Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Check for symmetry with respect to both axes and the origin. \(x^{4}-2 y=0\)

Step-by-Step Solution

Answer

The given function \(x^{4}-2 y=0\) is not symmetric about the x-axis or the origin, but it is symmetric about the y-axis.

Step 1: Checking Symmetry about x-axis

In order to check the symmetry with respect to the x-axis, replace every y in the equation with -y, the resulting equation is \( x^{4} - 2 (-y) = x^{4} + 2y = 0 \). Since the original equation and the new equation are not the same, the graph is not symmetric about the x-axis.

Step 2: Checking Symmetry about y-axis

In order to check the symmetry with respect to the y-axis, replace every x in the equation with -x, the resulting equation is \( (-x)^{4}-2 y = x^{4}-2 y=0 \). The original equation and the new equation are the same, so the graph is symmetric about the y-axis.

Step 3: Checking Symmetry about the origin

To check for symmetry about the origin, replace both x with -x and y with -y, the resulting equation is \((-x)^{4}-2 (-y) = x^{4} +2y \). Since the original equation and the new equation are not the same, the graph is not symmetric about the origin.

Key Concepts

Symmetry with respect to x-axis

Symmetry with respect to the x-axis means that if you fold the graph along the x-axis, both halves of the graph would match up perfectly.
To test this symmetry, replace each variable 'y' with '-y' in the equation and simplify.
In our example, starting with the equation \( x^4 - 2y = 0 \), we change 'y' to '-y'.
That gives us \( x^4 + 2y = 0 \).
This is not the same as the original equation, \( x^4 - 2y = 0 \).

• Conclusion: The graph is not symmetrical with respect to the x-axis.

Whenever the new equation matches the original, it confirms symmetry about the x-axis.
In this case, it does not match, meaning the graph does not have this type of symmetry.

Symmetry with respect to y-axis

Checking symmetry with respect to the y-axis involves folding the graph along the y-axis and seeing if it aligns.
For the y-axis, replace every 'x' in the equation with '-x'.
Given our equation \( x^4 - 2y = 0 \), replace 'x' with '-x' to get \( (-x)^4 - 2y = x^4 - 2y = 0 \).

Notice how \( (-x)^4 \) simplifies back to \( x^4 \) since raising \(-x\) to an even power (like 4) cancels the negative sign.
This yields exactly the same equation meaning:

• Conclusion: The graph is symmetrical with respect to the y-axis.

If after substituting and simplifying, the equation remains unchanged, the graph has y-axis symmetry.
This implies that the pattern or shape on one side of the y-axis is the mirror reflection on the other side.

Origin symmetry

For origin symmetry, imagine rotating the graph 180 degrees around the origin.
To test this, replace 'x' with '-x' and 'y' with '-y' in the equation.
Starting with \( x^4 - 2y = 0 \), replacing gives \( (-x)^4 - 2(-y) = x^4 + 2y = 0 \).
This does not replicate the original equation.

• Conclusion: The graph is not symmetrical with respect to the origin.

Origin symmetry exhibits a specific rotational balance where after spinning the graph halfway (180 degrees), it looks identical to its original position.
Since our derived equation is different from the original, this specific type of symmetry is not present.