Question

In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=\frac{x}{x^{2}-4} $$

Short answer

The graph of the function \( y=\frac{x}{x^{2}-4} \) passes through the origin and has vertical asymptotes at \(x = -2, 2\). There are no extrema. It is not symmetric and there are no horizontal asymptotes.

Step-by-step solution

Check for Symmetry

The function is not symmetric. Since \(y(-x)\) does not equal \(y(x)\), the function is not symmetric about the y-axis, nor is it symmetric about the origin.

Find the Intercepts

The x-intercepts are determined when \(y = 0\). This happens when \(x = 0\). The y-intercept is found when \(x = 0\), which also results in \(y = 0\). So, the graph intersects the origin (0, 0).

Find the Asymptotes

Vertical asymptotes occur when the denominator is zero. The function is undefined when \(x^2 - 4 = 0\), or when \(x = -2, 2\). There exists no horizontal asymptote because the degree of the numerator is less than the degree of the denominator.

Determine the Extrema

There are no local or global minima or maxima as the function does not turn around. An extrema would exist if the derivative of the function equals zero or undefined, but in this case, the graph of the function is an unbounded curve.

Sketch the Graph

The graph should reflect all these properties: pass through the origin, have vertical asymptotes at \(x = -2, 2\), and no extrema. It should be descending from \( (-\infty, -2) \), increasing from \( (-2, 2) \), and descending again from \( (2, \infty) \).

Verify with a Graphing Utility

After sketching the graph by hand considering all these factors, use a graphing utility to confirm the accuracy of the sketch. The graph on the utility should match the sketch created in the previous steps.

Key concepts

Asymptotes in Rational Functions

When graphing rational functions, identifying asymptotes is a critical step. These are lines to which the graph approaches but never actually touches or crosses.

Vertical Asymptotes

For the given function, \(y = \frac{x}{x^2 - 4}\), vertical asymptotes arise where the denominator equals zero, which is at \(x = -2\) and \(x = 2\). These lines, \(x = -2\) and \(x = 2\), define boundaries on the graph between which the function exists. It's vital to note that the graph cannot touch or cross these lines.

Horizontal Asymptotes

Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator. Since the degree of the numerator is less than the degree of the denominator in our function, we deduce that the horizontal asymptote is the x-axis, or \(y = 0\). However, that's a special case because normally, such a conclusion implies that the graph approaches the x-axis as \(x\) goes to infinity. In this function, even though the condition holds, there is no line where the graph levels off as it should with a horizontal asymptote.

Slant and Curvilinear Asymptotes

For some rational functions, there may also exist slant or curvilinear asymptotes. These are not present for our specific example but are another important feature to consider when the degrees of the polynomial in the numerator is exactly one more than the denominator or when both are of the same degree, respectively.
Recognizing these asymptotic behaviors provides a framework for sketching the accurate behavior of a rational function near its boundaries.

Intercepts of Rational Functions

The intercepts of a function are where the graph crosses the x-axis and y-axis. They provide crucial starting points for graphing.

Finding the X-Intercepts

To locate x-intercepts, set \(y = 0\) and solve for \(x\). For our function \(y = \frac{x}{x^2 - 4}\), this happens when \(x = 0\), which is also the y-intercept in this case.

Finding the Y-Intercepts

To find the y-intercept, set \(x = 0\) and solve for \(y\). As we already mentioned, the y-intercept also occurs at \(0,0\) for this particular function, which is somewhat unique since it's also the x-intercept.
Intercepts are like the anchors of the graph; they're among the initial points we determine, providing an essential guide in plotting the function on the Cartesian plane.

Symmetry in Graphs

The symmetry of a function's graph can significantly simplify the graphing process. Symmetry refers to the balance of the graph about a certain line or point.

Tests for Symmetry

To test for y-axis symmetry, replace \(x\) with \( -x\) and check if the function remains unchanged. If the function's value is the same, the graph is symmetric about the y-axis. To check for origin symmetry (rotational symmetry), replace \(y\) with \( -y\) and \(x\) with \( -x\); if the function is unchanged, the graph has origin symmetry.
For \(y = \frac{x}{x^2 - 4}\), the function fails both tests, meaning it is neither symmetric about the y-axis nor about the origin. Knowing the lack of symmetry helps to focus on plotting accurate points throughout the entire domain of the function rather than relying on reflective properties to infer one side of the graph from the other.

Extrema of Functions

When graphing functions, looking for the extrema—the highest and lowest points on a graph, also known as maxima and minima—is key to understanding its overall shape.

Local Extrema

These are the highest or lowest points within a certain interval of the function. You can find them by solving for where the derivative is equal to zero (critical points) or where it is undefined.

Global Extrema

These refer to the absolute highest or lowest points across the function's entire domain.
For the function \(y = \frac{x}{x^2 - 4}\), we don't have any extrema. This is because the derivative of the function never equals zero or becomes undefined in such a way that a maximum or minimum value is formed. Instead, the function continues to increase or decrease infinitely within its domain, indicating an absence of both local and global extrema. Understanding this helps in setting realistic expectations about the range of the function and indicates that the graph will not have distinct peaks or troughs.