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Chapter 1: Problem 26

Determine whether the graph of each equation is symmetric with respect to the \(y\) -axis, the \(x\) -axis, the origin, more than one of these, or none of these. $$x^{3}-y^{2}=5$$

Short Answer

Expert verified
The graph of the equation \(x^{3}-y^{2}=5\) is only symmetric with respect to the \(x\)-axis.

Step by step solution

01

Check Symmetry About the y-Axis

Replace \(x\) with \(-x\) in the equation to see if it yields the original equation. Doing this gives us: \[(-x)^{3}-y^{2}=5\] which simplifies to \[-x^{3}-y^{2}=5\]. Since this is not the original equation, the graph of the equation is not symmetric about the \(y\)-axis.
02

Check Symmetry About the x-Axis

Replace \(y\) with \(-y\) in the equation to see if it yields the original equation. Doing this gives us: \[x^{3}-(-y)^{2}=5\] which simplifies to \[x^{3}-y^{2}=5\]. Thus, this is the original equation. Therefore, the graph of the equation is symmetric about the \(x\)-axis.
03

Check Symmetry About the Origin

Replace both \(x\) and \(y\) with \(-x\) and \(-y\) in the equation respectively to see if it yields the original equation. Doing this gives us: \[(-x)^{3}-(-y)^{2}=5\] which simplifies to \[-x^{3}-y^{2}=5\]. Since this is not the original equation, the graph of the equation is not symmetric about the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding y-axis Symmetry
In the world of graphs and equations, y-axis symmetry is when a graph is a mirror image on either side of the y-axis. Imagine folding your graph on the y-axis, and both sides match perfectly. To check for this kind of symmetry, you would replace \(x\) with \(-x\) in the equation. If the resulting equation matches the original, then y-axis symmetry is present. However, if it changes, it means the graph does not have this symmetry.
For the equation \(x^{3} - y^{2} = 5\), substituting \(-x\) for \(x\) gives us \(-x^{3} - y^{2} = 5\). This is not the same as the original equation, therefore, this graph does not exhibit y-axis symmetry.
Grasping x-axis Symmetry
x-axis symmetry means that a graph is a mirror image above and below the x-axis. Just imagine flipping the graph over the x-axis. If both halves are identical, then the graph is symmetrical about the x-axis. To test this, you simply substitute \(y\) with \(-y\) in your equation. If the new equation is identical to the original one, x-axis symmetry is confirmed.
Let's apply this to our equation \(x^{3} - y^{2} = 5\). Replace \(y\) with \(-y\) to get \(x^{3} - (-y)^{2} = 5\). Simplifying, it becomes \(x^{3} - y^{2} = 5\), which matches the original equation perfectly. Therefore, the graph is symmetric about the x-axis.
Exploring Origin Symmetry
Origin symmetry is slightly more complex, as it means every part of the graph has a corresponding, opposite part across the origin. It鈥檚 like rotating the graph 180 degrees around the origin, giving a complete mirror image. To verify origin symmetry, replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation. If it matches the original after simplification, origin symmetry exists.
For \(x^{3} - y^{2} = 5\), replacing \(x\) with \(-x\) and \(y\) with \(-y\) gives us \(-x^{3} - (-y)^{2} = 5\). This simplifies to \(-x^{3} - y^{2} = 5\), differing from the original equation, thus illustrating that this graph is not symmetric about the origin.
Demystifying Equation Transformations
Equation transformations are like mathematical shape-shifting. They're techniques used to change the appearance of an equation, making it easier to analyze for symmetries or other properties. There are different types of transformations:
  • Translation: Shifting the graph horizontally or vertically without altering its shape.
  • Reflection: Flipping the graph across an axis to check for symmetry.
  • Rotation: Turning the graph around a point, typically the origin.
For symmetry checks, we specifically use reflection transformations by substituting values like \(-x\) for \(x\) or \(-y\) for \(y\). These reflections help determine if a graph folds over like a mirror on the y-axis, x-axis, or through the origin. They serve as essential tools in confirming or denying symmetry within a graph equation.

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