Problem 38 - 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. $$f(x)=\frac{x}{4-x^{2}}$$

Step-by-Step Solution

Answer

Vertical asymptotes at \(x = 2\) and \(x = -2\), horizontal asymptote at \(y = 0\), x-intercept at \((0, 0)\).

Step 1: Identify the Vertical Asymptotes

To find the vertical asymptotes, set the denominator equal to zero and solve for \(x\): \[ 4 - x^2 = 0 \] This can be solved as follows: \(x^2 = 4\) leads to \(x = 2\) and \(x = -2\). Thus, the vertical asymptotes are at \(x = 2\) and \(x = -2\).

Step 2: Identify the Horizontal Asymptote

For the rational function \(f(x) = \frac{x}{4-x^2}\), compare degrees of the polynomial in the numerator and the polynomial in the denominator. The numerator is of degree 1 and the denominator is of degree 2. Since the degree of the denominator is higher, there is a horizontal asymptote at \(y = 0\).

Step 3: Determine the Behavior at Asymptotes

Analyze the behavior of \(f(x)\) as \(x\) approaches the vertical asymptotes. Check both sides of each asymptote by considering values slightly less and slightly more than \(x = 2\) and \(x = -2\). For example, as \(x\) approaches 2 from the left, the value of \(f(x)\) becomes very negative, and from the right, it becomes very positive. This indicates a change in sign across the vertical asymptote.

Step 4: Find Intercepts

Set \(f(x) = 0\) to find the x-intercept. Since the numerator must be zero (\(x = 0\)), the x-intercept is at \((0, 0)\). The y-intercept is also at \((0,0)\) as found by evaluating \(f(x)\) at \(x = 0\).

Step 5: Sketch the Graph

Using the asymptotes, intercepts, and analyzed asymptotic behavior, sketch the graph. Draw vertical lines at \(x = 2\) and \(x = -2\) for vertical asymptotes and a horizontal line at \(y = 0\) for horizontal asymptote. Plot the intercept \((0, 0)\). Draw curves approaching the asymptotes, based on the slope and direction determined previously.

Key Concepts

Vertical Asymptotes

In rational functions, vertical asymptotes occur where the function is undefined.
This generally happens where the denominator of the function equals zero while the numerator does not. For the function \( f(x) = \frac{x}{4-x^2} \), the vertical asymptotes can be found by setting the denominator zero.

• Set \(4 - x^2 = 0\).
• Solve for \(x\): \(x^2 = 4\) leads to \(x = 2\) and \(x = -2\).

Thus, the vertical asymptotes occur at \(x = 2\) and \(x = -2\). These are the vertical lines that the graph approaches but never touches or crosses.
When plotting the graph, draw dotted or dashed lines at these points to indicate the asymptotes. The graph of the function will extend infinitely in the positive or negative direction as it approaches these lines, often flipping direction as it crosses them.

Horizontal Asymptotes

Horizontal asymptotes in rational functions provide information on the end behavior of a graph.
They show what value the function approaches as \(x\) goes to infinity or negative infinity. In the function \( f(x) = \frac{x}{4-x^2} \), the concept of degree of polynomials helps identify this asymptote.

• The degree of the numerator (top of the fraction) is 1.
• The degree of the denominator (bottom of the fraction) is 2.

When the degree of the denominator is greater than the numerator, the horizontal asymptote is \(y = 0\). This means that as \(x\) gets very large in the positive or negative direction, \(f(x)\) will approach but never reach 0. On the graph, this is represented by a horizontal line at \(y = 0\), which the function's curve will approach but not cross as it extends towards infinity.

X-Intercept

The x-intercept of a rational function is a point where the graph intersects the x-axis.
This occurs where the function value is zero, i.e., the numerator is zero while the denominator is non-zero. In our example, the function \( f(x) = \frac{x}{4-x^2} \) has its numerator \( x \) set to zero:

• Set \(x = 0\), thus the function \( f(x) = 0 \).
• Therefore, the x-intercept is at the point \((0, 0)\).

This point is significant on the graph as it indicates where the graph crosses the x-axis.
Additionally, it means that the y-intercept, where the graph crosses the y-axis, is also at \((0,0)\). This dual role of the point makes it a key feature when sketching the function.
Always remember to denote intercepts on your graph clearly to show where the function reaches zero.