# How do you write a rational function that has the following properties: a zero at x= 4, a hole at x= 7, a vertical asymptote at x= -3, a horizontal asymptote at y= 2/5?

Jan 31, 2018

Rational function is $\frac{2 {x}^{2} - 22 x + 56}{5 {x}^{2} - 20 x - 105}$

#### Explanation:

A zero at $x = 4$ means we have $\left(x - 4\right)$ as a factor in numerator;

a hole at $x = 7$ means, we have $x - 7$ a factor both in numerator as well as denominator;

a vertical asymptote at $x = - 3$ means $x + 3$ a factor in denominator only

a horizontal asymptote at $y = \frac{2}{5}$ means highesr degrees in both numerator and denominator are equal and their coefficients are in ratio of $2 : 5$

Hence desired rational function is $\frac{2 \left(x - 4\right) \left(x - 7\right)}{5 \left(x - 7\right) \left(x + 3\right)}$

i.e. $\frac{2 {x}^{2} - 22 x + 56}{5 {x}^{2} - 20 x - 105}$

See its graph down below. Observe vertical asymptote $x = - 3$ and horizontal asymptote $y = \frac{2}{5}$. We have a zero at $x = 4$ as functtion passes through $\left(4 , 0\right)$. Hole is not seen as $x - 7$ cancels out, but we know, the function is not definedat this point.

graph{(2x^2-22x+56)/(5x^2-20x-105) [-10.67, 9.33, -4.4, 5.6]} 1