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

1 Answer
Jul 9, 2017

Answer:

#"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]}