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Chapter 1: Problem 110

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

Short Answer

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

Step by step solution

01

Identify Symmetry Options

To test symmetry, we can check for three types: symmetry about the y-axis, symmetry about the x-axis, and symmetry about the origin. We need to assess each one.
02

Test for Y-axis Symmetry

For a function to have y-axis symmetry, replacing \(x\) with \(-x\) should yield the original function. Substitute \(-x\) into \(y = x^2 + |x|\) and simplify: \[ y = (-x)^2 + |-x| = x^2 + |x| \]The equation is identical, indicating that the function is symmetric with respect to the y-axis.
03

Test for X-axis Symmetry

For x-axis symmetry, replacing \(y\) with \(-y\) should yield the original x terms, which means after substitution, the equation should hold true for any values of \(x\) if it is symmetric with respect to the x-axis. Substitute \(-y\) into the equation:\[ -y = x^2 + |x| \]There's no simplification that will satisfy this equation for all \(x\), indicating that there is no x-axis symmetry.
04

Test for Origin Symmetry

To test origin symmetry, substitute both \(y\) and \(x\) with their negatives: \(-y = (-x)^2 + |-x| \). Simplify:\[-y = x^2 + |x| \]The negative sign in front of \(y\) prevents this equation from being equal to the original \(y = x^2 + |x|\), so it does not have origin symmetry.

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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 in an equation means that if you were to fold the graph along the y-axis, both halves would match perfectly. Mathematically, to determine if a function is symmetric about the y-axis, you replace every instance of \(x\) with \(-x\) in the equation. If the modified equation looks exactly like the original, then the function has y-axis symmetry.
  • Example check: If we take our given equation, \( y = x^2 + |x| \), and replace \(x\) with \(-x\), we get:
  • \[ y = (-x)^2 + |-x| \]
  • Simplified, this becomes \(y = x^2 + |x|\), which matches the original equation.
So, the equation \(y = x^2 + |x|\) is symmetric about the y-axis. This means every point \((x,y)\) on the graph has a corresponding point \((-x,y)\). Think of it graphically as mirror image symmetry across the y-axis.
Exploring X-axis Symmetry
X-axis symmetry is about reflection over the x-axis, which means that for every point \((x, y)\) we have a point \((x, -y)\). In terms of equations, symmetry about the x-axis means substituting \(y\) with \(-y\) should yield back the same x-expression or function.
  • Let's test our equation \(y = x^2 + |x|\).
  • By replacing \(y\) with \(-y\), we get:
  • \[ -y = x^2 + |x| \]
  • This equation does not simplify back to the original equation \(y = x^2 + |x|\).
Therefore, this function does not exhibit x-axis symmetry. Visually, this means that folding the graph over the x-axis will not perfectly align both sides.
Investigating Origin Symmetry
Origin symmetry implies that the graph would look the same if rotated 180 degrees about the origin (0,0). For a function to demonstrate origin symmetry, replacing \(x\) with \(-x\) and \(y\) with \(-y\) should give an equivalent to the original equation.
  • For our equation \(y = x^2 + |x|\), testing origin symmetry involves:
  • Substitute \(x\) with \(-x\) and \(y\) with \(-y\):
  • \[ -y = (-x)^2 + |-x| \]
  • This simplifies to \(-y = x^2 + |x|\), which does not match the original \(y = x^2 + |x|\).
As a result, the function lacks origin symmetry. This means if you rotate the graph around the origin, it will not match up with itself.

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