Problem 77 - Algebra and Trigonometry

Question

\(77-82\) . Test the equation for symmetry. $$ y=x^{4}+x^{2} $$

Step-by-Step Solution

Answer

The equation is symmetric about the y-axis only.

Step 1: Determine Symmetry about the y-axis

We need to test for symmetry about the y-axis by replacing every instance of \(x\) in the equation with \(-x\) and checking if the resulting equation is the same as the original. The given function is \(y = x^4 + x^2\). Substitute \(-x\) into the equation: \[ y = (-x)^4 + (-x)^2 = x^4 + x^2 \]Since the equation remains unchanged, the function is symmetric about the y-axis.

Step 2: Determine Symmetry about the x-axis

Testing for symmetry about the x-axis involves replacing \(y\) with \(-y\) in the equation and checking if the equation remains true. The original function is \(y = x^4 + x^2\). Substituting \(-y\) gives:\[ -y = x^4 + x^2 \]Solving for y, we get \( y = -(x^4 + x^2) \). This equation is not the same as the original, so the function does not have symmetry about the x-axis.

Step 3: Determine Symmetry about the Origin

For symmetry about the origin, we substitute \(x\) with \(-x\) and \(y\) with \(-y\) in the function. Starting from the original function, \(y = x^4 + x^2\):\[ -y = (-x)^4 + (-x)^2 \]Simplifying gives \(-y = x^4 + x^2\), or equivalently \(y = -(x^4 + x^2)\). This is different from the original equation, indicating that there is no origin symmetry.

Step 4: Conclusion: Identify Symmetries

Based on the tests from the previous steps, the function \(y = x^4 + x^2\) has symmetry about the y-axis only. It does not exhibit symmetry about the x-axis or the origin.

Key Concepts

y-axis symmetry

In mathematics, symmetry can help us understand how functions behave. One type of symmetry is y-axis symmetry. This means that if you folded the graph along the y-axis, both halves would match perfectly. To test for this symmetry in an equation, you replace every instance of \(x\) with \(-x\).

Our example is \(y = x^4 + x^2\). Let's see what happens when we substitute \(-x\) for \(x\): \[ y = (-x)^4 + (-x)^2 = x^4 + x^2 \]

The equation remains unchanged, indicating that it is symmetric about the y-axis. In simpler terms, if you look at the graph of this equation, one side mirrors the other across the y-axis.

• This type of symmetry is common in even-power functions.
• It helps predict the behavior of a function without drawing its entire graph.

Recognizing y-axis symmetry can be particularly useful in calculus when evaluating integrals or derivatives.

x-axis symmetry

X-axis symmetry is less common in mathematical equations that describe functions, but it's an important concept in geometry and other areas. It means that the graph of an equation will be the same above and below the x-axis.

To test a function for x-axis symmetry, you replace \(y\) with \(-y\) in the equation and see if the equation remains the same. Consider our equation: \(y = x^4 + x^2\).

Let's check for x-axis symmetry:
Substitute \(-y\) for \(y\): \[-y = x^4 + x^2\] Solving for \(y\), we find \(y = -(x^4 + x^2)\).

This result does not match the original equation, thus the function \(y = x^4 + x^2\) does not have x-axis symmetry.

• X-axis symmetry implies "upward" and "downward" symmetry.
• It is particularly useful when dealing with real-world scenarios that involve reflection over a horizontal plane.

Understanding whether a graph has this symmetry is helpful, though it mainly applies to relations rather than functions.

origin symmetry

Origin symmetry occurs when the graph of an equation is unchanged when flipped over both the x-axis and y-axis simultaneously.

This type of symmetry is tested by substituting \(-x\) for \(x\) and \(-y\) for \(y\). Our equation is \(y = x^4 + x^2\).
Let's check for origin symmetry:
Substitute \(-x\) for \(x\) and \(-y\) for \(y\):
\[-y = (-x)^4 + (-x)^2\] Simplifying gives:
\[-y = x^4 + x^2\], or equivalently \(y = -(x^4 + x^2)\).
This differs from the original equations, so there is no origin symmetry here.

• This symmetry is typically seen in graphs of odd-power polynomials (like \(y = x^3\)).
• It allows predicting function behavior across all four quadrants of the plane.

Origin symmetry helps in solving equations involving rotational symmetry transforming points \((x, y)\) to \((-x, -y)\).