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. $$ x y^{2}=4 $$

Short answer

The graph for the equation \(x y^{2} = 4\) is composed of two curves. One curve is in the first quadrant which represents the equation \(y = \sqrt{\frac{4}{x}}\), and another curve is in the fourth quadrant representing \(y = -\sqrt{\frac{4}{x}}\). Both curves approach the y-axis but don't touch it, which makes x=0 a vertical asymptote for the graph.

Step-by-step solution

Rearrange the Equation

Firstly, rearrange the equation to a more familiar form. It's a good approach to rewrite the given equation, \(x y^{2} = 4\), in terms of \(y\). That means we need to solve the equation for \(y\). To do this, divide both sides of the equation by \(x\) to isolate \(y^{2}\) on one side, then take square root on both sides to solve for \(y\). It will give us two solutions: \[y = \sqrt{\frac{4}{x}}; y = -\sqrt{\frac{4}{x}}\] So, we get two equivalent equations.

Identify Key Properties

For the first equation \(y = \sqrt{\frac{4}{x}}\), the graph will only exist when \(x > 0\), since we cannot take the square root of a negative number. The graph of \(y = \sqrt{\frac{4}{x}}\) is symmetric in the first quadrant (because \(y\) is positive). For the second equation \(y = -\sqrt{\frac{4}{x}}\), the graph will only exist when \(x > 0\), for the same reason. The graph of \(y = -\sqrt{\frac{4}{x}}\) is symmetric in the fourth quadrant (because \(y\) is negative).There are no x-intercepts because for any value of \(x\), \(y\) will not equal 0. The y-intercepts do not exist because the graph is undefined at \(x = 0\). There are no extrema because there are no local maximum or minimum values, given the indefinite nature of the function on its defined interval. The vertical asymptote is \(x = 0\), meaning the graph approaches but never touches the y-axis.

Sketch the Graph

Now that we have an understanding of the behavior of the graph, we can sketch it. Start by drawing the x and y axes. Mark the known points and the vertical asymptote. Draw a curve in the first quadrant for the first equation \(y = \sqrt{\frac{4}{x}}\) and another curve in the fourth quadrant for the second equation \(y = -\sqrt{\frac{4}{x}}\). Remember, both graphs should approach but never touch the y-axis.

Verify with a Graphing Utility

Finally, as per the problem requirements validate the sketch with a graphic calculator or an online graphing tool. It serves to ensure the sketch is accurate and to identify any possible errors.