Question

Find a rational function that satisfies the given conditions.

Vertical asymptotes x=-2,x=7

Horizontal asymptote y=7/2

​x-intercept ​(−5​, ​0)

Answer 1

vertical asymptotes: 1/((x+2)(x-7))
root at x = -5:

(x+5)/((x+2)(x-7))

horizontal asymptote. we need the degree to be equal, so try

7x(x+5)
--------------
2(x+2)(x-7)

The problem here is that now we have another x-intercept at (0,0)

So, let's work with

7(x+5)^2
--------------
2(x+2)(x-7)

That gives us an intercept, but it does not cross the x-axis.

We could go with something like

7(x^2+1)(x+5)
--------------
2(x+2)(x-7)^2

That gives us the same degree top and bottom, a single crossing at -5, and still the two vertical asymptotes.