Problem 98 - Precalculus Essentials

Question

Determine whether the graph of \(x^{2}-y^{3}=2\) is symmetric with respect to the \(y\) -axis, the \(x\) -axis, the origin, more than one of these, or none of these. (Section \(1.3,\) Examples 2 and 3)

Step-by-Step Solution

Answer

The graph of the equation \(x^{2}-y^{3}=2\) is symmetric with respect to the y-axis only.

Step 1: Check for symmetry with respect to y-axis

A graph is symmetric about the y-axis if replacing x with -x in the equation, results in the same equation. Here, replace \(x\) with \(-x\) in the equation to get \((-x)^{2}-y^{3}=2\), which simplifies to \(x^{2}-y^{3}=2\). Since it is the same as the original equation, the graph is symmetric about the y-axis.

Step 2: Check for symmetry with respect to x-axis

A graph is symmetric about the x-axis if replacing y with -y in the equation, results in the same equation. Here, replace \(y\) with \(-y\) in the equation to get \(x^{2}-(-y)^{3}=2\), which simplifies to \(x^{2}+y^{3}=2\). Since it is not the same as the original equation, the graph is not symmetric about the x-axis.

Step 3: Check for symmetry with respect to the origin

A graph is symmetric about the origin if replacing x with -x and y with -y in the equation, results in the exact original equation. Here, replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation to get \((-x)^{2}-(-y)^{3}=2\), which simplifies to \(x^{2}+y^{3}=2\). Since it is not the same as the original equation, the graph is not symmetric about the origin.

Key Concepts

Understanding Y-Axis Symmetry

When we talk about y-axis symmetry in a graph, we are interested in whether or not the graph looks the same on both sides of the y-axis. To test this, we replace every instance of \(x\) in the equation with \(-x\).
This means we flip the graph over the y-axis like a mirror reflection. The goal is to see if the equation remains unchanged after this substitution.

### How to Check Y-Axis Symmetry:

• Start with your equation.
• Replace \(x\) with \(-x\).
• Simplify the equation to see if it goes back to the original.

If it does, that means the graph is symmetric with respect to the y-axis.

For example, with the equation \(x^{2} - y^{3} = 2\), replacing \(x\) with \(-x\) gives us \((-x)^{2} - y^{3} = 2\), which simplifies back to \(x^{2} - y^{3} = 2\). Since this is the same as the original equation, the graph indeed has y-axis symmetry.

Exploring X-Axis Symmetry

X-axis symmetry means that if you were to fold the graph along the x-axis, the parts of the graph would match up. For testing x-axis symmetry, the method involves replacing every instance of \(y\) with \(-y\) in the equation.

### How to Check X-Axis Symmetry:

• Take your equation as given.
• Substitute \(y\) with \(-y\).
• Simplify the resultant equation and compare with the original.

If the simplified equation matches the original one, then the graph is x-axis symmetric.

In our example of \(x^{2} - y^{3} = 2\), substituting \(y\) with \(-y\) results in \(x^{2} - (-y)^{3} = 2\), which simplifies to \(x^{2} + y^{3} = 2\). Since the equations do not match, the graph lacks x-axis symmetry.

Deciphering Origin Symmetry

The concept of origin symmetry refers to symmetry that revolves around the origin point (0,0) on a graph. This type of symmetry would mean that if you rotate the graph 180 degrees around the origin, it would look the same.

To check this, both variables, \(x\) and \(y\), are replaced by their negative counterparts \(-x\) and \(-y\). This essentially checks if the graph can be "flipped" around both axes at once.

### How to Check Origin Symmetry:

• Begin with your initial equation.
• Replace \(x\) with \(-x\) and \(y\) with \(-y\) simultaneously.
• Simplify and see if it returns to the original equation.

In our example with \(x^{2} - y^{3} = 2\), making these substitutions results in \((-x)^{2} - (-y)^{3} = 2\), which simplifies to \(x^{2} + y^{3} = 2\). Since this is different from the original equation, the graph does not have origin symmetry.