Questions: Use the graph of each equation to test for symmetry with respect to the x-axis, y-axis, and the origin. Support the answer numerically. Then confirm algebraically.

Use the graph of each equation to test for symmetry with respect to the x-axis, y-axis, and the origin. Support the answer numerically. Then confirm algebraically.

Transcript text: Use the graph of each equation to test for symmetry with respect to the $x$-axis, $y$-axis, and the origin. Support the answer numerically. Then confirm algebraically.

Solution

Solution Steps

Step 1: Identify the given equation

The given equation is $ y = \frac{-2}{x} $.

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

To test for symmetry with respect to the x-axis, replace $ y $ with $ -y $ in the equation and see if the equation remains unchanged. \[ -y = \frac{-2}{x} \] This simplifies to: \[ y = \frac{2}{x} \] Since $ y = \frac{2}{x} $ is not the same as the original equation $ y = \frac{-2}{x} $, the graph is not symmetric with respect to the x-axis.

Step 3: Test for symmetry with respect to the y-axis

To test for symmetry with respect to the y-axis, replace $ x $ with $ -x $ in the equation and see if the equation remains unchanged. \[ y = \frac{-2}{-x} \] This simplifies to: \[ y = \frac{2}{x} \] Since $ y = \frac{2}{x} $ is not the same as the original equation $ y = \frac{-2}{x} $, the graph is not symmetric with respect to the y-axis.

Step 4: Test for symmetry with respect to the origin

To test for symmetry with respect to the origin, replace $ x $ with $ -x $ and $ y $ with $ -y $ in the equation and see if the equation remains unchanged. \[ -y = \frac{-2}{-x} \] This simplifies to: \[ -y = \frac{2}{x} \] Multiplying both sides by -1: \[ y = \frac{-2}{x} \] Since this is the same as the original equation $ y = \frac{-2}{x} $, the graph is symmetric with respect to the origin.