Question

Writing a Rational Function 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 with vertical asymptotes at \(x=6\) and \(x=-2\), and with a zero at \(x=3\) is \(f(x) = \frac{(x-3)}{(x+2)(x-6)}\).

Step-by-step solution

Identify the factors for the denominator

For the vertical asymptotes (roots of the denominator), we want the denominator to be 0. The factors are thus \(x-(-2)\) or \(x+2\) for \(x=-2\), and \(x-6\) for \(x=6\). Hence, the denominator D(x) becomes \((x+2)(x-6)\).

Identify the factors for the numerator

The zero of the function is the root of the numerator, so when \(x=3\), we want the numerator to be 0. So, the factor for the numerator N(x) becomes \((x-3)\).

Write the Rational Function

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

Key concepts

Vertical Asymptotes

Vertical asymptotes occur where a rational function's denominator equals zero, creating a division by zero scenario. This typically results in the graph of the function shooting straight up or down, indicating it approaches infinity or negative infinity.
To find vertical asymptotes for a rational function, solve the equation where the denominator equals zero.
In our exercise, we're asked for vertical asymptotes at \(x=6\) and \(x=-2\).
This means the denominator of our function should be zero when \(x\) takes these values.

• For \(x=6\): we factor the denominator as \((x-6)\) because \(x-6=0\) when \(x=6\).
• For \(x=-2\): we add the factor \((x+2)\) to the denominator because \(x+2=0\) when \(x=-2\).

We combine these to get the complete denominator, \((x+2)(x-6)\). This will create vertical asymptotes at the specified points.

Zero of the Function

The zero of a rational function is the point where its value becomes zero, which happens when the numerator is zero while the denominator is not zero.
A zero is a point where the graph of the function crosses the x-axis.
In our example, we're tasked to have a zero at \(x=3\).
This means our numerator must be zero when \(x=3\).
We achieve this by including a factor in the numerator that nullifies to zero at this point.
So, we add \((x-3)\) in the numerator.

• When \(x=3\), \(x-3=0\), resulting in the whole function becoming zero, as long as the denominator is non-zero.

This ensures that we have a zero at \(x=3\) for our rational function.

Denominator and Numerator Factors

Identifying and setting the proper factors for both the numerator and the denominator is crucial for constructing a rational function.
The factors of the denominator are related to the vertical asymptotes.
Each factor corresponds to an x-value where the denominator is zero, and therefore, where we have a vertical asymptote.

• In our example, the factors \((x+2)(x-6)\) ensure vertical asymptotes at \(x=-2\) and \(x=6\).

The numerator's factors are tied to the function's zero points.
When you're given a zero, include a factor in the numerator that becomes zero at the specified point.

• In our case, the factor \((x-3)\) provides a zero of the function at \(x=3\).

Always verify: combine the numerator and denominator into the rational function, resulting in the form \(f(x) = \frac{(x-3)}{(x+2)(x-6)}\). This structure reflects our desired asymptotes and zero.