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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. 35 (page 234)

In Problems $7 - 44$ , follow Steps 1 through 7 on page 228 to analyze the graph of each function.
$R (x) = \frac{6 x^{2} - 7 x - 3}{2 x^{2} - 7 x + 6}$

Short Answer

Expert verified

The graph is

Step by step solution

01

Given information

Given function $R (x) = \frac{6 x^{2} - 7 x - 3}{2 x^{2} - 7 x + 6}$

02

Factoring the numerator and denominator

Factoring, we get

$R (x) = \frac{6 x^{2} - 7 x - 3}{2 x^{2} - 7 x + 6} R (x) = \frac{6 x^{2} - 9 x + 2 x - 3}{2 x^{2} - 3 x - 4 x + 6} R (x) = \frac{3 x (2 x - 3) + 1 (2 x - 3)}{x (2 x - 3) - 2 (2 x - 3)} R (x) = \frac{(3 x + 1) (2 x - 3)}{(x - 2) (2 x - 3)} R (x) = \frac{3 x - 1}{x - 2}$

03

Calculate the domain

Setting denominator to zero, we get

$(x - 2) = 0 x = 2$

The domain is all real numbers except $2$

04

Calculate the x intercept

Setting numerator to zero, we get

$(3 x + 1) = 0 x = - \frac{1}{3}$

The $y$ intercept is $R (0) = - \frac{1}{2}$

05

Calculate the vertical asymptote

The zeroes of denominator is $x = 2$ so the vertical asymptote is $x = 2$

06

Calculating the horizontal asymptote

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

$y = \frac{a}{b} y = \frac{6}{2} y = 3$

$R (x) = \frac{6 x^{2} - 7 x - 3}{2 x^{2} - 7 x + 6} = 3 6 x^{2} - 7 x - 3 = 3 (2 x^{2} - 7 x + 6) 6 x^{2} - 7 x - 3 = 6 x^{2} - 21 x + 18 14 x = 21 x = \frac{2}{3}$

The intersection point is $(\frac{2}{3}, 3)$

07

Sketching the graph

The graph is

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Most popular questions from this chapter

Find the real zeros of f. Use the real zeros to factor f.

$f (x) = x^{3} + 2 x^{2} - 5 x - 6$

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

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

In Problems 49鈥� 60, for each polynomial function:

(a) List each real zero and its multiplicity.

(b) Determine whether the graph crosses or touches the x-axis at each x-intercept.

(c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of $| x |$

$f (x) = 3 (x - 7) {(x + 3)}^{2}$

Find the real zeros of f. Use the real zeros to factor f.

$f (x) = 3 x^{3} - 7 x^{2} + 12 x - 28$

In Problems $7 - 44$ , follow Steps $1$ through 7 on page $228$ to analyze the graph of each function.

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

See all solutions

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