Find an equation of a rational function

[DDRchick]

Find an equation of a rational function f(x)that satisfies the conditions. We're allowed to use a calculator, by the way. (:
Okay, conditions:
vertical asymptote: x=-5, x=0
Horizontal asymptote: y=0
x-intercept=7; f(1)=4




On the test i had no idea how to do it, but after seeing her key,I somewhat understand. I looked in the textbook and was able to see where many things came from.
a(x-7)/(x+5)(x)
for one, i have no idea where the a and the x came from. I do, however, understand that the (x-7) is from the x-int. and that the (x+5) is from the v.a.
4=a(1-7)/(1+5)(x) again, no idea where the a and x came from. I see that 4=f(1) which means that y=4 and x=1.
4=a(-6)/(6)(1)
I understand she plugged in that last x finally with a 1 and simplified everything else.
4=-a, a=-4
f(x)=-4(x-7)/x(x+5)
I understand the final equation. I'd just love to know were she got that a and the x (by iself on the denominator) from! Thanks (:

 

[LCKurtz]

The vertical asymptotes of x = -5 and 0 gave the factors of (x+5) and (x-0) in the denominator. The x intercept of 7 gives the (x-7) in the numerator.

That's where the fraction $\frac{x-7}{(x+5)(x)}$ comes from. The second degree in the denominator vs. first degree in the numberator gives the horizontal asymptote of 0 for free. You have one condition left and only need to note that multiplying the fraction by a constant a doesn't change any of the above features and let's you get the last constraint.

 

[DDRchick]

Ohhh okay thanks so much! :]