Q9. - Algebra 1

Question

Identify the asymptotes of the function.

$\mathit{y}\mathbf{=}\frac{\mathbf{2}}{\mathbf{x}\mathbf{-}\mathbf{4}}$

Step-by-Step Solution

Answer

The vertical asymptote: $\mathit{x}\mathbf{=}\mathbf{4}$

The horizontal asymptote: $\mathit{y}\mathbf{=}\mathbf{0}$

Step 1. Define a rational function.

A function of the form $y=\frac{p}{q}$ is called a rational function, where $p$ and $q$ are the polynomials and $q\ne 0$.

Step 2. Define the asymptotes for the rational function in the form y = a x − b + c .

A rational function in the form $y=\frac{a}{x-b}+c,a\ne 0$, has a vertical asymptote at the $x$-value that makes the denominator equal zero, $x=b$ and it has a horizontal asymptote at $y=c$.

Step 3. Calculate the asymptotes for the function y = 2 x − 4 .

Observe the rational function $y=\frac{2}{x-4}$.

For vertical asymptote,

Denominator $=0$

$\begin{array}{l}x-4=0\\ x=4\end{array}$

So, $x=-2$ is the vertical asymptote.

For horizontal asymptote,

$y=c$

Since, $c=0$.

So, $y=0$ is the horizontal asymptote. 

Therefore,

The vertical asymptote: $x=4$

The horizontal asymptote: $y=0$