# How do you graph y=(3x^2)/(x^2-25) using asymptotes, intercepts, end behavior?

Jan 25, 2017

No intercepts. Yet, origin is a point on the graph.
Horizontal asymptote : $\leftarrow y = 3 \rightarrow$
Vertical asymptote : $\uparrow x = \pm 5 \downarrow$

#### Explanation:

x-intercept ( y = 0 ) : 0, giving origin as a point on the graph.

y-intercept ( x = 0 ): 0

$y = \frac{3}{1 - \frac{25}{x} ^ 2} \to 3$, as x to +-3.#

By actual division,

$y = P + \frac{Q}{R} = 3 + \frac{75}{{x}^{2} - 25}$

The asymptotes are given by

$y = P = 3 \mathmr{and} R = {x}^{2} - 25 = 0$ that gives $x = \pm 5$

An asymptotes-inclusive Socratic graph is inserted.

graph{(3x^2/(x^2-25)-y)(y-3)(x-5+.01y)(x+5+.01y)=0 [-40, 40, -20, 20]}