Problem 37 - 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. $$f(x)=\frac{5 x}{x^{2}-1}$$

Step-by-Step Solution

Answer

Vertical asymptotes: \(x = 1\), \(x = -1\); horizontal asymptote: \(y = 0\). Graph behaves as outlined.

Step 1: Identify Asymptotes

To find vertical asymptotes, set the denominator equal to zero: \(x^2 - 1 = 0\). Solving gives \(x^2 = 1\), so the vertical asymptotes are \(x = 1\) and \(x = -1\). The horizontal asymptote for the function \(f(x) = \frac{5x}{x^2 - 1}\) can be found by comparing the degrees of the numerator and the denominator. Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is \(y = 0\).

Step 2: Determine Key Points

Evaluate the function at key points to sketch the graph. For the y-intercept, set \(x = 0\), giving \(f(0) = 0\). For \(x o 1^+\) and \(x o 1^-\), \(f(x)\to \,\pm\infty\), and for \(x o -1^+\) and \(x o -1^-\), \(f(x)\to \,\mp\infty\). Check behavior as \(x\to \pm\infty\): since the horizontal asymptote is \(y = 0\), \(f(x)\to 0\).

Step 3: Examine Function Behavior Around Asymptotes

Check the sign of \(f(x)\) around the vertical asymptotes. As \(x\) approaches 1 from the left \((x\to 1^-)\), \(f(x)\to -\infty\), and from the right \((x\to 1^+)\), \(f(x)\to +\infty\). At \(x = -1\), as \(x\to -1^-\), \(f(x)\to +\infty\), and as \(x\to -1^+\), \(f(x)\to -\infty\).

Step 4: Sketch the Graph

Draw the vertical asymptotes at \(x = 1\) and \(x = -1\), and the horizontal asymptote at \(y = 0\). Mark the y-intercept at (0,0). Sketch the graph approaching the asymptotes according to the sign and behavior: as \(x\) approaches -1 and 1 from each side and \(f(x)\to \pm\infty\). The curve should approach \(y=0\) as \(x\to \pm\infty\).

Key Concepts

Vertical Asymptotes

A vertical asymptote occurs in a rational function when the value of the denominator is zero, but the numerator is not zero at the same point. This results in the function getting infinitely large or small as it approaches this specific x-value. For the function \( f(x) = \frac{5x}{x^2 - 1} \), to find the vertical asymptotes, set the denominator equal to zero: \( x^2 - 1 = 0 \). Solving this equation gives \( x^2 = 1 \), which means \( x = 1 \) and \( x = -1 \) are the vertical asymptotes.

These vertical lines show that as the graph approaches \( x = 1 \) from the left, \( f(x) \to -\infty \), and from the right, \( f(x) \to +\infty \). Similarly, for \( x = -1 \), approaching from the left \( f(x) \to +\infty \), and from the right \( f(x) \to -\infty \). Vertical asymptotes are critical for sketching the graph because they signify sharp changes in the behavior of the function.

Horizontal Asymptotes

Horizontal asymptotes help describe the end behavior of rational functions as \( x \) approaches \( \pm \infty \). For a rational function like \( f(x) = \frac{5x}{x^2 - 1} \), you can determine if there is a horizontal asymptote by comparing the degrees of the polynomial in the numerator and the denominator.

In this function, the degree of the numerator (\( 5x \)) is 1, while the degree of the denominator (\( x^2 - 1 \)) is 2. When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).

This means that as \( x \to \pm\infty \), the value of \( f(x) \) approaches zero. The presence of a horizontal asymptote offers insight into how the function behaves over a long range, which is crucial when sketching the graph.

Graph Sketching

Creating a sketch of a rational function provides a visual representation of the function's behavior. To begin, mark the vertical asymptotes you identified at \( x = 1 \) and \( x = -1 \). Draw dashed vertical lines at these points, indicating they are boundaries where the function will not touch.

Next, draw the horizontal asymptote \( y = 0 \) as a dashed line, showing where the function's value tends to flatten out as \( x \to \pm \infty \).

Identify key points like the y-intercept. For this function, when \( x = 0 \), \( f(x) = 0 \), so mark this point on your graph.

While sketching, ensure the graph approaches the asymptotes correctly. As \( x \to 1^- \), \( f(x) \to -\infty \), meaning the curve should swoop downward towards the vertical asymptote. From \( x \to 1^+ \), \( f(x) \to +\infty \) so the curve should rise upward. Similarly, at \( x = -1 \), the function behaves oppositely.

Lastly, complete the sketch by connecting the curves along the asymptotes and ensure smooth, continuous transitions between sections.

Function Behavior

Understanding function behavior involves examining how a function behaves near its asymptotes and beyond. For \( f(x) = \frac{5x}{x^2 - 1} \), we focus on its behavior near the vertical asymptotes \( x = 1 \) and \( x = -1 \), and as \( x \to \pm \infty \) regarding the horizontal asymptote.

As \( x \to 1^- \), the function quickly drops toward negative infinity, reflecting that just before reaching \( x = 1 \) from the left, values of \( f(x) \) become very large negatively. Conversely, as \( x \to 1^+ \), it shoots up towards positive infinity. A mirror image of this behavior occurs at \( x = -1 \) but reversed in sign.

At infinity, the function becomes negligible, approaching the horizontal asymptote \( y = 0 \). This indicates that far away from the critical points defined by the asymptotes, the graph behaves calmly with its peaks and troughs flattening out. Recognizing these properties helps provide a complete picture when sketching rational functions.