Question

Test the equation $y=5x^2-1$for symmetry with respect to the x-axis, the y-axis, and the origin.

Short answer

The graph of the equation is not symmetric with respect to the x-axis.

The graph of the equation is symmetric with respect to the y-axis.

The graph of the equation is not symmetric with respect to the origin.

Step-by-step solution

Given Information 

Test the equation $y=5x^2-1$for symmetry with respect to the x-axis, the y-axis, and the origin.

 x axis symmetry 

To test for symmetry with respect to the x-axis, replace y by -y

$$\begin{gathered} y=5x^2-1 \\ -y=5x^2-1isnotequivalenttoy=5x^2-1 \end{gathered}$$

The graph of the equation is not symmetric with respect to the x-axis.

y-axis symmetry 

To test for symmetry with respect to the y-axis, replace x by -x.

$$\begin{gathered} y=5x^2-1 \\ Replacexwith-x \\ y=5{(-x)}^2-1y=5x^2-1isequivalenttoy=5x^2-1 \end{gathered}$$

The graph of the equation is symmetric with respect to the y-axis.

symmetric about origin

To test for symmetry with respect to the origin, replace x by -x and y by -y

$$\begin{gathered} y=5x^2-1 \\ Replacexwith-xandywith-y \\ -y=5{(-x)}^2-1-y=5x^2-1isnotequivalenttoy=5x^2-1 \end{gathered}$$

The graph of the equation is not symmetric with respect to the origin.