Problem 7 - Algebra for College Students

Question

For Problems \(1-20\), graph each rational function. Check first for symmetry, and identify the asymptotes. $$ f(x)=\frac{x}{x^{2}+x-6} $$

Step-by-Step Solution

Answer

The function is symmetrical about the origin with vertical asymptotes at \( x = 2 \) and \( x = -3 \), and a horizontal asymptote at \( y = 0 \).

Step 1: Identify Symmetry

To determine the symmetry of the function \( f(x) = \frac{x}{x^2 + x - 6} \), we test for symmetry with respect to the y-axis and origin. A function \( f(x) \) is symmetric about the y-axis if \( f(-x) = f(x) \) and symmetric about the origin if \( f(-x) = -f(x) \). Compute \( f(-x) = \frac{-x}{(-x)^2 + (-x) - 6} = -\frac{x}{x^2 - x - 6} \). Since \( f(-x) e f(x) \) and \( f(-x) = -f(x) \), the function is symmetric about the origin.

Step 2: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Set \( x^2 + x - 6 = 0 \) and solve \( (x - 2)(x + 3) = 0 \). Thus, the vertical asymptotes are at \( x = 2 \) and \( x = -3 \).

Step 3: Determine Horizontal Asymptote

For the function \( f(x) = \frac{x}{x^2 + x - 6} \), compare the degrees of the numerator and denominator. The numerator has degree 1 and the denominator degree 2. Since the denominator's degree is greater, the horizontal asymptote is at \( y = 0 \).

Step 4: Graph the Function

Given the symmetry about the origin, vertical asymptotes at \( x = 2 \) and \( x = -3 \), and a horizontal asymptote at \( y = 0 \), draw the graph. The function will approach the vertical asymptotes as it diverges and cross the y-axis at the origin, going toward the horizontal asymptote as \( x \to \pm \infty \).

Key Concepts

Symmetry of Functions

When analyzing a function like \( f(x) = \frac{x}{x^2 + x - 6} \), it's essential to check for symmetry as it simplifies the graphing process. Functions can be symmetric about the y-axis or the origin.

We check for y-axis symmetry by seeing if \( f(-x) = f(x) \). For symmetry about the origin, we need \( f(-x) = -f(x) \). In our function, calculating \( f(-x) \) gives us \(-\frac{x}{x^2 - x - 6}\), which equals \(-f(x)\). Therefore, this function is symmetric about the origin.

This symmetry tells us that the graph will be a mirror image across the origin. Understanding symmetry can help predict the behavior of a graph quickly without plotting numerous points.

Vertical Asymptotes

Vertical asymptotes are critical in understanding where a graph shoots off to infinity or negative infinity. These occur where the function's denominator is zero, as long as the numerator isn't also zero.

In the example, our denominator is \(x^2 + x - 6\). We find the roots by factoring it to \((x - 2)(x + 3) = 0\), giving us \(x = 2\) and \(x = -3\). Hence, these are the locations of our vertical asymptotes.

• Vertical asymptote at \(x = 2\)
• Vertical asymptote at \(x = -3\)

As you approach these points, the function value tends to infinity or negative infinity, creating a visual divide in the graph. This helps us understand the function's "no-go zones."

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a graph as it stretches toward infinity. This happens when you assess the degrees of the polynomials in the numerator and denominator.

Our numerator, \(x\), has a degree 1, and the denominator, \(x^2 + x - 6\), has a degree 2. Since the degree of the numerator is less than that of the denominator, the horizontal asymptote is at \(y = 0\).

As \(x\) approaches positive or negative infinity, the function \(f(x)\) gets closer to zero. This concept guides us in understanding the flattening behavior of the graph.

Rational Functions

Rational functions are the ratio of two polynomials, like \( f(x) = \frac{x}{x^2 + x - 6} \). Such functions can have unique features, like asymptotes and symmetry, impacting their graph shapes.

They are defined everywhere apart from where the denominator is zero, which results in vertical asymptotes. Horizontal asymptotes form depending on the degrees of the numerator and denominator, guiding the end behavior of the graph.

• Watch for undefined points due to zero denominators.
• Identify symmetries to simplify graphing.
• Check degrees for horizontal asymptotes understanding.

Understanding the components and behavior of rational functions enables problem-solving with ease. It tells us where the function values skyrocket or dive and what the graph looks like far from the origin.