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Chapter 1: Problem 75

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. $$x^{2 / 3}+y^{2 / 3}=1$$

Short Answer

Expert verified
Question: Determine the symmetry of the graph of the equation \(x^{2/3} + y^{2/3} = 1\). Answer: The graph is symmetric about the x-axis, y-axis, and the origin.

Step by step solution

01

Test for symmetry about the x-axis

Replace all the y values with -y: $$x^{2/3} + (-y)^{2/3} = 1$$ Since \((-y)^{2/3} = y^{2/3}\), this means that the equation is symmetric about the x-axis.
02

Test for symmetry about the y-axis

Replace all the x values with -x: $$(-x)^{2/3} + y^{2/3} = 1$$ Since \((-x)^{2/3} = x^{2/3}\), the equation is also symmetric about the y-axis.
03

Test for symmetry about the origin

Replace both x and y values with -x and -y: $$(-x)^{2/3} + (-y)^{2/3} = 1$$ Since \((-x)^{2/3} = x^{2/3}\) and \((-y)^{2/3} = y^{2/3}\), the equation is symmetric about the origin.
04

Check the symmetry by graphing

Now, graph the equation \(x^{2/3} + y^{2/3} = 1\). Observe that the graph is symmetric about the x-axis, y-axis, and the origin, verifying our results from the previous steps.

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Key Concepts

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

Symmetry About x-axis
Graph symmetry about the x-axis means the graph remains unchanged if flipped over the x-axis. Think of it as a mirror image where the x-axis is the mirror.

To test for x-axis symmetry, substitute every instance of the variable \(y\) with \(-y\) in the equation. If the new equation is identical to the original one, then the graph is symmetric about the x-axis.

For example, in the equation \(x^{2/3} + y^{2/3} = 1\), when you replace \(y\) with \(-y\), it becomes \(x^{2/3} + (-y)^{2/3} = 1\). Because \((-y)^{2/3} = y^{2/3}\), the equation does not change, proving x-axis symmetry.

  • This concept is crucial for understanding how graphs reflect around horizontal lines.
  • Such symmetry might simplify calculations by reducing the amount of graph required to analyze manually.
Symmetry About y-axis
When a graph is symmetric about the y-axis, it looks the same on both sides of the y-axis. This means if you fold the graph along the y-axis, both halves will align perfectly.

To determine y-axis symmetry in an equation or function, substitute \(x\) with \(-x\). If the equation remains unchanged, the graph is symmetric about the y-axis.

Looking at the equation \(x^{2/3} + y^{2/3} = 1\), replacing \(x\) with \(-x\) yields \((-x)^{2/3} + y^{2/3} = 1\). Here, \((-x)^{2/3} = x^{2/3}\), confirming the y-axis symmetry.

  • Y-axis symmetry is often present in even functions such as \(x^2\).
  • This type of symmetry is visually pleasing and can make graphing easier.
Symmetry About the Origin
A graph is symmetric about the origin if rotating it 180 degrees around the origin leaves the graph unchanged.

Testing for origin symmetry involves replacing \(x\) with \(-x\) and \(y\) with \(-y\). If the resulting equation looks the same, the graph has origin symmetry.

In \(x^{2/3} + y^{2/3} = 1\), doing these replacements results in \((-x)^{2/3} + (-y)^{2/3} = 1\). Since both \((-x)^{2/3} = x^{2/3}\) and \((-y)^{2/3} = y^{2/3}\), the original form is retained, confirming origin symmetry.

  • Origin symmetry implies that for every point \((x, y)\), the point \((-x, -y)\) also lies on the graph.
  • This characteristic is common in odd functions like \(x^3\).

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