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. $$ y=\frac{2 x}{9-x^{2}} $$

Short answer

The graph of the function \( y=\frac{2x}{9-x^{2}} \) is symmetric with respect to the origin, has intercepts at the origin (0,0), has vertical asymptotes at \( x = 3 \) and \( x = -3 \), has a horizontal asymptote at \( y = 0 \), and increases without bound as it nears the vertical asymptotes, resembling a hyperbola.

Step-by-step solution

Identify the Intercepts

The x-intercept happens where the graph crosses the x-axis, meaning \( y = 0 \). Since \( y = 0 \) when \( x = 0 \), the x-intercept is at point (0,0). The y-intercept happens where the graph crosses the y-axis, this is also at point (0,0) because when \( x = 0, y = 0 \).

Check for Symmetry

The graph of the function is symmetric about the origin if replacing \( x \) with \( -x \) yields the same function. The original function is \( y=\frac{2x}{9-x^{2}} \), replacing \( x \) with \( -x \) gives \( y=\frac{-2x}{9-(-x)^{2}} = -y \). This indicates that the function is symmetric with respect to the origin.

Find the Asymptotes

Vertical asymptotes occur when the denominator of a fraction is zero. Set \( 9 - x^{2} = 0 \) to solve for \( x \). This happens when \( x = 3 \) and \( x = -3 \), so these are the vertical asymptotes. A horizontal asymptote is where the function heads for as \( x \) goes to \( \pm \) infinity. Using the rule that when the degree of the denominator is greater than the degree of the numerator, the x-axis (y = 0) is the horizontal asymptote.

Identify Extrema

The extrema for a function occur at the critical points, where the first derivative equals zero or does not exist. For the function \( y=\frac{2x}{9-x^{2}} \), the derivative \( y' = \frac{18}{(9-x^{2})^{2}} \) does not equal zero anywhere but is undefined at \( x = 3 \) and \( x = -3 \), the same as the asymptotes. These are not extrema, but discontinuities.

Create the Graph

Now with all this information, create a graph. Mark the intercept at the origin, the vertical asymptotes at \( x = 3 \) and \( x = -3 \), and the horizontal asymptote at \( y = 0 \). Draw the function, making sure to curve toward the asymptotes as needed. The graph should look like a hyperbola, increasing without bound as it approaches the vertical asymptotes, and symmetric about the origin because of the origin symmetry.