Question

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

Answer 1

No intercepts. Yet, origin is a point on the graph.
Horizontal asymptote : `larr y = 3 rarr`
Vertical asymptote : `uarr x = +-5 darr`

Explanation:

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

y-intercept ( x = 0 ): 0

`y= 3/(1-25/x^2) to 3`, as x to +-3.#

By actual division,

`y=P+Q/R= 3+75/(x^2-25)`

The asymptotes are given by

`y = P=3 and R=x^2-25=0` that gives `x=+-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]}