Problem 75 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Find any slant or vertical asymptotes. (b) Graph \(y=f(x) .\) Show all asymptotes. $$ f(x)=\frac{0.5 x^{2}-2 x+2}{x+2} $$

Step-by-Step Solution

Answer

Vertical asymptote: \(x = -2\). Slant asymptote: \(y = 0.5x - 3\).

Step 1: Identify the function type

The function is a rational function in the form \(f(x) = \frac{P(x)}{Q(x)}\) where \(P(x) = 0.5x^2 - 2x + 2\) and \(Q(x) = x + 2\). To find asymptotes, consider the degrees of the numerator (2) and the denominator (1).

Step 2: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero but not the numerator. Solve \(x + 2 = 0\), resulting in \(x = -2\). This is the vertical asymptote, as \(P(-2)eq0\).

Step 3: Check for Slant Asymptotes

Since the degree of the numerator is one more than the degree of the denominator, perform polynomial long division of \(0.5x^2 - 2x + 2\) by \(x+2\) to find the slant asymptote. The quotient is \(0.5x - 3\), which is the equation of the slant asymptote.

Step 4: Graph the function and asymptotes

Graph \(f(x)\). Draw the vertical asymptote \(x = -2\) as a dashed vertical line. Draw the slant asymptote \(y = 0.5x - 3\) as a dashed diagonal line. The graph of \(f(x)\) will approach these lines but never cross them. Plot a few points for accuracy around these asymptotes.

Key Concepts

Understanding Asymptotes in Rational Functions

Asymptotes are lines that the graph of a function approaches but never quite reaches. For rational functions, there are typically two kinds of asymptotes: vertical and slant (or oblique).

Vertical asymptotes occur when the function's denominator is equal to zero, causing the function's value to approach infinity. These vertical lines indicate points where the function is undefined. To find them, set the denominator equal to zero and solve for the variable. For example, in the function given, the vertical asymptote is found by solving the equation \(x + 2 = 0\), giving \(x = -2\). This means as \(x\) approaches \(-2\), the function grows indefinitely.

Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They differ from horizontal asymptotes, which are not present in our example, as horizontal asymptotes appear when the degree of the numerator is less than or equal to the degree of the denominator. Slant asymptotes provide a linear expression (line) that the graph approaches for extreme values of \(x\). In our function, because the numerator has a higher degree, a slant asymptote is formed by performing polynomial division.

Polynomial Division to Find Slant Asymptotes

Polynomial division is a method to divide one polynomial by another, similar to numerical long division. This technique is particularly useful in finding slant asymptotes for rational functions.

In our current problem, the degree of the numerator \(0.5x^2 - 2x + 2\) is higher by one compared to the denominator \(x+2\). Therefore, we use polynomial long division to find the slant asymptote. By dividing \(0.5x^2 - 2x + 2\) by \(x+2\), we find the quotient to be \(0.5x - 3\).

This expression \(y = 0.5x - 3\) represents the slant asymptote. As \(x\) becomes very large (positive or negative), the values of \(f(x)\) will approach, but not precisely meet, this line. The remainder becomes less significant, and the function's graph veers towards the slant asymptote as \(x\) moves away from zero.

Graphing Rational Functions with Asymptotes

Graphing rational functions involves plotting both the function itself and its asymptotes to understand the graph's behavior. Asymptotes serve as guides, indicating the behavior of a graph as \(x\) moves toward the edges of the graph.

To graph our function \(f(x) = \frac{0.5x^2 - 2x + 2}{x+2}\), begin by drawing its asymptotes. The vertical asymptote at \(x = -2\) should be shown as a dashed vertical line. This visually represents that the function approaches but doesn’t cut through this line. Next, draw the slant asymptote \(y = 0.5x - 3\) as a dashed line extending diagonally. This line shows the graph's trajectory for large values of \(x\).

After the asymptotes are in place, plot a few key points in different regions divided by the asymptotes to determine the function's path around these lines. The graph will approach these dashed lines but will never intersect them. Understanding where and how the graph moves in relation to the asymptotes is crucial in accurately sketching rational functions.