Problem 120
Question
Given an equation in \(x\) and \(y,\) how do you determine if its graph is symmetric with respect to the origin?
Step-by-Step Solution
Verified Answer
To determine if a given equation in \(x\) and \(y\) is symmetric with respect to the origin, a check is made to see if the equation remains the same when both \(x\) and \(y\) are replaced by their negatives \(-x\) and \(-y\). If it does, then it is symmetric about the origin.
1Step 1: Understand Symmetry About The Origin
A graph is said to be symmetric about the origin, if rotating it by 180 degrees about the origin leaves the graph unchanged. Meaning, replace every point \((x, y)\) in the graph with point \((-x, -y)\) and the graph remains unchanged.
2Step 2: Apply The Test For Symmetry About The Origin
Assign the given equation and replace \(x\) by \(-x\) and \(y\) by \(-y\). If the resultant equation is equivalent to the original equation, it is symmetric about the origin.
3Step 3: Verify The Symmetry
Check if the transformed equation (after substituting \(x\) with \(-x\) and \(y\) with \(-y\)) is the same as the original equation. If they match, then the graph is symmetrical about the origin.
Other exercises in this chapter
Problem 119
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