Problem 27 - 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. $$x^{2}+y^{2}=100$$

Step-by-Step Solution

Answer

The graph of the equation \(x^2 + y^2 = 100\) is symmetric with respect to the y-axis, the x-axis, and the origin.

Step 1: Test symmetry with respect to the y-axis

To test for symmetry in relation to the y-axis, \(x\) in the equation can be replaced with \(-x\). If the resulting equation is identical to the original equation, it is symmetric in relation to the y-axis. The resulting equation: \[(-x)^2 + y^2 = 100\]\[\rightarrow x^2 + y^2 = 100\] This is identical to the original equation, therefore the graph of the equation is symmetric in relation to the y-axis.

Step 2: Test symmetry with respect to the x-axis

To test for symmetry in relation to the x-axis, \(y\) in the equation should be replaced with \(-y\). If the resulting equation is identical to the original equation, then it is symmetric in relation to the x-axis. The resulting equation: \[x^2 + (-y)^2 = 100\]\[\rightarrow x^2 + y^2 = 100\] This is identical to the original equation, therefore the graph of the equation is symmetric in relation to the x-axis.

Step 3: Test symmetry with respect to the origin

To test for symmetry in relation to the origin, both \(x\) and \(y\) in the equation should be replaced with \(-x\) and \(-y\), respectively. If the resulting equation is identical to the original equation, then it is symmetric in relation to the origin. The resulting equation: \[(-x)^2 + (-y)^2 = 100\]\[\rightarrow x^2 + y^2 = 100\] This is identical to the original equation, therefore the graph of the equation is symmetric in relation to the origin.