Question 1.3.3: Testing for Symmetry Determine whether the graph of y = x³ i...
Testing for Symmetry
Determine whether the graph of y = x³ is symmetric with respect to the y-axis, the x-axis, or the origin.
Test for symmetry with respect to the y-axis. Replace x with –x and see if this results in an equivalent equation.
\begin{matrix} \text{This is the given equation.}\\\text{Replace x with -x.}\\\text{(-x)³ = (-x)(-x)(-x) = -x³}\end{matrix}
Replacing x with –x does not give the original equation. Thus, the graph of y = x³ is not symmetric with respect to the y-axis.
Test for symmetry with respect to the x-axis. Replace y with –y and see if this results in an equivalent equation.
\begin{matrix} \text{This is the given equation.}\\\text{Replace y with -y.}\\\text{Multiply both sides by -1 and solve for y.}\end{matrix}
Replacing y with –y does not give the original equation. Thus, the graph of y = x³ is not symmetric with respect to the x-axis.
Test for symmetry with respect to the origin. Replace x with –x and y with –y and see if this results in an equivalent equation.
\begin{matrix} \text{This is the given equation.}\\\text{Replace x with -x and y with -y.}\\\text{(-x)³ = (-x)(-x)(-x) = -x³}\\\text{Multiply both sides by -1 and solve for y.}\end{matrix}
Replacing x with –x and y with –y gives the original equation. Thus, the graph of y = x³ is symmetric with respect to the origin.
Our symmetry tests reveal that the graph of y = x³ is symmetric with respect to the origin only. The symmetry is shown in Figure 1.33.
