Question

what are the equations of the asymptotes of the graph of the function $f(x)=\frac{3x^2 - 2x - 1}{x^2 + 3x - 10}$?
\bigcirc $x = -5, x = 2$ and $y = 3$
\bigcirc $x = -2, x = 5$ and $y = 3$
\bigcirc $x = 3, y = -5$, and $y = 2$
\bigcirc $x = 3, y = -2$, and $y = 5$

Explanation:

Step1: Find vertical asymptotes

Set denominator \(x^{2}+3x - 10=0\). Factor: \((x + 5)(x-2)=0\). So \(x=-5\) and \(x = 2\) (vertical asymptotes).

Step2: Find horizontal asymptote

Since degree of numerator (\(n = 2\)) and denominator (\(m=2\)) are equal. For \(f(x)=\frac{ax^{n}+...}{bx^{m}+...}\), \(y=\frac{a}{b}\). Here \(a = 3\), \(b = 1\), so \(y = 3\).

Answer:

A. \(x=-5,x = 2\) and \(y = 3\)