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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 4: Q. 36 (page 234)

In Problems 7-44, follow Steps 1 through 7 on page 228 to analyze the graph of each function.
$R (x) = \frac{8 x^{2} + 26 x + 15}{2 x^{2} - x - 15}$

Short Answer

Expert verified

The graph is

Step by step solution

01

Given information

Given function $R (x) = \frac{8 x^{2} + 26 x + 15}{2 x^{2} - x - 15}$

02

Factoring the numerator and denominator

Factoring, we get

$R (x) = \frac{8 x^{2} + 26 x + 15}{2 x^{2} - x - 15} R (x) = \frac{8 x^{2} + 30 x + 4 x + 15}{2 x^{2} - 6 x + 5 x - 15} R (x) = \frac{2 x (4 x + 15) + 1 (4 x + 15)}{2 x (x - 3) + 5 (x - 3)} R (x) = \frac{(2 x + 1) (4 x + 15)}{(2 x + 5) (x - 3)}$

03

Calculating the domain

Setting the denominator to zero, we get

$(2 x + 5) (x - 3) = 0 x = - \frac{5}{2}, 3$

The domain is all real numbers except $- \frac{5}{2}, 3$

04

Calculating the x intercept

Setting numerator to zero, we get

$(2 x + 1) (4 x + 15) = 0 x = - \frac{1}{2}, \frac{- 4}{15}$

The $y$ intercept is $R (0) = - 1$ i.e. $(0, - 1)$

05

Calculating the vertical asymptote

The zeroes of denominator is $x = - \frac{5}{2}, 3$ so the vertical asymptote is $x = - \frac{5}{2}, 3$

06

Calculating the horizontal asymptote

The degree of numerator is equal to denominator so the horizontal asymptote is

$y = \frac{a}{b} y = \frac{1}{1} y = 1$

$R (x) = \frac{8 x^{2} + 26 x + 15}{2 x^{2} - x - 15} = 4 8 x^{2} + 26 x + 15 = 4 (2 x^{2} - x - 15) 8 x^{2} + 26 x + 15 = 8 x^{2} - 8 x - 60 x = - \frac{15}{7}$

The intersection point is $(- \frac{15}{7}, 4)$

07

Sketching the graph

The graph is

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Most popular questions from this chapter

In Problems63鈥�72, find the real solutions of each equation.

$2 x^{3} - 11 x^{2} + 10 x + 8 = 0$

Solve the inequality $3 - 4 x > 5$ . Graph the solution set.

True or False : The graph of $f (x) = \frac{x}{x - 3}$ is above the x-axis for $x < 0$ or $x > 3$ , so the solution set of the inequality $\frac{x}{x - 3} 鈮� 0$ is ${x : x 鈮� 0 o r x 鈮� 3}$ ..

Find the domain of the rational function.

$F (x) = \frac{3 x (x - 1)}{2 x^{2} - 5 x - 3}$

Use the Factor Theorem to prove that $x - c$ is a factor of

$x^{n} - c^{n}$ for any positive integer n.

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