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. $$ y=\frac{2 x^{2}}{x^{2}-4} $$

Short answer

The graph of \(y=\frac{2 x^{2}}{x^{2}-4}\) is symmetric about the y-axis. The vertical asymptotes are at \(x = \pm 2\) and the horizontal asymptote is at \(y = 2\). The only intercept is at the origin as \(x = 0\). The minimum point of the function is at \(x = 0\). The graph is a parabola opening upwards.

Step-by-step solution

Determine the symmetry

For even symmetry, substitute \(-x\) for \(x\) , if it produces the same equation then it is symmetric about the y-axis. Plug in \(-x\) into the equation: \(y=\frac{2(-x)^{2}}{(-x)^{2}-4} = \frac{2x^{2}}{x^{2}-4}\) which is the same equation, so the graph is symmetric about the y-axis.

Determine the asymptotes

Find the vertical asymptotes by setting the denominator equal to zero: \(x^{2}-4=0\). This gives \(x^{2}=4\), so \(x = \pm 2\). The function is undefined at \(x = 2\) and \(x = -2\), so they are the vertical asymptotes. Horizontal asymptotes happen as \(x\) approaches \(\pm \infty\). As \(x\) grows larger, the \(-4\) becomes insignificant so the equation approaches \(y = \frac{2x^{2}}{x^{2}} = 2\). Therefore, \(y=2\) is the horizontal asymptote.

Locate the intercepts

The x-intercepts occur when \(y=0\), so set \(y = 0\), thus \(x^{2} = 0\), giving \(x = 0\). The y-intercept also occurs when \(x = 0\), hence \(y = \frac{2(0)}{0-4} = 0\). Therefore, the graph intersects at the origin.

Find the extrema

Because the graph is symmetric about the y-axis and there is a horizontal asymptote at y=2, the extrema occur at the vertex of the parabola. In this case, \(x = 0\) is the minimum of the function.

Sketch the graph

Draw the graph, marking the points and features calculated. Add the vertical asymptotes at \(x = \pm 2\) , the horizontal asymptote at \(y = 2\), and the x and y intercepts at \(x = y = 0\). Sketch a parabolic curve with a minimum point at \(x = 0\) respectful to all these features.