Write an equation for the rational function, 𝑓, which has the following characteristics: 𝑓 has: a vertical asymptote of 𝑥 = 4 : a horizontal asymptote of 𝑦 = 0 a y-intercept at −1 no x – intercept

Question

Can I please get some help with this question with a step by step

Answer 1

For a vertical asymptote at x = 4, (x-4) must be a factor of the denominator.

For a horizontal asymptote at y = 0, the degree of the denominator should be greater than the degree of the numerator.

Since the graph cannot cross the x-axis, we probably have to square the factor in the denominator to ensure that all x values (other than 4) generate either positive or negative y values. We need the graph to lie below the x-axis in order to have a negative y-intercept, so put a negative constant in the numerator. The question is what should that constant be?

So far we have:

c/(x-4)². When x = 0 we want a y-value of -1 (y-intercept).

So, c/(0-4)²= -1

c=-16.

This should do it:

f(x) = -16/(x-4)²

Check it out:

desmos.com/calculator/hjjj4urfpv

Answer 2

4/(x-4)

Vertical asymptote - division by 0 when x=4

Horizontal asymptote - the power of the top should be less then the power of the bottom

Y-intercept - when you plugin x-0

No x-intercept - top is never = 0