Problem 23 - 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. $$y=2 x+3$$

Step-by-Step Solution

Answer

The graph of the equation \(y=2x+3\) is not symmetric with respect to the x-axis, the y-axis, or the origin.

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

Replace \(x\) with \(-x\) in the equation, if the equation remains the same then it is symmetric with respect to the y-axis. Replacing \(x\) with \(-x\) gives: \(y=-2x+3\). Thus, the equation is not symmetric about the y-axis because the resulting equation is different from the original.

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

Replace \(y\) with \(-y\) in the equation, if the equation remains the same then it is symmetric with respect to the x-axis. Replacing \(y\) with \(-y\) gives: \(-y=2x+3\). Thus, the equation is not symmetric about the x-axis because the resulting equation is different from the original.

Step 3: Test for symmetry with respect to the origin

Replace both \(x\) and \(y\) with \(-x\) and \(-y\) respectively in the equation, if the equation remains the same then it is symmetric with respect to the origin. Replacing \(x\) with \(-x\) and \(y\) with \(-y\) gives: \(-y=-2x-3\). Thus, the equation is not symmetric about the origin because the resulting equation is different from the original.