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Chapter 4: Q. 34 (page 234)

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

R(x)=x2+3x-10x2+8x+15

Short Answer

Expert verified

The graph is

Step by step solution

01

Given information

Given functionR(x)=x2+3x-10x2+8x+15

02

Factoring the numerator and denominator

Factoring, we get

R(x)=x2+3x-10x2+8x+15R(x)=x2+5x-2x-10x2+5x+3x+15R(x)=x(x+5)-2(x+5)x(x+5)+3(x+5)R(x)=(x-2)(x+5)(x+3)(x+5)R(x)=x-2x+3

03

Calculate the domain

Setting denominator to zero, we get

(x+3)=0x=-3

The domain is all real numbers except-3

04

Calculate the x intercept 

Setting numerator to zero, we get

(x-2)=0x=2

Theyintercept isR(0)=-23

05

Calculating the vertical asymptote

The zeroes of denominator is x=-3so the vertical asymptote isx=-3

06

Calculate the horizontal asymptote

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

y=aby=11y=1

R(x)=x2+3x-10x2+8x+15=1x2+3x-10=x2+8x+15-5x=25x=-5

The point of intersection is(-5,1)

07

Sketching the graph

The graph is

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