Problem 24 - 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. $$y=2 x+5$$

Step-by-Step Solution

Answer

The graph of the equation \(y=2x+5\) is not symmetric with respect to the \(y\)-axis, the \(x\)-axis or the origin.

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

Symmetry with respect to the \(y\)-axis happens if substituting \(x\) with \(-x\) in equation results in the same equation. That is, replace \(x\) with \(-x\). So, \(y=2(-x) + 5\) which simplifies to \(y=-2x + 5\). This equation is not the same as the original. Hence, the graph is not symmetric with respect to the \(y\)-axis.

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

Symmetry with respect to the \(x\)-axis occurs if replacing \(y\) with \(-y\) in equation yields the same equation. Hence, replace \(y\) with \(-y\). We get \(-y = 2x + 5\), which simplifies to \(y=-2x-5\). This equation is not the same as the original, therefore the graph is not symmetric with respect to the \(x\)-axis.

Step 3: Test for symmetry with respect to the origin

Symmetry with respect to the origin happens if substituting \(x\) with \(-x\) and \(y\) with \(-y\) in equation results in the original equation. Therefore, we substitute \(x\) with \(-x\) and \(y\) with \(-y\) in the given equation. Thus, \(-y=2(-x) + 5\) which simplifies to \(-y=-2x + 5\) or \(y=2x - 5\). This equation is not the same as the original. Hence, the graph is not symmetric with respect to the origin.