Question

How do you graph `f(x)=(3x+8)/(x-2)` using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

see explanation

Explanation:

`color(blue)"Asymptotes"`

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve : `x-2=0rArrx=2" is the asymptote"`

Horizontal asymptotes occur as

`lim_(xto+-oo),f(x)toc" ( a constant)"`

divide terms on numerator/denominator by x

`f(x)=((3x)/x+8/x)/(x/x-2/x)=(3+8/x)/(1-2/x)`

as `xto+-oo,f(x)to(3+0)/(1-0)`

`rArry=3" is the asymptote"`

Holes occur when there is a duplicate factor on the numerator/denominator. This is not the case here, hence there are no holes.

`color(blue)"Intercepts"`

`x=0tof(0)=8/(-2)=-4larr" y-intercept"`

`y=0rArr3x+8=0rArrx=-8/3larr" x-intercept"`
graph{(3x+8)/(x-2) [-20, 20, -10, 10]}