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

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

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

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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
so maybe \[f(x)=\frac{2x}{x-3}\] would work
unfortunately it does not, because \[f(1)=\frac{2}{-2}=-1\] and you want \[f(1)=0\] so maybe try \[f(x)=\frac{2x-2}{x-3}\]
now at least \[f(1)=\frac{2-2}{1-3}=0\] how about \[f(-\frac{1}{2})\]?
nope that doesn't work, i am a moron numerator should be \[(2x+1)(x-1)\]
that will give you the correct zeros,
so now maybe \[f(x)=\frac{(2x+1)(x-1)}{(x-3)^2}\]
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 !

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