Problem 58 - College Algebra and Calculus: An Applied Approach

Question

Write a rational function \(f\) that has the specified characteristics. (There are many correct answers.) Vertical asymptotes: \(x=0, x=\frac{5}{2}\) Horizontal asymptote: \(y=-3\)

Step-by-Step Solution

Answer

The rational function that has the specified characteristics is \(f(x) = \frac{-6x^2}{2x^2 - 5x}\).

Step 1: Identify Vertical Asymptotes

First step to writing the rational function is to incorporate the vertical asymptotes. A vertical asymptote appears at points where the denominator of the function becomes zero. In this case, the function has asymptotes at \(x = 0\) and \(x = \frac{5}{2}\). Starting with a simple denominator \(q(x)\) that gives these asymptotes, we can choose \(q(x) = 2x(x - \frac{5}{2}) = 2x^2 - 5x\).

Step 2: Identify Horizontal Asymptote

The horizontal asymptote depends on the degree of the polynomial in the numerator relative to the degree in the denominator. Since the numerator and the denominator must be the same degree, and the horizontal asymptote is \(y = -3\), we need the ratio of the leading coefficients to be -3. We can achieve this by letting the numerator \(p(x)\) be \(p(x) = -6x^2\).

Step 3: Write the Rational Function

We combine these into our rational function \(f(x) = \frac{-6x^2}{2x^2 - 5x}\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never actually touches or crosses. They occur in rational functions where the denominator is zero but the numerator is non-zero. This is because division by zero is undefined, causing the function's value to spike or dip infinitely.
For example, the rational function in our exercise needs vertical asymptotes at \(x=0\) and \(x=\frac{5}{2}\). To ensure these asymptotes, the denominator of the function itself must be zero at these \(x\) values.

• For \(x=0\), a factor of "x" must be present in the denominator.
• For \(x=\frac{5}{2}\), the expression \(2x - 5\) or \(x - \frac{5}{2}\) must be incorporated.

Combine these factors into the denominator, resulting in: \[q(x) = 2x(x - \frac{5}{2}) = 2x^2 - 5x\]By including these expressions in the denominator, the condition for vertical asymptotes is fulfilled.
It is crucial to distinguish vertical asymptotes from holes in the graph, which occur when both the numerator and denominator are zero at the same \(x\) value.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a rational function as \(x\) approaches infinity or negative infinity. They indicate the value that the function approaches as \(x\) becomes extremely large. The determination of a rational function's horizontal asymptote is heavily linked to the degrees of its numerator and denominator.
In the exercise, the horizontal asymptote is given as \(y = -3\). The degree of the numerator, \(p(x)\), and the degree of the denominator, \(q(x)\), need to be the same for a horizontal asymptote that isn't present at \(y = 0\) or represented as another form.

• If the numerator's degree is less than the denominator's, the horizontal asymptote will be at \(y=0\).
• If the degrees are equal, the horizontal asymptote is defined by the ratio of the leading coefficients of the numerator and denominator.
• In our case, to have \(y = -3\) as an asymptote, the ratio \( \frac{-6}{2} \) should equal \(-3\). Hence, coefficients were chosen appropriately.

This asymptotic behavior ensures that even as \(x\) exponentially increases or decreases, the function's value steers towards \(y = -3\), balancing over the x-axis at this constant height.

Polynomial Degree

Polynomial degree is crucial in determining various characteristics of a rational function, such as asymptotes and end behavior. The degree of a polynomial is the highest power or exponent of the variable in the expression.
In rational functions, understanding the relative degrees of the polynomials in the numerator and the denominator is vital:

• The degree in the denominator dictates the potential occurrence of vertical asymptotes.
• If the numerator and denominator have the same degree, as seen in our exercise, horizontal asymptotes are determined by the ratio of leading coefficients.
• If the numerator's degree exceeds the denominator's by \(1\), an oblique (or slant) asymptote may occur.

In the rational function from the exercise, the numerator \(-6x^2\) and the denominator \(2x^2 - 5x\) were both constructed to have a degree of 2. This setup establishes that the dominant terms in each polynomial steer the end behavior of the function, in this case, forming the horizontal asymptote defined earlier at \(y = -3\). Additionally, understanding polynomial degrees helps ensure that the function is shaped accurately according to its guidelines.