# How do you graph y=(3x)/(2x-4) using asymptotes, intercepts, end behavior?

Dec 9, 2016

see explanation.

#### Explanation:

The denominator of y cannot be zero as this would make y 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: $2 x - 4 = 0 \Rightarrow x = 2 \text{ is the asymptote}$

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

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

as $x \to \pm \infty , y \to \frac{3}{2 - 0}$

$\Rightarrow y = \frac{3}{2} \text{ is the asymptote}$

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1) Hence there are no oblique asymptotes.

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

$x = 0 \Rightarrow y = \frac{0}{-} 4 = 0 \Rightarrow \left(0 , 0\right)$

$y = 0 \Rightarrow 3 x = 0 \Rightarrow \left(0 , 0\right)$

$\Rightarrow \text{There is only 1 intercept at the origin}$
graph{(3x)/(2x-4) [-10, 10, -5, 5]}