1. ## Rational Function Formula

The graph of y = f(x) has one vertical asymptote, at x= -7 , and a horizontal asymptote at y = 21 . The graph of f crosses the x-axis once, at x = 6 , and the y-axis at y = -18 . Find the simplest possible formula for the rational function.

2. Ok, we have 1 horizontal asymptote and 1 vertical asymptote, so this is a rectangular hyperbola, in the form:

$(y - b) = \frac{c}{x - a}$

The asymptotes, and thus the function, have shifted up 21 (=b) and shifted left -7 (=a).

Thus,
$
(y - 21) = \frac{c}{x + 7}$

To find c, simply substitute in a known coordinate: (6,0) or (0,-18).
Substituting in (6,0), we get

$
(0 - 21) = \frac{c}{6 + 7}$

$

-21 = \frac{c}{13}$

$c = -21 \times 13 = -273$

Thus the equation is:
$

(y - 21) = \frac{-273}{x + 7}$

3. ## Very good.....

Originally Posted by nzmathman
Ok, we have 1 horizontal asymptote and 1 vertical asymptote, so this is a rectangular hyperbola, in the form:

$(y - b) = \frac{c}{x - a}$

The asymptotes, and thus the function, have shifted up 21 (=b) and shifted left -7 (=a).

Thus,
$
(y - 21) = \frac{c}{x + 7}$

To find c, simply substitute in a known coordinate: (6,0) or (0,-18).
Substituting in (6,0), we get

$
(0 - 21) = \frac{c}{6 + 7}$

$

-21 = \frac{c}{13}$

$c = -21 \times 13 = -273$

Thus the equation is:
$

(y - 21) = \frac{-273}{x + 7}$
I thank you for your time and effort.