Question

How do you find the domain, VA, HA, Zeros and intercepts of `y=(x^2-16)/(x^2-4)`?

Answer 1

`"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 values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.

`"solve " x^2-4=0to(x-2)(x+2)=0`

`rArrx=+-2" are the asymptotes"`

`rArr"domain is " x inRR,x!=+-2`

`"horizontal asymptotes occur as"`

`lim_(xto+-oo),ytoc" (a constant )"`

divide terms on numerator/denominator by the highest power of x, that is `x^2`

`y=(x^2/x^2-16/x^2)/(x^2/x^2-4/x^2)=(1-16/x^2)/(1-4/x^2)`

as `xto+-oo,yto(1-0)/(1-0)`

`rArry=1" is the asymptote"`

`color(blue)"Intercepts"`

`x=0toy=(-16)/(-4)=4larrcolor(red)" y-intercept"`

`y=0tox^2-16=0to(x-4)(x+4)=0`

`rArrx=+-4larrcolor(red)" x-intercepts"`

`"which, of course are also the zeros"`
graph{(x^2-16)/(x^2-4) [-10, 10, -5, 5]}