Question

In Exercises \(11-18,\) identify and sketch the quadric surface. Use a computer algebra system to confirm your sketch. $$ 3 z=-y^{2}+x^{2} $$

Short answer

The given equation represents a hyperbolic paraboloid. The sketch would display a saddle shape that opens along the x-axis and closes along the y-axis.

Step-by-step solution

Identify the type of quadric surface

Looking at the equation, \(3z = -y^{2} + x^{2}\), it corresponds to the standard form of a hyperbolic paraboloid, which is usually written as \(z = ax^{2} - by^{2}\) or \(z = -ax^{2} + by^{2}\). In this particular equation, \(a = 1/3\) and \(b = -1/3\). The signs for the square terms in the equation determine that it opens up along the x-axis and down along the y-axis, hence forming a saddle shape.

Setup the sketch

To set up the sketch, create a three-dimensional \(x\), \(y\), and \(z\) grid. Remove the coefficient \(3\) from \(z\) by dividing the entire equation by \(3\), which leads to \(z = \frac{1}{3}x^{2} - \frac{1}{3}y^{2}\). Mark some points where the surface intersects the axes. This would be where \(x = 0\), \(y = 0\), and \(z = 0\). Hint: This hyperbolic paraboloid intersects the \(z\) axis at 0 and opens up/down around it.

Sketch the quadric surface

Draw the hyperbolic paraboloid. As identified before, this figure should appear as a saddle shape that opens along the x-axis and down along the y-axis. It is symmetric with respect to the \(z\) axis.

Use a computer algebra system

As an optional step, using a computer algebra system or graphing calculator to plot the function can confirm whether the sketch is accurate or not. Plot the equation \(z = \frac{1}{3}x^{2} - \frac{1}{3}y^{2}\) to confirm.