Question 1.3.2: Testing for Symmetry Determine whether the graph of x = y² -...
Testing for Symmetry
Determine whether the graph of
x = y² – 1
is symmetric with respect to the y-axis, the x-axis, or the origin.
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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{Multiply both sides by -1 and solve for x.} \end{matrix}
Replacing x with –x does not give the original equation. Thus, the graph of
x = y² – 1 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{(-y)² = (-y)(-y) = y²}\end{matrix}
Replacing y with –y gives the original equation. Thus, the graph of x = y² – 1 is 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{(-y)² = (-y)(-y) = y²}\\\text{Multiply both sides by -1 and solve for x.}\end{matrix}
Replacing x with –x and y with –y does not give the original equation. Thus, the graph of x = y² – 1 is not symmetric with respect to the origin.
Our symmetry tests reveal that the graph of x = y² – 1 is symmetric with respect to the x-axis only