Book edition 6th

Author(s) Sullivan

Pages 1200 pages

ISBN 9780321795465

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Short Answer

Find a rational function that might have the given graph.

One possibility of the rational function is $R (x) = \frac{3 (x + 2) {(x - 1)}^{2}}{(x + 3) {(x - 4)}^{2}}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

Consider the given graph,

The numerator of a rational function $R (x) = \frac{p (x)}{q (x)}$

in lowest terms determines the x-intercepts of its graph.

The graph has x-intercept, localid="1646072236844" $- 2$ , as this touches the x-axis and is an even multiplicity and has x-intercept, localid="1646072278870" $1$ , as this crosses the x-axis and is an odd multiplicity.

So, one possibility for the numerator is given below,

localid="1646073763896" $p (x) = (x + 2) {(x - 1)}^{2}$ .

Step 2. Determine the vertical asymptotes.

Consider the given question,

The denominator of a rational function in lowest terms determines the vertical asymptotes of its graph.

The vertical asymptotes are $x = - 3, 4$ .

As $R (x)$ approaches $\infty$ to the left of $x = - 3$ and $R (x)$ approaches $- \infty$ to the right of localid="1646072581381" $x = - 3$ , then $x + 3$ is a factor of odd multiplicity of $q (x)$ .

Similarly, localid="1646073819637" $(x - 4)$ is also a factor of odd multiplicity in $q (x)$ .

So, one possibility for the denominator is given below,

$q (x) = {(x - 4)}^{2} (x + 3)$ .

Step 3. Form the rational function.

Consider the given question,

The horizontal asymptote of the given graph is $y = 3$ . Thus, the degree of the numerator must be equal to the degree of the denominator and that the quotient of the leading coefficients must be $3$ .

Hence, a possible rational function is given below,

localid="1646145213137" $R (x) = \frac{3 (x + 2) {(x - 1)}^{2}}{(x + 3) {(x - 4)}^{2}}$

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Precalculus Enhanced with Graphing Utilities

Answers without the blur.

Short Answer

Find a rational function that might have the given graph.

Step by Step Solution

Step 1. Given information.

Step 2. Determine the vertical asymptotes.

Step 3. Form the rational function.

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Precalculus Enhanced with Graphing Utilities

Answers without the blur.

Short Answer

Find a rational function that might have the given graph.

Step by Step Solution

Step 1. Given information.

Step 2. Determine the vertical asymptotes.

Step 3. Form the rational function.

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