Question

Give an example of a rational function that has a horizontal asymptote at y = 1 and a vertical asymptote at
x = 4.

Answer 1

what a great question
what must the denominator look like if the vertical asymptote is \(x=4\) ?

Answer 2

(x-4)?

Answer 3

exactly

Answer 4

and horizonal 1/x?

Answer 5

well, it could also be \(2x-8\) but why not keep it simple and write what you did

Answer 6

so you know it looks like
\[\frac{something}{x-4}\] all you need is the "something"

Answer 7

the horizontal asymptote is \(y=1\) and the degree of the denominator is \(1\) so what must the degree of the numerator be?

Answer 8

x-1?

Answer 9

that would work, yes, but maybe not for the reason you think

Answer 10

the degree of the numerator must be the same as the degree of the denominator, and the degree of the denominator is 1, so it could be
\[\frac{x}{x-4}\]

Answer 11

or it could be
\[\frac{x-1}{x-4}\]if you like

Answer 12

and that would answer both questions?

Answer 13

either way
the degree of the numerator must be 1, and the leading coefficient must also be 1
so you can use any example you like, so long as the denominator is \(x-4\) and the numerator starts with \(x\)

Answer 14

your example would work fine
so would
\[\frac{x+5}{x-4}\] anything reallyl

Answer 15

alright thanks again!