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Chapter 2: Problem 49

45鈥�50 ? Test the equation for symmetry. $$ y=x^{3}+10 x $$

Short Answer

Expert verified
The function is symmetric with respect to the origin.

Step by step solution

01

Identify Symmetry Types

To test the equation for symmetry, we check for three types: symmetry with respect to the x-axis, y-axis, and the origin. Let's identify which type of symmetry we will test.
02

Symmetry with Respect to the x-axis

Replace all instances of \( y \) with \( -y \) in the equation. The equation becomes:\[ -y = x^3 + 10x \]Since this equation is not the same as the original equation, the function is not symmetric with respect to the x-axis.
03

Symmetry with Respect to the y-axis

Replace all instances of \( x \) with \( -x \). The equation becomes:\[ y = (-x)^3 + 10(-x) \]Simplifying this gives:\[ y = -x^3 - 10x \]The derived equation is not identical to the original equation, indicating no y-axis symmetry.
04

Symmetry with Respect to the Origin

To test for origin symmetry, replace both \( x \) with \( -x \) and \( y \) with \( -y \). This yields:\[ -y = (-x)^3 + 10(-x) \]Simplifying gives:\[ -y = -x^3 - 10x \]or, equivalently,\[ y = x^3 + 10x \]Since the resulting equation is the same as the original one, the function exhibits symmetry with respect to the origin.

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Key Concepts

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

Symmetry with Respect to the X-Axis
In algebra, determining if a function or equation is symmetric with respect to the x-axis involves substituting every occurrence of the variable \( y \) with \( -y \). If the resulting equation is equivalent to the original, it indicates x-axis symmetry.

Let鈥檚 consider the equation \( y = x^3 + 10x \).
  • Replace \( y \) with \( -y \) to get \( -y = x^3 + 10x \).
  • Observe the new equation.
  • If it mirrors the original equation, symmetry is confirmed.
For this case, the modified equation \( -y = x^3 + 10x \) is not the same as \( y = x^3 + 10x \) when all signs are considered. Therefore, the equation does not exhibit symmetry with respect to the x-axis.
Symmetry with Respect to the Y-Axis
Testing for symmetry with respect to the y-axis involves replacing the variable \( x \) with \( -x \) in the equation. After substitution, if the new equation is identical to the original one, then it reveals y-axis symmetry.

Examining the function \( y = x^3 + 10x \):
  • Replace \( x \) with \( -x \) to transform the equation into \( y = (-x)^3 + 10(-x) \).
  • Simplify this to \( y = -x^3 - 10x \).
  • Compare it with the original equation.
In this case, \( y = -x^3 - 10x \) differs from \( y = x^3 + 10x \). Therefore, the equation does not show symmetry with respect to the y-axis.
Symmetry with Respect to the Origin
A function demonstrating symmetry with respect to the origin will have the same expression when both \( x \) and \( y \) are replaced by \( -x \) and \( -y \), respectively. This test checks if \( f(-x, -y) = f(x, y) \).

Consider our equation: \( y = x^3 + 10x \).
  • Substitute \( x \) with \( -x \) and \( y \) with \( -y \) yielding \( -y = (-x)^3 + 10(-x) \).
  • Simplify it to \( -y = -x^3 - 10x \).
  • Rearrange it to \( y = x^3 + 10x \).
Notice that the adjusted equation, \( y = x^3 + 10x \), matches the original equation. Thus, this confirms the equation has symmetry with respect to the origin.

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