Problem 64 - Calculus of a Single Variable

Question

Sketching a Graph In Exercises \(59-74\) , sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. $$ y=\frac{2 x^{2}}{x^{2}-4} $$

Step-by-Step Solution

Answer

The graph of \(y = \frac{2x^2}{x^2 - 4}\) will be a symmetric graph with y-intercept at \(0\), and x-intercept at \(0\). Vertical asymptotes are at \(x = -2\) and \(x = 2\). There is an oblique asymptote at \(y=2\). The sketch of the graph should be verified using a graphing utility.

Step 1: Find the Y-intercept

The y-intercept of a function is found by setting \(x = 0\). Substituting \(x = 0\) into \(y = \frac{2x^2}{x^2 - 4}\) yields \(y = 0\). Therefore, the y-intercept is \(0\).

Step 2: Find the X-intercepts

The x-intercepts are found by setting \(y = 0\). This occurs when the numerator of the fraction equals zero, so \(2x^2 = 0\)=> \(x = 0\). Therefore, the x-intercept is \(x = 0\).

Step 3: Check for Symmetry

The function is symmetric about the y-axis if replacing x with -x doesn't change the function. Replacing x with -x gives us \(y = \frac{2(-x)^2}{(-x)^2-4} = \frac{2x^2}{x^2 - 4}\), which is just our original function. Therefore, the function is symmetric about the y-axis.

Step 4: Find the Asymptotes

Vertical asymptotes occur where the denominator equals zero and the numerator doesn’t. So, setting \(x^2 - 4 = 0\) gives \(x^2 = 4\) => \(x = 2, -2\). So, vertical asymptotes are at \(x = -2 \) and \(x = 2\). The function has no horizontal asymptote, but has an oblique asymptote, which can be found by polynomial division or noting when \(x-> \pm \infty\), \(y->2\). So \(y=2\) is an oblique asymptote.

Step 5: Sketching the Graph

Plot the intercepts, the asymptotes, and a few points around the vertical asymptotes for accuracy. Based on this data and the symmetry of the function, sketch a mirror image across the y-axis to complete the graph. Verify the result using a graphing utility.

Key Concepts

Intercepts

Intercepts are crucial points where the graph of a function meets the axes. Understanding intercepts helps in easily identifying the position of a graph in relation to the coordinate axes.

**Y-intercepts** are found by setting \(x = 0\). For the function \(y = \frac{2x^2}{x^2 - 4}\), substituting \(x = 0\) results in \(y = 0\). This means the graph intersects the y-axis at the origin, point (0,0).

• Y-intercept: Occurs where the graph crosses the y-axis.
• Determine by setting \(x = 0\).

**X-intercepts** require setting the function \(y = 0\). This involves solving \(2x^2 = 0\), yielding \(x = 0\). Thus, the graph touches the x-axis at the origin, as well.

• X-intercept: Where the graph meets the x-axis.
• Solve by setting \(y = 0\), focusing on the numerator of the function.

Symmetry

Investigating symmetry in a graph can make the sketching process simpler, allowing for mirroring across certain lines.

For a function symmetric about the y-axis, changing \(x\) to \(-x\) in the equation should leave the equation unchanged. In this case, when we replace \(x\) with \(-x\), the resulting function \(y = \frac{2(-x)^2}{(-x)^2-4}\) simplifies back to the original \(y = \frac{2x^2}{x^2 - 4}\). This confirms that the given function is symmetric about the y-axis.

• Symmetric about the y-axis: If \(f(x) = f(-x)\).
• Simplifies graphing by ensuring both sides of the y-axis are identical.

Recognizing symmetry aids in accurately sketching the rest of the graph with less effort.

Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity as \(x\) approaches a certain value, indicating a point of discontinuity.

Vertical asymptotes are found by identifying the values that make the denominator zero while the numerator is not zero. For our function, set \(x^2 - 4 = 0\) to find \(x^2 = 4\). Solving this gives \(x = 2\) and \(x = -2\), thus revealing vertical asymptotes at these points.

• Vertical Asymptotes: Occur when the denominator of a rational function equals zero.
• Critical in understanding where the function is undefined.
• Indicate where the graph tends towards infinity.

This understanding helps plot the graph accurately, as the function will steep infinitely close to these lines.

Oblique Asymptotes

Oblique asymptotes, also called slant asymptotes, are lines that the curve approaches as \(x\) moves towards plus or minus infinity. When dealing with rational functions, if the degree of the numerator is exactly one higher than the degree of the denominator, an oblique asymptote often appears.

For the given function, since the numerator's degree equals the denominator's, there's no typical oblique asymptote. However, by performing polynomial division or observing the behavior at infinity, it's noted that as \(x\) becomes very large or very small, the function approaches \(y = 2\). Thus, a kind of horizontal behavior with oblique-like traits emerges in this rational function.

• Oblique Asymptotes: Present when numerator's degree surpasses the denominator's by one.
• Help define the behavior of the function at extreme values of \(x\).
• For this function: Approaches \(y = 2\) as \(x\) tends to infinity.

Understanding this concept aids in plotting long-term behavior on graphs.