Problem 119
Question
Given an equation in \(x\) and \(y,\) how do you determine if its graph is symmetric with respect to the \(x\) -axis?
Step-by-Step Solution
Verified Answer
To determine the graph's symmetry with respect to the x-axis for a given equation in \(x\) and \(y,\) substitute \(y\) for \(-y\). If the resulting equation is identical to the original one, this indicates that the graph of the equation is symmetric with respect to the x-axis.
1Step 1: Understand the Symmetry with Respect to x-Axis
To check if the given equation of a graph is symmetrical with respect to the x-axis, one must understand that a function is symmetric to the x-axis when every point (x,y) on the graph has its mirror image point (x,-y) on the graph as well.
2Step 2: Apply the Rule to Check Symmetry
To check the symmetry, substitute '-y' in place of 'y' in the given equation. If after simplifying, the modified equation is identical to the initial equation, it can be said that the graph of the equation is symmetric to the x-axis.
3Step 3: Cross Verification
While it's enough to prove mathematically, visualizing it using graph plotting can also help in understanding the concept better. Plot the original equation and see if the points bear the mirroring property according to the x-axis.
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