Problem 6 - Algebra and Trigonometry

Question

The following questions are about the rational function $$ I(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} $$ The function \(r\) has horizontal asymptote \(y=\) ___________

Step-by-Step Solution

Answer

The horizontal asymptote is \(y=1\).

Step 1: Identify the Degree of the Numerator

The first step is to determine the degree of the polynomial in the numerator of the rational function. The numerator is \((x+1)(x-2)\), which expands to \(x^2 - x - 2\). The degree of this polynomial is 2.

Step 2: Identify the Degree of the Denominator

Next, identify the degree of the polynomial in the denominator. The denominator is \((x+2)(x-3)\), which expands to \(x^2 - x - 6\). The degree of this polynomial is also 2.

Step 3: Compare Degrees to Find Horizontal Asymptote

For a rational function, if the degrees of the numerator and the denominator are equal, as they are in this case (both are 2), the horizontal asymptote is determined by the leading coefficients of the numerator and denominator. The leading coefficient in both the numerator (\(x^2\)) and the denominator (\(x^2\)) is 1.

Step 4: Determine the Horizontal Asymptote

Since the leading coefficients of both the numerator and the denominator are 1, the horizontal asymptote is \(y = \frac{1}{1} = 1\).

Key Concepts

Horizontal Asymptote

A horizontal asymptote in the context of rational functions is a horizontal line that the graph of the function approaches as the input variable, usually denoted as \(x\), goes towards infinity or negative infinity. It is important to note that a function might touch or cross its horizontal asymptote, but it will eventually return to it.
The horizontal asymptote is especially helpful in understanding the end-behavior of rational functions, which are expressions of the form \(\frac{N(x)}{D(x)}\), where \(N(x)\) and \(D(x)\) are polynomials.

• If the degree of the numerator \(N(x)\) is less than the degree of the denominator \(D(x)\), the horizontal asymptote is \(y = 0\).
• If the degrees are equal, as in our example \(I(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)}\) with both degrees being 2, the horizontal asymptote is \(y = \text{the ratio of the leading coefficients of the numerator and the denominator}\).
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote, but there could be an oblique asymptote instead.

It's important to understand horizontal asymptotes as they provide us with a clear picture of how the function behaves in the extremes of its domain.

Polynomial Degree

The degree of a polynomial gives us insight into the most significant term that will affect the function's behavior for large values of \(x\). It is defined as the highest power of \(x\) that appears with a non-zero coefficient. Understanding the degree of both the numerator and the denominator in a rational function helps determine the existence and nature of horizontal asymptotes.
In our example, the rational function \(I(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)}\) has both a numerator and a denominator degree of 2. This is because, when expanded, they both result in quadratic polynomials:

• Numerator: \((x+1)(x-2)\) simplifies to \(x^2 - x - 2\)
• Denominator: \((x+2)(x-3)\) simplifies to \(x^2 - x - 6\)

Both of these expansions confirm a polynomial degree of 2, which plays a crucial role in comparing the degrees to determine the horizontal asymptote.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It helps us understand the polynomial's growth rate and can play a pivotal role when determining horizontal asymptotes in the context of rational functions.
In the rational function \(I(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)}\), the polynomials in the numerator and the denominator both have a degree of 2, and their leading term is \(x^2\). Thus, the leading coefficient for both is 1.
When the degrees of the numerator and denominator are equal, as they are here, the horizontal asymptote is determined by taking the ratio of the leading coefficients:

• For this example: Numerator's leading coefficient is 1.
• Denominator's leading coefficient is 1.

Therefore, the horizontal asymptote is \(y = \frac{1}{1} = 1\). This makes the leading coefficient a key factor in predicting and verifying the horizontal asymptote's location in the graph of the function.