Q26. - Algebra 1

Question

Identify the asymptotes of each function.

$y=\frac{4}{x^2+3x-28}+2$

Step-by-Step Solution

Answer

The asymptotes of the given function are $x=4$, $x=-7$ and $y=2$.

Step 1. Define asymptotes.

A rational function which is in the form of $y=\frac{a}{x-b}+c$, $a\ne 0$ has a horizontal asymptote at $y=c$ and a vertical asymptote at $x=b$.

Step 2. Set a denominator equal to zero.

Set a denominator of a given expression equal to zero. 

$x^2+3x-28=0$

Step 3. Factor and simplify.

Factor the above equation and simplify further.

$$\begin{gathered} \left(x-4\right)\left(x+7\right)=0 \\ \left(x-4\right)=0\text{ or }\left(x+7\right)=0 \\ x=4x=-7 \end{gathered}$$

Therefore, the vertical asymptotes are $x=4$ and $x=-7$. Horizontal asymptote is $y=2$.