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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. 32 (page 234)

In Problems 7-44, follow Steps 1 through 7 on page 228 to analyze the graph of each function.
$R (x) = \frac{(x - 1) (x + 2) (x - 3)}{x (x - 4)^{2}}$

Short Answer

Expert verified

The graph is

Step by step solution

01

Given information

Given function $R (x) = \frac{(x - 1) (x + 2) (x - 3)}{x (x - 4)^{2}}$

02

Calculate the domain

Setting denominator to zero, we get

$x {(x - 4)}^{2} = 0 x = 0, 4$

The domain is all real numbers except $0, 4$

03

Calculate the x intercept

Setting numerator to zero, we get

$(x - 1) (x + 2) (x - 3) = 0 x = 1, - 2, 3$

There is no $y$ intercept

04

Calculate the vertical asymptote

The zeroes of denominator is $x = 0, 4$ so the vertical asymptote is $x = 0, 4$

05

Calculate the horizontal asymptote

The degree of numerator is same as denominator so the horizontal asymptote is

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

$R (x) = \frac{x^{3} - 2 x^{2} - 5 x + 16}{x^{3} - 8 x^{2} + 16} = 1 x^{3} - 2 x^{2} - 5 x + 16 = x^{3} - 8 x^{2} + 16 6 x^{2} - 5 x = 0 x = \frac{5}{6}$

It intersects at $(\frac{5}{6}, 1)$

06

Sketching the graph

The graph is

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Most popular questions from this chapter

Find the domain of the rational function.

$\frac{- 4 x^{2}}{(x - 2) (x + 4)}$

Find the real zeros off. Use the real zeros to factorf. $f (x) = 2 x^{4} + 17 x^{3} + 35 x^{2} - 9 x - 45$

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

Graph each rational function using transformations.

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

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function.

$R (x) = \frac{8 x^{2} + 26 x - 7}{4 x - 1}$

See all solutions

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