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Chapter 0: Problem 30

Test for symmetry with respect to each axis and to the origin. $$y=x^{3}+x$$

Short Answer

Expert verified
The function \(y=x^{3}+x\) is symmetric with respect to the origin, but not symmetric with respect to either the x-axis or y-axis.

Step by step solution

01

Test for y-axis symmetry

To test for y-axis symmetry (even function), replace \(x\) in the equation with \(-x\), and then simplify. If the resulting equation is identical to the original equation, then the graph is symmetrical about the y-axis. Hence, replace \(x\) with \(-x\) in the equation \(y=x^{3}+x\), yielding \(y=(-x)^{3}+(-x)\). After simplifying, the resulting equation is \(y=-x^{3}-x\), which is not identical to the original equation, so the function is not symmetric with respect to the y-axis.
02

Test for x-axis symmetry

To test for x-axis symmetry (odd function), replace \(y\) in the equation with \(-y\). Unfortunately, this test is not applicable here since Resolved equation includes both x and y, so we skip this step.
03

Test for origin symmetry

To test for symmetry with respect to the origin, replace \(x\) and \(y\) in the equation with \(-x\) and \(-y\) respectively. If the resulting equation is identical to the original equation, then the graph is symmetrical about the origin. Hence, replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation \(y=x^{3}+x\), yielding \(-y=(-x)^{3}+(-x)\). After simplifying, the resulting equation is \(-y=-x^{3}-x\), which is, if we multiply both sides by -1, identical to the original equation, so the function 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.

Y-Axis Symmetry
Y-axis symmetry is when a function's graph mirrors itself across the y-axis. This occurs in even functions. To test for y-axis symmetry in any function, you replace all instances of the variable \(x\) with \(-x\) and simplify the equation.The goal is to see if the new equation looks exactly like the original.
  • If the equation stays the same after this replacement, the function is symmetrical with respect to the y-axis.
  • If the equation changes, as it did for the equation \(y=x^3+x\), where we got \(y=-x^3-x\), then it is not y-axis symmetrical.
This knowledge helps in understanding the graph's layout and can simplify integration and other calculus operations.
Origin Symmetry
Origin symmetry is present when rotating a graph 180 degrees about the origin yields an identical graph. Calculus often checks this with odd functions.The test for origin symmetry involves replacing \(x\) with \(-x\) and \(y\) with \(-y\) in the equation.Simplifying this equation will show whether or not it matches the original.
  • If it does, like in the equation \(y = x^3 + x\) where substituting gives \(-y = -x^3 - x\) and eventually simplifies back to the original equation, the function is origin-symmetrical.
  • Functions with origin symmetry have balanced behavior around the axis and origin, creating a harmony in function behavior as \(y = f(x)\) and \(-y = f(-x)\).
Recognizing this symmetry aids in graph sketching and simplifies analyses of function properties.
Graphical Symmetry
Graphical symmetry relates to overall patterns in a graph relative to its axes and the origin. Recognizing y-axis symmetry or origin symmetry can significantly impact how we approach problems.
  • Y-axis symmetry simplifies problem-solving by indicating that operations on one side of the y-axis mirror on the other.
  • Origin symmetry tells us the function behaves consistently around the graph's center, easing integration and differentiation tasks.
When studying calculus, identifying graphical symmetry can help in predicting behavior, transforming functions, and setting boundary conditions. Even when symmetry is absent, the process of testing helps solidify understanding of function behavior and graphing skills.

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Most popular questions from this chapter

Use the Vertical Line Test to determine whether \(y\) is a function of \(x\). To print an enlarged copy of the graph, select the MathGraph button. $$y=\left\\{\begin{aligned} x+1, & x \leq 0 \\\\-x+2, & x>0 \end{aligned}\right.$$

Determine whether the function is even, odd, or neither. Use a graphing utility to verify your result. $$f(x)=x^{2}\left(4-x^{2}\right)$$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=f(-x)\) for all \(x\) in the domain of \(f\), then the graph of \(f\) is symmetric with respect to the \(y\) -axis.

Automobile Aerodynamies The horsepower \(H\) required to overcome wind drag on a certain automobile is approximated by \(H(x)=0.002 x^{2}+0.005 x-0.029, \quad 10 \leq x \leq 100\) where \(x\) is the speed of the car in miles per hour. (a) Use a graphing utility to graph \(H\). (b) Rewrite the power function so that \(x\) represents the speed in kilometers per hour. [Find \(H(x / 1.6) .]\)

Find the domain of the function. $$h(x)=\frac{1}{\sin x-\frac{1}{2}}$$
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