Question

The rational function has a y-intercept of 7. What is the equation for this function?

Answer 1

Ummm... There are infinitely many. Can you narrow it down a bit?

Answer 2

this is the graph. ^

Answer 3

Do you recognize that the function is shifted by -2, 5 from the origin? Do you know the form of the function if centered on the origin?

Answer 4

It looks like this: \(y = \dfrac{ax+b}{cx+d}\)

y-intercept is 7, substitute x = 0

\(7 = \dfrac{b}{d} \implies b = 7d\)

Now, it looks like this: \(y = \dfrac{ax+7d}{cx+d}\)

Vertical Asymptote at x = -2, the denominator must be zero at x = 2

\(c(-2) + d = 0 \implies d = 2c\)

Now, it looks like this: \(y = \dfrac{ax+7(2c)}{cx+2c} = \dfrac{ax+14c}{c(x+2)}\)

Horizontal Asymptote at y = 5. This one is a little trickier.

\(a = 5c\)

Now, it looks like this: \(y = \dfrac{(5c)x+14c}{c(x+2)} = \dfrac{c(5x + 14)}{c(x+2)} = \dfrac{5x+14}{x+2}\).

I think it's done!

Answer 5

Thats really confusing. . . .

Answer 6

All at once, I agree. One step at a time and it's a beautiful thing. Think through each step. Don't try to do it all at the same time.

Answer 7

* Whoops! The denominator must be zero at x = -2.