Question

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

$R\left(x\right)=\frac{x^2+3x-10}{x^2+8x+15}$

Short answer

The graph is

Step-by-step solution

Given information

Given function$R\left(x\right)=\frac{x^2+3x-10}{x^2+8x+15}$

Factoring the numerator and denominator

Factoring, we get

$$\begin{gathered} R\left(x\right)=\frac{x^2+3x-10}{x^2+8x+15} \\ R\left(x\right)=\frac{x^2+5x-2x-10}{x^2+5x+3x+15} \\ R\left(x\right)=\frac{x(x+5)-2(x+5)}{x(x+5)+3(x+5)} \\ R\left(x\right)=\frac{(x-2)(x+5)}{(x+3)(x+5)} \\ R\left(x\right)=\frac{x-2}{x+3} \end{gathered}$$

Calculate the domain

Setting denominator to zero, we get

$$\begin{gathered} (x+3)=0 \\ x=-3 \end{gathered}$$

The domain is all real numbers except$-3$

Calculate the x intercept 

Setting numerator to zero, we get

$$\begin{gathered} (x-2)=0 \\ x=2 \end{gathered}$$

The$y$intercept is$R\left(0\right)=-\frac{2}{3}$

Calculating the vertical asymptote

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

Calculate the horizontal asymptote

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

$$\begin{gathered} y=\frac{a}{b} \\ y=\frac{1}{1} \\ y=1 \end{gathered}$$

$$\begin{gathered} R\left(x\right)=\frac{x^2+3x-10}{x^2+8x+15} \\ =1 \\ {x}^2+3x-10=x^2+8x+15 \\ -5x=25 \\ x=-5 \end{gathered}$$

The point of intersection is$(-5,1)$

Sketching the graph

The graph is