Question

In Exercises 39-42, use a computer algebra system to graph the function. $$ z=y^{2}-x^{2}+1 $$

Short answer

The graph of \(z = y^{2} - x^{2} + 1\) results in a saddle-shaped surface, where the graph shows a minimum along the y-axis and a maximum along the x-axis, while curving down away from the axes.

Step-by-step solution

Software translation

Initially, the function needs to be correctly inputted into the software. Depending on the software used, this may require specific syntax. However, in general, the formula can be directly inputted as \(z = y^{2} - x^{2} + 1\) into the algebra system.

3D plotting

As we're dealing with a function of two variables (x and y), this will lead to a 3D plot. Ensure that the software is set to produce a 3-dimensional graphical output. Learn the exact way of how the software is managing multivariable functions.

Graph generation

After correctly setting up your function and ensuring the software is set to produce a 3-dimensional plot, generate the graph. The resulting 3D graph should represent how z varies according to the combined variations of x and y.

Graph interpretation

The graph of this function will show a saddle point at (0,0), meaning there are minima and maxima on two different axes. All points not on the axes will curve down, showing that the function reduces away from the x-y plane.