Question 2.7.4: Testing for Symmetry with Respect to the Origin Determine wh...

(a) Replace x with -x and y with -y.

\left.\begin{matrix} x²+ y2 = 16 \\ \\ (-x)² + (-y)²= 16\\ \\ x² + y² = 16\end{matrix} \right\} \text{Equivalent}

The graph, which is the circle shown in Figure 77 in Example 3(c), is symmetric with respect to the origin.

(b) Replace x with -x and y with -y.

\left.\begin{matrix} y= x³\\ \\  -y = (-x)³ \\  \\-y = -x³ \\  \\y= x³ \end{matrix} \right\} \text{Equivalent}

The graph, which is that of the cubing function, is symmetric with respect to the origin and is shown in Figure 80.