Problem 33 - A Graphical Approach to College Algebra

Question

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{4-2 x}{8-x}$$

Step-by-Step Solution

Answer

The graph of \( f(x) = \frac{4-2x}{8-x} \) has a vertical asymptote at \( x = 8 \), a horizontal asymptote at \( y = 2 \), x-intercept at \( x = 2 \), and y-intercept at \( y = \frac{1}{2} \).

Step 1: Identify Vertical Asymptotes

To find the vertical asymptotes of the function \( f(x)=\frac{4-2x}{8-x} \), we need to set the denominator equal to zero. Thus, solve for \( x \) in the equation \( 8-x = 0 \), giving \( x = 8 \). This is the vertical asymptote.

Step 2: Identify Horizontal Asymptotes

Evaluate the degrees of the numerator and the denominator. Both are linear (degree 1). Thus, the horizontal asymptote is found by dividing the leading coefficients: \( \frac{-2}{-1} = 2 \). So, the horizontal asymptote is \( y = 2 \).

Step 3: Find the x-intercept

To find the x-intercept, set the numerator equal to zero and solve for \( x \): \( 4-2x = 0 \). Solving gives \( x = 2 \). Thus, the graph intercepts the x-axis at \( (2,0) \).

Step 4: Find the y-intercept

Substitute \( x = 0 \) into the function: \( f(0) = \frac{4-2(0)}{8-0} = \frac{4}{8} = \frac{1}{2} \). So, the y-intercept is \( \left(0, \frac{1}{2}\right) \).

Step 5: Analyze the Behavior Around Asymptotes

As \( x \to 8^+ \), \( f(x) \) tends to \(-\infty\) and as \( x \to 8^- \), \( f(x) \to +\infty\). As \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to 2 \).

Step 6: Sketch the Graph

Plot the vertical asymptote at \( x = 8 \) and the horizontal asymptote at \( y = 2 \). Plot the intercepts \( (2, 0) \) and \( (0, \frac{1}{2}) \). Use the behavior analysis to sketch the general layout of the curve.

Key Concepts

Understanding Asymptotes in Rational Functions

An asymptote is a line that a graph approaches but never touches or crosses. In rational functions, asymptotes provide essential information about the behavior and direction of the graph. There are two main types: vertical and horizontal asymptotes.

A vertical asymptote occurs at values of \( x \) where the function becomes undefined. For the given function \( f(x)=\frac{4-2x}{8-x} \), setting the denominator to zero, \( 8-x=0 \), gives us \( x=8 \). Therefore, the vertical asymptote is \( x=8 \). The function's graph will get infinitely close to this line but will never cross it.

Horizontal asymptotes demonstrate the end-behavior of a function as \( x \) approaches positive or negative infinity. By comparing the degrees of the polynomial in the numerator and the denominator (both linear here), you determine the horizontal asymptote by the ratio of their leading coefficients. Thus, the horizontal asymptote for this function is \( y=2 \). This means that as \( x \) goes towards large positive or negative numbers, the function value will approach 2.

Finding x-intercepts of Rational Functions

The x-intercepts of a function occur when the output value \( f(x) = 0 \). For rational functions, this involves setting the numerator equal to zero. Taking \( f(x)=\frac{4-2x}{8-x} \), set \( 4-2x = 0 \) and solve for \( x \). This results in \( x=2 \). Thus, the x-intercept is the point \( (2, 0) \), where the graph crosses the x-axis.

This point is crucial to the graph as it indicates where the function changes sign on the horizontal axis. It's worth noting that unlike vertical asymptotes, x-intercepts are actual points on the graph.

Understanding y-intercepts of Rational Functions

Y-intercepts occur where the graph crosses the y-axis. In rational functions, you find this by substituting \( x=0 \) into the function. For \( f(x)=\frac{4-2x}{8-x} \), substitute \( x=0 \): \( f(0) = \frac{4-2(0)}{8-0} = \frac{4}{8} = \frac{1}{2} \). Therefore, the y-intercept is \( (0, \frac{1}{2}) \).

This point tells you where the graph starts on the vertical axis. Unlike x-intercepts, which occur at roots of the numerator, y-intercepts involve a simple evaluation of the function at \( x=0 \). Understanding intercepts is essential for tracing the initial part of the graph.

Skilled Graph Sketching of Rational Functions

Graph sketching is a rewarding part of analyzing rational functions. With the x-intercept, y-intercept, and asymptotes determined, you have a framework to build the graph's shape. Start by marking these points and lines on a graph:

• Vertical asymptote at \( x=8 \)
• Horizontal asymptote at \( y=2 \)
• X-intercept at \( (2,0) \)
• Y-intercept at \( (0, \frac{1}{2}) \)

This setup guides how you connect the dots.

With these reference points, observe the behavior around the vertical asymptote (\( x=8 \)). As \( x \to 8^+ \), the function \( f(x) \to -\infty \) and as \( x \to 8^- \), the function \( f(x) \to +\infty \). This means the graph will dip sharply downward as it approaches from the right and steep upward from the left.

Check the end behavior using the horizontal asymptote, confirming that as \( x \to \pm\infty \), \( f(x) \to 2 \). Finally, connect these sections smoothly, maintaining consistency with the asymptotic and intercept analysis. This approach lands you an accurate sketch of the rational function's graph.