for x . 0 = 4 x 3 − 4 x 2 − x + 1 4 x 2 ( x − 1 ) − 1 ( x − 1 ) = 0 ( x − 1 ) ( 4 x 2 − 1 ) = 0 By the zero property, x − 1 = 0 and 4 x 2 − 1 = 0 Taking, x − 1 = 0 Add 1 on both sides, x − 1 + 1 = 0 + 1 x = 1 Taking, 4 x 2 − 1 + 1 = 0 + 1 4 x 2 = 1 Divide 4 on both sides, 4 x 2 4 = 1 4 x 2 = 1 4 By the square root property, x = ± 1 4 = 1 2 x = ± 0.5 x = 0.5 and x = − 0.5 Thus, the x -intercepts are ( 1 , 0 ) , ( 0.5 , 0 ) and ( − 0.5 , 0 ) . To find the y -intercepts, we substitute 0 for x and solve for y . Here, f ( x ) = y y = 4 x 3 − 4 x 2 − x + 1 (i.e.) y = 4 ( 0 ) 3 − 4 ( 0 ) 2 − ( 0 ) + 1 y = 4 ( 0 ) − 4 ( 0 ) − 0 + 1 y = 0 − 0 − 0 + 1 y = 1 Thus, the y -intercept is ( 0 , 1 ) . Step 2: Determine the end behavior. The degree of the function f ( x ) = 4 x 3 − 4 x 2 − x + 1 is 3, which is odd, and the leading co-efficient is 4, which is positive. Thus, the graph falls on the left and rises 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 zeros of the polynomial function divide the x - axis into four intervals. ( − ∞ , − 0.5 ) , ( − 0.5 , 0 ) , ( 0 , 0 .5 ) , and ( 0.5 , ∞ ) . Table the test value of x from each interval and determine the sign. f ( x ) = 4 x 2 − 4 x 2 − x + 1 Interval Test point of f ( x ) Sign of f ( x ) Graph of f ( x ) ( − ∞ , − 0.5 ) f ( − 1 ) = 4 ( − 1 ) 3 − 4 ( − 1 ) 2 − ( − 1 ) + 1 = 4 ( − 1 ) − 4 ( 1 ) + 1 + 1 = − 4 − 4 + 1 + 1 = − 8 + 2 f ( − 1 ) = − 6 − below the x -axis ( − 0.5 , 0 ) f ( − 0.1 ) = 4 ( − 0.1 ) 3 − 4 ( − 0.1 ) 2 − ( − 0.1 ) + 1 f ( − 0.1 ) = 4 ( − 0.001 ) − 4 ( 0.01 ) + 0.1 + 1 f ( − 0.1 ) = − 0.004 − 0.04 + 0.1 + 1 f ( − 0.1 ) = 1.02 + above the x - axis ( 0 , 0.5 ) f ( 0.1 ) = 4 ( 0.1 ) 3 − 4 ( 0.1 ) 2 − ( 0.1 ) + 1 = 4 ( 0.001 ) − 4 ( 0

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ISBN: 9781305878747

Author: Gustafson

Publisher: CENGAGE L

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Chapter 4.2, Problem 58E

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Chapter 4.2, Problem 57E

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To graph: The polynomial function.

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