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Chapter 1: Problem 119

Given an equation in \(x\) and \(y,\) how do you determine if its graph is symmetric with respect to the \(x\) -axis?

Short Answer

Expert verified
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.

Step by step solution

01

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.
02

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.
03

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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