Problem 49 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The function is symmetric with respect to the origin.

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.

Key Concepts

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’s 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.