Question

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=\frac{x}{\sqrt{x^{2}-4}}\)

Short answer

The graph of the given function \(y=\frac{x}{\sqrt{x^{2}-4}}\) can be plotted using a graphing utility. The function is undefined when \(x = -2\) and \(x = 2\). By choosing an appropriate window, all the relative extrema and points of inflection can be identified visually from the graph.

Step-by-step solution

Identify the type of function

The provided function, \(y=\frac{x}{\sqrt{x^{2}-4}}\), is a rational function where the numerator is a linear function and the denominator is a square root function.

Determine the domain of function

The domain of this function is all real numbers except those that make the denominator equal to zero or negative. In this case, \(x^{2}-4\) must be greater than 0, so \(x<-2\) or \(x>2\). The function is undefined at \(x = -2\) and \(x = 2\).

Graph the function

Use the graphing utility to graph the function \(y=\frac{x}{\sqrt{x^{2}-4}}\). Ensure that the window size is chosen such that it captures all relative extrema and points of inflection. Typically a window of x = [-5, 5] and y = [-5, 5] can provide a useful initial view point.

Identify the relative extrema and points of inflection

Look at the graph. Relative extrema are the highest or lowest points (peaks or valleys) on the graph. Points of inflection are where the graph changes from a curve up to a curve down, or vice versa. Identify these points on the graph.