Problem 18 - Precalculus Essentials

Question

Determine whether the graph of each equation is symmetric with respect to the \(y\) -axis, the \(x\) -axis, the origin, more than one of these, or none of these. $$y=x^{2}-2$$

Step-by-Step Solution

Answer

The graph of the given equation \(y=x^{2}-2\) is symmetric with respect to the y-axis.

Step 1: Test for symmetry about the y-axis

To determine if a graph is symmetric about the y-axis, we replace \(x\) with \(-x\) in the equation. If we get the original equation, then it is symmetric about the y-axis. In the given equation, replace \(x\) with \(-x\) => \(y=(-x)^{2}-2 => y=x^{2}-2\). As we can see, we obtained the original equation. So, the graph is symmetric about the y-axis.

Step 2: Test for symmetry about the x-axis

To determine if a graph is symmetric about the x-axis, we replace \(y\) with \(-y\) in the equation. If we get the original equation, then it is symmetric about the x-axis. In the given equation, replace \(y\) with \(-y\) => \(-y = x^{2}-2\). We didn't get back the original equation, so it's not symmetric about the x-axis.

Step 3: Test for symmetry about the origin

To determine if a graph is symmetric about the origin, we replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation. If we get the original equation, then it is symmetric about the origin. In the given equation, replace \(x\) with \(-x\) and \(y\) with \(-y\) => \(-y = (-x)^{2} - 2 = x^{2} - 2 \). We didn't get back the original equation, so it's not symmetric about the origin.