Question

How do you graph `y=(3x)/(2x-4)` using asymptotes, intercepts, end behavior?

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 value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve: `2x-4=0rArrx=2" is the asymptote"`

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

`y=((3x)/x)/((2x)/x-4/x)=3/(2-4/x)`

as `xto+-oo,yto3/(2-0)`

`rArry=3/2" is the asymptote"`

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1) Hence there are no oblique asymptotes.

`color(blue)"Intercepts"`

`x=0rArry=0/-4=0rArr(0,0)`

`y=0rArr3x=0rArr(0,0)`

`rArr"There is only 1 intercept at the origin"`
graph{(3x)/(2x-4) [-10, 10, -5, 5]}