Question

Determine whether the graphs of the following equations and functions have symmetry about the \(x\) -axis, the \(y\) -axis, or the origin. Check your work by graphing. $$x^{3}-y^{5}=0$$

Short answer

Question: Determine the types of symmetry (x-axis, y-axis, or origin) for the equation \(x^3 - y^5 = 0\). Answer: The equation \(x^3 - y^5 = 0\) has symmetry about the x-axis only.

Step-by-step solution

Testing symmetry about the x-axis

To check for symmetry about the x-axis, we will replace \(y\) with \(-y\). If the equation remains the same, it has symmetry about the x-axis. $$x^3 - (-y)^5 = 0$$ $$x^3 -y^5 = 0$$ Since replacing \(y\) with \(-y\) does not change the equation, it has symmetry about the x-axis.

Testing symmetry about the y-axis

To check for symmetry about the y-axis, we will replace \(x\) with \(-x\). If the equation remains the same, it has symmetry about the y-axis. $$(-x)^3 - y^5 = 0$$ $$-x^3 - y^5 = 0$$ Replacing \(x\) with \(-x\) changes the equation, so it does not have symmetry about the y-axis.

Testing symmetry about the origin

To check for symmetry about the origin, we will replace \(x\) with \(-x\) and \(y\) with \(-y\). If the equation remains the same, it has symmetry about the origin. $$(-x)^3 - (-y)^5 = 0$$ $$-x^3 - y^5 = 0$$ Since the equation changes when both \(x\) and \(y\) are replaced by their negative counterparts, it does not have symmetry about the origin.

Graphing to verify

To check our work, we can graph the equation \(x^3 - y^5 = 0\). The graph shows symmetry about the x-axis as determined earlier, but not about the y-axis or the origin. In conclusion, the equation \(x^3 - y^5 = 0\) has symmetry about the x-axis only.

Key concepts

Testing Symmetry

Testing symmetry is a great way to understand how an equation behaves geometrically. It's like using a mirror to see if the equation's graph looks the same when flipped.

Here's how you can test it for different axes or points:

• **Symmetry about the x-axis**: Replace each instance of \(y\) with \(-y\) in the equation. If the resulting equation is the same as the original, the graph has symmetry here.


• **Symmetry about the y-axis**: Do the same but replace each \(x\) with \(-x\). Again, if the equation doesn't change, there's symmetry here.


• **Symmetry about the origin**: Check by replacing \((x, y)\) with \((-x, -y)\). If the equation remains identical, the graph is symmetric about the origin.

Don't worry if an equation doesn't show symmetry with all checks. Many graphs exhibit only certain symmetries, or even none, which affects their shapes and features.

Graphing Techniques

Graphing is a useful way to visualize the function or equation at hand. It's helpful not only for checking symmetry but for understanding the overall behavior of the graph.

There are a few basic techniques to keep in mind:

• **Plot Key Points**: Start by calculating and plotting key points where the graph changes direction or crosses the axes.


• **Look For Patterns**: Symmetries or repetitive patterns in the graph make it easier to sketch accurately.


• **Use Graphing Tools**: Software or calculators can be invaluable, especially for complex graphs or where precision is key. These tools can automatically show symmetries you might miss by hand.

Remember, graphing isn't just about drawing lines and curves; it's about interpreting the story that the equation is telling.

Equation Analysis

Analyzing an equation helps you to break down what each part does to contribute to the graph. Understanding this makes it easier to predict symmetry or changes in direction.

Here's a simple approach to get started:

• **Identify the Highest Power**: The highest exponent of \(x\) or \(y\) often dictates the broad shape and symmetry of the graph. Odd powers might suggest asymmetry, while even powers hint at symmetric patterns.


• **Check Coefficients**: These determine stretch or compression. Pay attention to negative coefficients as well—they may flip the graph about an axis!


• **Combine Insights**: Use these individual components to predict how changing one part of an equation might alter the graph's look or symmetry.

When you understand these elements, deciphering what an equation's graph looks like becomes easier, allowing more accurate predictions without always relying on a plotted graph.

Symmetry Types

There are three main types of symmetry that graphs of equations exhibit, each with its own distinct characteristics.

Here's a closer look at each type:

• **X-Axis Symmetry**: If you reflect the graph over the x-axis, it looks the same. In terms like \(y\) and \(-y\), they hold the same relationship owing to this symmetry. Many physical systems, like waves, illustrate this form.


• **Y-Axis Symmetry**: This is achieved if the left side of the graph is a mirror image of the right side. It's like a butterfly or a heart shape, balanced on either side of the \(y\)-axis. Functions like even polynomials commonly show this form.


• **Origin Symmetry**: The graph flipped around the origin remains unchanged. Each part has an identical counterpart diagonally across the graph. Odd functions often exhibit this balance.

Understanding these types will not only help with graphing, but it also deepens comprehension of the underlying mathematical principles. This understanding is crucial for exploring more advanced topics in algebra and calculus.