Problem 8 - Calculus of a Single Variable

Question

In Exercises 5–24, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ y=\frac{x^{2}+1}{x^{2}-4} $$

Step-by-Step Solution

Answer

The graph of \(y = \frac{x^2 + 1}{x^2 - 4}\) has a y-intercept at \(-0.25\) and vertical asymptotes at \(x = -2\) and \(x = 2\). The graph has a horizontal asymptote at \(y = 1\). A sketch of the graph can be verified by a graphing utility.

Step 1: Identify x-intercepts and y-intercepts

To find the x-intercepts, set the numerator equal to zero and solve for x: \(x^2 + 1 = 0\). This has no solutions, so there are no x-intercepts. To find the y-intercept, replace x by 0 in the equation: \( y = \frac{0^2 + 1}{0^2 - 4} = -\frac{1}{4}\) . So the y-intercept is \(y = -0.25\).

Step 2: Identify vertical and horizontal asymptotes

To find the vertical asymptotes, set the denominator equal to zero and solve for x: \(x^2 - 4 = 0 \rightarrow x = ±2\). So the vertical asymptotes are \(x = -2\) and \(x = 2\). Since the degrees of the numerator and the denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients, which is 1.

Step 3: Identify relative extrema and points of inflection

Relative extrema and points of inflection can be found by taking the first and second derivatives, respectively. However, for the given function, this process is complicated and is best done with the aid of a graphing utility.

Step 4: Sketch the graph and verify with a graphing utility

Plot the x- and y-intercepts, and draw the asymptotes. The graph approaches \(y = 1\) as \(x \rightarrow ±\infty\), and it approaches \(y = -2\) and \(y = 2\) for \( x = ±2\), respectively. Use a graphing utility to verify your results.

Key Concepts

Graph Analysis

When you analyze a rational function graph, you're looking for key features that define the behavior of the function. Essential aspects include intercepts, asymptotes, and points of inflection. This can determine how a graph curves and where it crosses or approaches certain lines. Begin by examining the overall shape and direction of the graph. Rational functions like \( y = \frac{x^2 + 1}{x^2 - 4} \), often have one or more asymptotes and might be divided into distinct sections by these asymptotes. Graph analysis also involves understanding the direction and curvature of the graph at different sections. These changes are particularly important around critical points such as intercepts and asymptotes, as they signal significant shifts in the graph's behavior.

Asymptotes

Asymptotes are critical lines that a graph approaches but never quite touches. For rational functions, these are usually vertical or horizontal lines.

• Vertical Asymptotes: These occur where the denominator of the function is zero because dividing by zero is undefined. For this function, solve \(x^2 - 4 = 0\) to find \(x = -2\) and \(x = 2\). Therefore, the vertical asymptotes are at these lines.
• Horizontal Asymptotes: To find the horizontal asymptote, compare the degrees of the numerator and denominator. If they are equal, like in this example, the horizontal asymptote is the ratio of their leading coefficients. Here, both coefficients are 1, giving us \(y = 1\).

These lines guide us in understanding how the graph approaches infinity. For example, as \(x\) becomes very large or very small, the values of \(y\) will get closer and closer to the asymptotic values.

Intercepts

Intercepts are the points where the graph crosses the axes, and they provide foundational insights into the behavior of the graph.

• X-Intercepts: These occur where the graph crosses the x-axis. For rational functions, set the numerator to zero. In this case \(x^2 + 1 = 0\) has no real solutions – there are no x-intercepts.
• Y-Intercept: These occur where the graph crosses the y-axis. Substitute \(x = 0\) into the function: \( y = \frac{0^2 + 1}{0^2 - 4} = -\frac{1}{4}\). This tells us the graph meets the y-axis at \( y = -0.25\).

These intercepts are useful for sketching the initial outlines of the graph, helping to plot points as part of the graph's behavior.

Extrema and Inflection Points

These points tell us about the peaks, valleys, and changes in the curve of the graph, offering deeper insights into its overall shape.

• Relative Extrema: To find maxima and minima, we use the first derivative of the function. However, for complex rational functions, using graphing utilities might be more efficient.
• Inflection Points: These are points where the concavity of the graph changes, usually found using the second derivative. Again, graphing tools can simplify this process.

Identifying these points can be complex due to the intricate nature of derivatives for rational functions. Using a graphing utility to visualize and confirm these points on the graph is often a practical approach. This allows you to see the transitions in the graph's behavior more clearly.