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

Short answer

The graph of the function \(y=\frac{x}{\sqrt{x^{2}-4}}\) has no intercepts, vertical asymptotes at \(x=-2\) and \(x=2\), and is symmetric about the origin. There are no extrema for this function.

Step-by-step solution

Calculate the intercepts

Intercepts are the points where the graph intersects the x and y axes. For the y-intercept, substitute \(x = 0\) in the equation; and for x-intercept, substitute \(y = 0\). As observed in the given function, one cannot substitute \(x = 0\) as it leads to the square root of a negative number. So, no y-intercept exists. Substituting \(y = 0 \), we find that x=0 is not a solution so again no x-intercept exists.

Determine the asymptotes

Vertical asymptotes occur at the values of \(x\) that make the denominator equal to zero without affecting the numerator. Setting the denominator equal to zero to find such values, \(\sqrt{x^{2}-4}=0\), gives us \(x^{2}=4\). The square root of 4 in \(x^{2}=4\) can be +2 or -2. Therefore, the given function has vertical asymptotes at \(x=-2\) and \(x=2\).

Analyze for symmetry

The function is symmetric if it's even function \(f(x) =f(-x)\) or odd function \(f(x) =-f(-x)\). Here, replacing \(x\) with \(-x\) we obtain \(-f(x)\) indicating it's an odd function and symmetric about the origin.

Identify the extrema

For rational functions, extrema can be either local (occurring at a particular interval) or global (highest or lowest point of the entire function). But for this function, the critical points where extrema can occur are not defined.

Sketch the graph

The x-axis consists of vertical asymptotes at \(x=-2\) and \(x=2\), with intercepts at none. The function is symmetric about the origin. Thus, the function on one side of the y-axis is just a reflection of the other side. Also, the graph will approach (but never cross) the vertical asymptotes.