Explanation Given: The function: f ( x ) = − x 2 + x + 2 x − 1 . Formula used: To find the asymptotes,: Let f ( x ) = P ( x ) / Q ( x ) be a rational function with degree of Q ( x ) at least 1. (1) The vertical asymptote of the graph of f corresponds to each distinct root of Q ( x ) = 0 . (2) When the degrees of P ( x ) < Q ( x ) , then the x − axis is a horizontal asymptote. (3) When the degrees of P ( x ) = Q ( x ) , then the horizontal asymptote found by the ratio of the leading coefficients. (4) When the degree of P ( x ) > Q ( x ) , then rewrite the function as quotient+ remainder divisor . The horizontal/oblique asymptote graph is y equal to the quotient. To determine the intercepts, The x − intercepts are found by assigning zero to y in the equation and the point ( 0 , x ) is the x − intercept for the function. The y − intercepts are found by assigning zero to x in the equation and the point ( y , 0 ) is the y − intercept for the function. Calculation: For the given function, f ( x ) = − x 2 + x + 2 x − 1 , the vertical asymptotes are given by, x − 1 = 0 This implies, x = 1 is the vertical asymptote. As the power of the numerator is greater than that of the denominator, for the horizontal asymptote, proceed as follow, f ( x ) = − x 2 + x + 2 x − 1 1 − 1 1 2 − 1 0 − 1 0 2 f ( x ) = − x ( x − 1 ) + 2 x − 1 = − x ( x − 1 ) x − 1 + 2 x − 1 = − x + 2 x − 1 Then, y = − x is an asymptote. Therefore, y = − x is an oblique asymptote for the given function. Put, y = 0 , in the function, then x = − 1 , 2

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ISBN: 9780321916600

Author: Mark Dugopolski

Publisher: PEARSON

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Chapter 3, Problem 80RE

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To calculate: All the asymptotes, x− intercepts, y− intercepts for the graph of each rational function and sketch the graph of the function,f(x)=−x2+x+2x−1.

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Chapter 3, Problem 79RE

Chapter 3, Problem 81RE

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All the asymptotes, x − intercepts, y − intercepts for the graph of each rational function and sketch the graph of the function, f ( x ) = − x 2 + x + 2 x − 1 .

Solution Summary: The author explains how to calculate the asymptotes for each rational function.

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To calculate: All the asymptotes, x− intercepts, y− intercepts for the graph of each rational function and sketch the graph of the function,f(x)=−x2+x+2x−1.

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Chapter 3 Solutions