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