Answer:
[tex]f(x)=-\dfrac{14(x-3)(x-6)}{9(x+1)(x-4)}[/tex]
Step-by-step explanation:
The vertical asymptotes will be found where the denominator is zero. The x-intercepts will be found where the numerator is zero. The y-intercept can be made what you like using a suitable vertical scale factor.
Denominator factors: (x+1), (x-4) . . . . make the vertical asymptotes
Numerator factors: (x-6), (x-3) . . . . make the x-intercepts.
Now, our function is ...
f(x) = a(x -6)(x -3)/((x +1)(x -4))
The y-intercept for this is ...
f(0) = a(-6)(-3)/(1(-4)) = a(-18/4) = -9a/2
We want to choose "a" so this is 7:
7 = -9a/2
-14/9 = a . . . . multiply by -2/9
The rational function you want is ...
[tex]\boxed{f(x)=-\dfrac{14(x-3)(x-6)}{9(x+1)(x-4)}}[/tex]
