How do you graph y= x/(x-4)?

1 Answer
Gió
Sep 17, 2015

Have a look:

Explanation:

Let us study our function in steps:

1) Domain (=x values allowed)
we need the denominator different from zero:
x-4!=0
x!=4
x=4 will be a vertical asymptote.

2) Intercepts:
set: x=0 then y=0
set: y=0 then x=0

3) Limits:
lim_(x->4^+)y=+oo
lim_(x->4^-)y=-oo
lim_(x->+oo)y=1
lim_(x->+oo)y=1
y=1 will be a horizontal asymptote.

4) Derivatives:
FIRST: y'=(x-4-x)/(x-4)^2=-4/(x-4)^2 (Quotient Rule)
set: y'=0 then -4=0 never so we have NO max or min.
SECOND: y''=(-2(-4)(x-4))/(x-4)^4=8/(x-4)^3
set y''>0
you get:
8>0 always
(x-4)^3>0 when x>4
graphically:
enter image source here

graph{x/(x-4) [-10, 10, -5, 5]}

Related questions
See all questions in Graphs of Rational Functions
Impact of this question
2294 views around the world
You can reuse this answer
Creative Commons License