Problem 19
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. $$x=y^{2}+6$$
Step-by-Step Solution
Verified Answer
The graph of the equation \(x=y^{2}+6\) is symmetric only with respect to the x-axis.
1Step 1: Check for Symmetry with respect to the Y-axis
Replace \(x\) with \(-x\) in the equation \(x=y^{2}+6\), the new equation becomes \(-x = y^{2} + 6\), which is not the same as the original. Therefore, the graph of the equation is not symmetric with respect to the y-axis.
2Step 2: Check for Symmetry with respect to the X-axis
Replace \(y\) with \(-y\) in the equation \(x=y^{2}+6\), the equation becomes \(x=(-y)^{2}+6\). After simplifying, we get \(x = y^{2} + 6\), which is the same as the original equation. Therefore, the graph of the equation is symmetric with respect to the x-axis.
3Step 3: Check for Symmetry with respect to the Origin
Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation \(x=y^{2}+6\), the new equation becomes \(-x = (-y)^{2} + 6\), which is not the same as the original equation. Therefore, the graph of the equation is not symmetric with respect to the origin.
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