Graphs of Rational Functions

Question

Write a rational function that satisfies the given criteria.

Vertical asymptote:x=5

Slant asymptote:y=x + 3

Zero of the function:x=−5

f(x)=?

Answer 1

The vertical asymptote of x=5 means that x-5 is in the denominator

The zero x=-5 means that x+5 is in the numerator

So far, this gives us f(x) = (x+5)/(x-5) for our rational function. If the slant asymptote is y=x+3, then there is some polynomial P multiplying the rational function to get the slant asymptote disregarding the remainder:

P(x+5)/(x-5) = x+3

P(x+5) = x²-2x-15

P = (x²-2x-15)/(x+5)

Now, we can use synthetic division to divide the polynomials on the right-hand side:

-5 | 1 -2 -15

____ -5_35

1 -7 | 20

Hence, our polynomial P is x-7. We can then determine the rational function that meets all the criteria:

f(x) = (x-7)(x+5)/(x-5)

f(x) = (x²-2x-35)/(x-5)

Thus, the answer is f(x) = (x²-2x-35)/(x-5)

Hope this helped!

Answer 2

f(x) = (x² -2x -35) / (x - 5)