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Chapter 1: Problem 40

Use the algebraic tests to check for symmetry with respect to both axes and the origin. $$ x y=4 $$

Short Answer

Expert verified
According to the tests, the graph of the equation \(xy = 4\) is only symmetric with respect to the origin and not symmetric with respect to either the x or y axis.

Step by step solution

01

x-axis Symmetry

Replace \(y\) with \(-y\) in the equation \(xy = 4\) to test for symmetry with respect to the x-axis. This results in \(x(-y) = 4\) or \(-xy = 4\). Since this is not the same as the original equation, the graph of the equation is not symmetric with respect to the x-axis.
02

y-axis Symmetry

The next step is to replace \(x\) with \(-x\) in the equation \(xy = 4\) to test symmetry regarding the y-axis. We get \(-xy = 4\), which again, is not the same as the original equation, so the graph is not symmetric with respect to the y-axis.
03

Origin Symmetry

Lastly, replace both \(x\) with \(-x\) and \(y\) with \(-y\) in the equation \(xy = 4\) for testing symmetry about the origin. The result is \(-x(-y) = 4\) or \(xy = 4\), which is equivalent to the original equation. Therefore, the graph of the equation is symmetric with respect to the origin.

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Key Concepts

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

Understanding X-Axis Symmetry
Symmetry with respect to the x-axis means the graph remains unchanged when flipped over the x-axis. To check for x-axis symmetry in an algebraic equation, we replace every instance of the variable \(y\) with \(-y\).
For example, in the equation \(xy = 4\), we test for x-axis symmetry by substituting \(-y\) for \(y\), resulting in \(-xy = 4\).
  • If the new equation is the same as the original, the graph is symmetric with respect to the x-axis.
  • In this case, \(-xy = 4\) is not the same as \(xy = 4\), so there is no x-axis symmetry.
Exploring Y-Axis Symmetry
Y-axis symmetry occurs when flipping a graph over the y-axis leaves it unchanged. To test this kind of symmetry mathematically, replace \(x\) with \(-x\) in the equation.
Using \(xy = 4\), we substitute \(-x\) for \(x\) resulting in \(-xy = 4\).
  • If the transformed equation matches the original, it means the graph has y-axis symmetry.
  • Here, since \(-xy = 4\) doesn鈥檛 match the original equation \(xy = 4\), the graph lacks symmetry with the y-axis.
Discovering Origin Symmetry
Origin symmetry is present when a graph stays the same if rotated 180 degrees around the origin. To check for this, replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation and see if you arrive back at the original equation.
In the equation \(xy = 4\), replace \(x\) with \(-x\) and \(y\) with \(-y\), resulting in \(-x(-y) = 4\), which simplifies to \(xy = 4\).
  • If the equation after substitutions matches the original, the graph is symmetric about the origin.
  • Since we returned to \(xy = 4\), the equation shows origin symmetry, meaning the graph would look the same if flipped upside down.

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Most popular questions from this chapter

Use the given value of \(k\) to complete the table for the direct variation model $$y=k x^{2}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=k x^{2} & & & & & \\ \hline \end{array}$$ $$ k=\frac{1}{4} $$

An \(r\) -value of a set of data, also called a _____ _____ ,gives a measure of how well a model fits a set of data.

The mathematical model \(y=\frac{\kappa}{x}\) is an example of _____ variation.

Find a mathematical model for the verbal statement. \(h\) varies inversely as the square root of \(s\).

Find a mathematical model for the verbal statement. \(F\) varies directly as \(g\) and inverselv as \(r^{2}\).
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