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Chapter 0: Problem 34

Test for symmetry with respect to each axis and to the origin. $$x y-\sqrt{4-x^{2}}=0$$

Short Answer

Expert verified
The given equation, \(xy-\sqrt{4-x^2}\) is not symmetric about the x-axis or the y-axis, but it is symmetric about the origin.

Step by step solution

01

Test for Symmetry about the Y-axis

For y-axis symmetry, replace \(x\) with \(-x\) in the equation and simplify. The equation becomes: \(-xy -\sqrt{4-(-x)^2} = 0 \), which simplifies to: \(-xy - \sqrt{4-x^2} = 0\). This equation is not identical to the original equation, therefore, it does not have y-axis symmetry.
02

Test for Symmetry about the X-axis

For x-axis symmetry, replace \(y\) with \(-y\) in the equation and simplify. The equation becomes: \( -xy - \sqrt{4-x^2} = 0\). This is not identical to the original equation, so it does not have x-axis symmetry.
03

Test for Symmetry about the Origin

For origin symmetry, replace both \(x\) and \(y\) with \(-x\) and \(-y\) respectively. The equation becomes: \( xy - \sqrt{4-(-x)^2} = 0\), which simplifies to: \(xy - \sqrt{4-x^2} = 0\). This equation is identical to the original equation, thus the given equation exhibits symmetry 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
Y-axis symmetry in equations is all about ensuring that the left side of the graph mirrors the right side.
When an equation is symmetric with respect to the y-axis, it means that for every point (x, y) on the graph, there is a corresponding point (-x, y) as well. This is similar to how reflections work on a vertical mirror.

To test for y-axis symmetry, we replace the variable x with -x in the equation. Then, we simplify to see if we get the original equation. If the resulting equation is identical to the original, we have y-axis symmetry.
  • If the equation does not revert to its original form, there's no y-axis symmetry.
In our exercise, replacing x with -x did not yield the same equation, which confirmed the absence of y-axis symmetry. Remember, symmetry about the y-axis can give valuable clues about the overall shape and properties of the graph.
Exploring X-Axis Symmetry
X-axis symmetry deals with making sure the top half of the graph mirrors the bottom half.
If an equation is symmetric about the x-axis, then for every point (x, y) present, there should be a point (x, -y) as well.

This means imagining folding the graph over the x-axis, ensuring the top and bottom parts line up perfectly.
  • To check for this symmetry, the variable y is replaced with -y in the equation.
  • If simplification returns the original equation, x-axis symmetry is present.
  • If it doesn't, there is no symmetry about the x-axis.
In our specific example, substituting y with -y did not produce the same equation, indicating that there is no x-axis symmetry. Understanding this symmetry can assist in predicting how graphs behave in relation to the x-axis.
Discovering Origin Symmetry
Origin symmetry is a bit different; it involves a central point, the origin (0,0), where the graph is symmetric around it.
For equations that are symmetric concerning the origin, a point (x, y) implies there should be a point (-x, -y) too.

Imagine spinning the graph 180 degrees around the origin; it should look the same after the rotation.
  • Checking this symmetry involves substituting both x with -x and y with -y.
  • If the equation after this transformation matches the original, then the equation possesses origin symmetry.
In the given exercise, doing this replacement resulted in an equation identical to the original. Therefore, our equation demonstrated origin symmetry. Understanding symmetry about the origin is particularly useful for analyzing functions with central rotational symmetry.

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Most popular questions from this chapter

Reimbursed Expenses A company reimburses its sales representatives \(\$ 150\) per day for lodging and meals plus \(34 \mathrm{c}\) per mile driven. Write a linear equation giving the daily cost \(C\) to the company in terms of \(x\), the number of miles driven. How much does it cost the company if a sales representative drives 137 miles on a given day?

Evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(g(x)=x^{2}(x-4)\) (a) \(g(4)\) (b) \(g\left(\frac{3}{2}\right)\) (c) \(g(c)\) (d) \(g(t+4)\)

Show that the distance between the point \(\left(x_{1}, y_{1}\right)\) and the line \(A x+B y+C=0\) is Distance \(=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\)

Prove that the function is even. \(f(x)=a_{2 n} x^{2 n}+a_{2 n-2} x^{2 n-2}+\cdots+a_{2} x^{2}+a_{0}\)

Energy Consumption and Gross National Product The data show the per capita electricity consumptions (in millions of Btu) and the per capita gross national products (in thousands of \(\mathrm{U} . \mathrm{S} .\) dollars) for several countries in \(2000 .\) (Source: U.S. Census Bureau) $$ \begin{aligned} &\begin{array}{|l|c|} \hline \text { Argentina } & (73,12.05) \\ \hline \text { Chile } & (68,9.1) \\ \hline \text { Greece } & (126,16.86) \\ \hline \text { Hungary } & (105,11.99) \\ \hline \text { Mexico } & (63,8.79) \\ \hline \text { Portugal } & (108,16.99) \\ \hline \text { Spain } & (137,19.26) \\ \hline \text { United Kingdom } & (166,23.55) \\ \hline \end{array}\\\ &\begin{array}{|l|c|} \hline \text { Bangladesh } & (4,1.59) \\ \hline \text { Egypt } & (32,3.67) \\ \hline \text { Hong Kong } & (118,25.59) \\ \hline \text { India } & (13,2.34) \\ \hline \text { Poland } & (95,9) \\ \hline \text { South Korea } & (167,17.3) \\ \hline \text { Turkey } & (47,7.03) \\ \hline \text { Venezuela } & (113,5.74) \\ \hline \end{array} \end{aligned} $$ (a) Use the regression capabilities of a graphing utility to find a linear model for the data. What is the correlation coefficient? (b) Use a graphing utility to plot the data and graph the model. (c) Interpret the graph in part (b). Use the graph to identify the three countries that differ most from the linear model. (d) Delete the data for the three countries identified in part (c). Fit a linear model to the remaining data and give the correlation coefficient.
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