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

Factor the denominator \(x^2 + 3x - 10\). We need two numbers that multiply to \(-10\) and add to \(3\), which are \(5\) and \(-2\). So, \(x^2 + 3x - 10=(x + 5)(x - 2)\). Vertical asymptotes occur where the denominator is zero (and numerator is not zero). Set \((x + 5)(x - 2)=0\), so \(x=-5\) or \(x = 2\).

Step2: Find Horizontal Asymptote

For a rational function \(\frac{ax^n+...}{bx^m+...}\), if \(n = m\), the horizontal asymptote is \(y=\frac{a}{b}\). Here, numerator degree \(n = 2\), denominator degree \(m = 2\), \(a = 3\), \(b = 1\), so \(y=\frac{3}{1}=3\).

Answer:

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