Problem 17 - 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. $$y=x^{2}+6$$

Step-by-Step Solution

Answer

The graph of the equation \(y = x^{2} + 6\) is symmetric with respect to the y-axis, but it is not symmetric with respect to the x-axis or the origin.

Step 1: Check Symmetry with respect to Y-Axis

The equation \(y = x^{2}+6\) becomes \(y = (-x)^{2}+6\) when \(x\) is replaced with \(-x\). This simplifies to \(y = x^{2}+6\), which is the original equation, so the graph is symmetric with respect to the y-axis.

Step 2: Check Symmetry with respect to X-Axis

Replace \(y\) in the equation \(y = x^{2}+6\) with \(-y\). This gives \(-y = x^{2}+6\), which is not identical to the original equation after simplifying, so the graph is not symmetric with respect to the x-axis.

Step 3: Check Symmetry with respect to the Origin

Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation \(y = x^{2}+6\). This results in \(-y = (-x)^{2}+6\), which simplifies to \(-y = x^{2}+6\). This is not identical to the original equation, so the graph is not symmetric with respect to the origin.