# How do you graph f(x)=(3x+8)/(x-2) using holes, vertical and horizontal asymptotes, x and y intercepts?

Jan 8, 2017

see explanation

#### Explanation:

$\textcolor{b l u e}{\text{Asymptotes}}$

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve : $x - 2 = 0 \Rightarrow x = 2 \text{ is the asymptote}$

Horizontal asymptotes occur as

${\lim}_{x \to \pm \infty} , f \left(x\right) \to c \text{ ( a constant)}$

divide terms on numerator/denominator by x

$f \left(x\right) = \frac{\frac{3 x}{x} + \frac{8}{x}}{\frac{x}{x} - \frac{2}{x}} = \frac{3 + \frac{8}{x}}{1 - \frac{2}{x}}$

as $x \to \pm \infty , f \left(x\right) \to \frac{3 + 0}{1 - 0}$

$\Rightarrow y = 3 \text{ is the asymptote}$

Holes occur when there is a duplicate factor on the numerator/denominator. This is not the case here, hence there are no holes.

$\textcolor{b l u e}{\text{Intercepts}}$

$x = 0 \to f \left(0\right) = \frac{8}{- 2} = - 4 \leftarrow \text{ y-intercept}$

$y = 0 \Rightarrow 3 x + 8 = 0 \Rightarrow x = - \frac{8}{3} \leftarrow \text{ x-intercept}$
graph{(3x+8)/(x-2) [-20, 20, -10, 10]}