Explanation Procedure: “To graph a rational function f ( x ) = p ( x ) q ( x ) , where p ( x ) and q ( x ) have no common factors other than constants: 1. Find any real zeros of the denominator. Determine the domain of the function and sketch any vertical asymptotes. 2. Find the horizontal asymptote or the oblique asymptote, if there is one, and sketch it. 3. Find any zeros of the function. The zeros are found by determining the zeros of the numerator. These are the first coordinates of the x -intercepts of the graph. 4. Find f ( 0 ) . This gives the y - intercept, ( 0 , f ( 0 ) ) , of the function. 5. Find other function values to determine the general shape. Then draw the graph.” Definition used: “For a rational function f ( x ) = p ( x ) q ( x ) , where p ( x ) and q ( x ) are polinomials with no common factors other than constants, if a is a zero of the denominator, then the line x = a is a vertical asymptote for the graph of the function.” “When the numerator and the denominator of a rational function have the same degree, the line y = a / b is the horizontal asymptote, where a and b are the leading coefficients of the numerator and the denominator, respectively. When the degree of the numerator of a rational function is less than the degree of the denominator, the x -axis, or y = 0 , is the horizontal asymptote. When the degree of the numerator of a rational function is greater than the degree of the denominator, there is no horizontal asymptote.” Calculation: The given function is f ( x ) = x 2 + 2 x − 3 x 2 + 4 x + 3 . The equation f ( x ) = x 2 + 2 x − 3 x 2 + 4 x + 3 can be written as, f ( x ) = x 2 + 2 x − 3 x 2 + 4 x + 3 = x 2 + 3 x − x − 3 x 2 + 3 x + x + 3 = ( x + 3 ) ( x − 1 ) ( x + 3 ) ( x + 1 ) The numerator and denominator of the equation f ( x ) = x 2 + 2 x − 3 x 2 + 4 x + 3 have common factor ( x + 3 ) To determine To check: The graph f ( x ) = x 2 + 2 x − 3 x 2 + 4 x + 3 obtained in part (a), by using a graphing calculator.

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ISBN: 2810000027802

Author: BITTINGER

Publisher: PEARSON

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Chapter 4.5, Problem 49E

To determine

To draw: The hand drawn graph of the function f(x)=x2+2x−3x2+4x+3 with all the asymptotes and list the domain, x-intercepts and y-intercepts.

To determine

To check: The graph f(x)=x2+2x−3x2+4x+3 obtained in part (a), by using a graphing calculator.

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Chapter 4.5, Problem 48E

Chapter 4.5, Problem 50E

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Chapter 4 Solutions

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To draw: The hand drawn graph of the function f ( x ) = x 2 + 2 x − 3 x 2 + 4 x + 3 with all the asymptotes and list the domain, x -intercepts and y -intercepts.

Solution Summary: The author explains how to draw the graph of a rational function.

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To draw: The hand drawn graph of the function f(x)=x2+2x−3x2+4x+3 with all the asymptotes and list the domain, x-intercepts and y-intercepts.

To check: The graph f(x)=x2+2x−3x2+4x+3 obtained in part (a), by using a graphing calculator.

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Chapter 4 Solutions