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Chapter 1: Problem 82

Test the equation for symmetry. $$y=x^{2}+|x|$$

Short Answer

Expert verified
The equation is symmetric about the y-axis.

Step by step solution

01

Test for Symmetry About the Y-Axis

To test symmetry about the y-axis, replace \(x\) with \(-x\) in the equation and simplify:\[ y = (-x)^2 + |-x| \]Simplifying gives:\[ y = x^2 + |x| \]Since the equation remains the same, the equation is symmetric about the y-axis.
02

Test for Symmetry About the X-Axis

To test symmetry about the x-axis, replace \(y\) with \(-y\) in the equation:\[ -y = x^2 + |x| \]This does not simplify to the original equation as there are no values other than zero for which both sides are equal, so the equation is not symmetric about the x-axis.
03

Test for Symmetry About the Origin

To test symmetry about the origin, replace \(x\) with \(-x\) and \(y\) with \(-y\):\[ -y = (-x)^2 + |-x| \]Simplifying gives:\[ -y = x^2 + |x| \]As this is not the original equation, the equation is not symmetric about the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding y-axis symmetry
Y-axis symmetry occurs when a graph appears the same on both sides of the y-axis. For equations, this characteristic is confirmed if replacing \(x\) with \(-x\) does not change the equation. In mathematical terms, for an equation \(y = f(x)\), the condition for y-axis symmetry is \(f(x) = f(-x)\).
In this exercise, we check if the equation \(y = x^2 + |x|\) is symmetric about the y-axis. By substituting \(-x\) for \(x\), we get:
  • Start with: \(y = (-x)^2 + |-x|\)
  • Simplify: \(y = x^2 + |x|\)
Since the equation remains unchanged, it is symmetric about the y-axis. You can think of it as folding the graph along the y-axis without any mismatch between the two halves.
Exploring x-axis symmetry
An equation has x-axis symmetry if its graph looks identical above and below the x-axis. To check this mathematically, replace \(y\) with \(-y\) in the equation. If the original equation is restored after simplification, then the equation is symmetric with respect to the x-axis. For instance, if the equation is \(y = f(x)\), x-axis symmetry exists if \(-f(x) = f(x)\).
Our task is to evaluate if \(y = x^2 + |x|\) exhibits x-axis symmetry. We perform the following steps:
  • Start by substituting: \(-y = x^2 + |x|\)
  • Notice there is no transformation that returns the equation to its original form unless \(y = 0\)
Because the equation doesn't simplify to the original, the graph lacks x-axis symmetry. This illustrates that the shape of the graph cannot be mirrored perfectly across the x-axis.
Delving into origin symmetry
Origin symmetry suggests that a function looks the same if rotated 180 degrees around the origin. In terms of equations, this property demands that replacing \(x\) with \(-x\) and \(y\) with \(-y\) results in the same equation. Mathematically, an equation \(y = f(x)\) has origin symmetry if \(-f(-x) = f(x)\).
For the equation \(y = x^2 + |x|\), we need to test for origin symmetry:
  • First, substitute: \(-y = (-x)^2 + |-x|\)
  • Simplify to: \(-y = x^2 + |x|\)
As the transformed equation is different from the original, the graph does not exhibit symmetry about the origin. This tells us that if you rotate the entire graph 180 degrees, it won't look like the original.

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