Question

Create a function that has a graph with features that include
-vertical retricemptote at y axis x=3
-horizontal retricemptote at y=2
-x intercept of (-1/2, 0) & (1,0)
QUESTION IS REGAURDING RATIONAL FUNCTIONS/QUADRATIC FUNCTIONS ...

Answer 1

asymptotes

Answer 2

lol

Answer 3

vertical at x = 3 means denominator should have a factor of
\[x-3\] in it

Answer 4

horizontal asymptote at
\[y=2\] means the degree of the numerator and denominator must be the same, and the ratio of leading coefficients has to be 2

Answer 5

so maybe
\[f(x)=\frac{2x}{x-3}\] would work

Answer 6

unfortunately it does not, because
\[f(1)=\frac{2}{-2}=-1\] and you want
\[f(1)=0\] so maybe try
\[f(x)=\frac{2x-2}{x-3}\]

Answer 7

now at least
\[f(1)=\frac{2-2}{1-3}=0\] how about
\[f(-\frac{1}{2})\]?

Answer 8

nope that doesn't work, i am a moron
numerator should be
\[(2x+1)(x-1)\]

Answer 9

that will give you the correct zeros,

Answer 10

so now maybe
\[f(x)=\frac{(2x+1)(x-1)}{(x-3)^2}\]

Answer 11

that one will have the correct zeros, and also the correct horizontal and vertical asymptote. i was forgetting that if you have two zeros you need a polynomial of degree 2 !