Problem 20 - Algebra for College Students

Question

For Problems \(1-20\), graph each rational function. Check first for symmetry, and identify the asymptotes. $$ f(x)=\frac{x^{3}+8}{x^{2}} $$

Step-by-Step Solution

Answer

The graph of \( f(x) = \frac{x^3 + 8}{x^2} \) has a vertical asymptote at \( x = 0 \) and a slant asymptote \( y = x \); it has no symmetry.

Step 1: Analyze Symmetry

To determine symmetry, check if the function is even, odd, or neither. A function is even if \( f(-x) = f(x) \) and odd if \( f(-x) = -f(x) \). Substitute \(-x\) into the function \( f(x) = \frac{x^{3}+8}{x^{2}} \) giving \( f(-x) = \frac{(-x)^{3}+8}{(-x)^{2}} = \frac{-x^3 + 8}{x^2} \). This expression is neither equal to \( f(x) \) nor \(-f(x) \), so the function is neither even nor odd and has no symmetry across the y-axis or origin.

Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. The denominator \( x^2 \) is zero when \( x = 0 \). Since the numerator \( x^3 + 8 \) is non-zero at \( x=0 \), there is a vertical asymptote at \( x = 0 \).

Step 3: Identify Horizontal Asymptotes

Horizontal asymptotes depend on the degrees of the numerator and denominator. Here, the degree of the numerator \( x^3 + 8 \) is 3, and the degree of the denominator \( x^2 \) is 2. Since the degree of the numerator is greater, there is no horizontal asymptote.

Step 4: Identify Slant Asymptotes

Since the degree of the numerator (3) is one more than the degree of the denominator (2), the function has a slant (oblique) asymptote. Perform polynomial long division of \( x^3 + 8 \) by \( x^2 \) to find it. Dividing, we get \( x + 0 \) with a remainder. Therefore, the slant asymptote is the line \( y = x \).

Step 5: Sketch the Graph

Using the identified asymptotes, sketch a rough graph. Draw a vertical line at \( x=0 \), representing the vertical asymptote. Draw the slant asymptote \( y = x \). The graph approaches the slant asymptote as \( x \to \pm \infty \), and the function is undefined at \( x = 0 \). The graph reflects crossing behavior at non-zero intercepts.

Key Concepts

Asymptotes in Rational Functions

Asymptotes are lines that a graph approaches but never touches. They are crucial for understanding the behavior of rational functions. There are three main types:

• Vertical Asymptotes: These occur where the function becomes undefined, typically where the denominator is zero. For our function, \( f(x) = \frac{x^3 + 8}{x^2} \), the denominator \( x^2 \) is zero when \( x = 0 \). Thus, there is a vertical asymptote at \( x = 0 \). This informs us that the graph will not exist at \( x = 0 \), and the graph will tend to infinity as it approaches this asymptote from either side.
• Horizontal Asymptotes: These refer to the behavior of the graph as \( x \) approaches infinity. If the degree of the numerator is less than or equal to the degree of the denominator, there might be a horizontal asymptote. In our example, since the degree of the numerator is larger than the degree of the denominator (3 vs. 2), there is no horizontal asymptote.
• Slant (Oblique) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. In our example, this is the case. We find the slant asymptote by performing polynomial long division, resulting in the line \( y = x \).

Understand these concepts by identifying where each type of asymptote exists in a graph. They help in sketching an accurate graph of the function by predicting specific behaviors of the graph at extreme values.

Symmetry in Functions

Symmetry in functions helps determine if a graph mirrors itself about an axis or the origin.

• Even Functions: These have symmetry about the y-axis. Mathematically, they satisfy \( f(-x) = f(x) \). Even functions result in graphs that are identical on both sides of the y-axis.
• Odd Functions: These possess symmetry about the origin, satisfying \( f(-x) = -f(x) \). This means if one part of the graph is rotated 180 degrees, it looks identical.
• Neither: If a function does not satisfy either of the above conditions, it is neither even nor odd, meaning it does not have symmetry across the y-axis or about the origin.

For the function \( f(x) = \frac{x^3 + 8}{x^2} \), checking symmetry involved substituting \( -x \) into the function, leading to \( f(-x) = \frac{-x^3 + 8}{x^2} \), which is neither equal to \( f(x) \) nor \( -f(x) \). Thus, the function has no symmetrical properties. Recognizing such characteristics can influence how you visualize the graph.

Polynomial Division for Slant Asymptotes

Polynomial division is a technique to simplify rational functions, especially when identifying slant asymptotes. When the degree of the numerator is one more than the degree of the denominator, we use long division to express the function as a linear equation plus a remainder.In our function, \( f(x) = \frac{x^3 + 8}{x^2} \), we perform polynomial division:

• Divide \( x^3 \) by \( x^2 \) to get \( x \).
• Multiply the quotient \( x \) by the divisor \( x^2 \), resulting in \( x^3 \).
• Subtract this product from the original dividend \( x^3 + 8 \), simplifying to \( 0 + 8 \).
• The process reveals the slant asymptote \( y = x \), with 8 as the remainder, indicating a slight offset but not affecting the slant asymptote's behavior.

Grasping polynomial division is essential for complex rational functions. It assists in graphing by correcting for any upward or downward drift and focusing on the linear trend as the graph extends to infinity.