Question

Write a rational function with vertical asymptotes at \(x=6\) and \(x=-2,\) and with a zero at \(x=3\).

Short answer

The rational function is \(f(x) = \frac{x-3}{(x-6)(x+2)}\).

Step-by-step solution

Form the Denominator

The denominator of our rational function should be zero at \(x=6\) and \(x=-2\). This is achieved when the denominator is on the form \(q(x)=(x-a)(x-b)\). Since \(a=6\) and \(b=-2\) we will get \(q(x)=(x-6)(x+2)\), which is the denominator of the function.

Form the Numerator

The numerator needs to be zero for \(x=3\). This occurs when \(p(x)=c(x-d)\), where d is 3. The constant c doesn't affect the zeroes of the function, so we can let \(c=1\). Therefore, we get \(p(x)=(x-3)\).

Construct the Rational Function

Now combine the numerator and denominator to write the final rational function \(f(x) = \frac{p(x)}{q(x)} = \frac{x-3}{(x-6)(x+2)}\). This function has vertical asymptotes at \(x=6\) and \(x=-2,\) and a zero at \(x=3\).