Chapter 4.3, Problem 34AYU

In Problems 7-50, follow Steps 1 through 7 on page 234 to analyze the graph of each function.

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

To determine

To find: Analyze the given function.

Answer to Problem 34AYU

Explanation of Solution

Given:

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

Step 1: Factor $R$ to get

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

$R\left(x\right)=\frac{\left(x+5\right)\left(x-2\right)}{\left(x+5\right)\left(x+3\right)}$

That is $x=-5,-3$.

The domain $\left\{x\text{|}x\ne -5,x\ne -3\right\}$

Step 2: $R$ is in lowest terms $\frac{\left(x-2\right)}{\left(x+3\right)}$

Step 3: Locate the intercepts of the graph. Use multiplicity to determine the bahaviour of the graph of $R$ at each $x\text{-intercept}$.

$y\text{-intercept}$ $-\frac{2}{3}$.

$x\text{-intercept}$ is $2$.

Step 4: Locate the vertical asymptotes. Determine the behaviour of the graph on either side of each vertical asymptote.

$x=-3$, hole at $\left(-5,\frac{7}{2}\right)$.

Step 5: Locate the horizontal or oblique asymptote. Determine points, if any, at which the graph of $R$ intersects this asymptote.

Here horizontal asymptote $y=1$, not intersected.

Step 6: Graph $R$ using a graphing utility.

Step 7: