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Chapter 1: Problem 77

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. $$f(x)=x|x|$$

Short Answer

Expert verified
Answer: The graph of the function \(f(x) = x|x|\) is symmetric about the y-axis.

Step by step solution

01

Test for x-axis Symmetry

To check for x-axis symmetry, we will test if \(f(x) = f(-x)\). \(f(-x)=-x|-x|= -x|x|\). Since \(f(-x) \neq f(x)\), the function is not symmetric about the x-axis.
02

Test for y-axis Symmetry

To check for y-axis symmetry, we will test whether \(f(-x) = -f(x)\). As we found before, \(f(-x) = -x|x|\). So now we check if this is equal to \(-f(x)\): \(-f(x) = -(x|x|) = -x|x|\). Since \(f(-x) = -f(x)\), the function is symmetric about the y-axis.
03

Test for Symmetry about the Origin

To check for symmetry about the origin, we will test if \(f(-x) = -f(-x)\). We already found that \(f(-x) = -x|x|\). Let's now find \(-f(-x)\): \(-f(-x) = -(-x|x|) = x|x|\). Since \(f(-x) \neq -f(-x)\), the function is not symmetric about the origin. Now that we've tested for all symmetries, we found that the graph of the function \(f(x)=x|x|\) is symmetric about the y-axis. To verify this, you can graph the function and check if it reflects the same way across the y-axis.

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Key Concepts

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

Symmetry about x-axis
When examining symmetry about the x-axis, we want to assess if the graph reflects over this axis. This reflection would mean that for every point \((x, y)\) on the graph, the point \((x, -y)\) must also exist.

In simpler terms, it implies that changing the y-value to a negative equivalent should yield the same function output:
  • Mathematically, this is tested as: If \(f(x) = f(-x)\), then the function is symmetric about the x-axis.
  • Symmetry about the x-axis suggests that if you folded the graph along this axis, the two halves would match perfectly.

For instance, the equation \(f(x) = x|x|\) doesn't satisfy this as changing \(x\) to \(-x\) changes the function entirely. It doesn't reflect a mirror image across the x-axis, hence there's no x-axis symmetry here.
Symmetry about y-axis
Y-axis symmetry means that when you substitute \(-x\) into the function, the output remains identical to the function with \(x\).

This idea focuses on reflecting over the vertical y-axis. Thus, every point \((x, y)\) must have a corresponding point \((-x, y)\) for a y-axis symmetry:
  • In equation form, this is represented as: If \(f(-x) = f(x)\).
  • This checks if the function is unchanged even when the input x is changed to its negative. If it holds true, the graph shows y-axis symmetry.

In the specific case of \(f(x)=x|x|\), we found that \(f(-x) = -x|x|\) was not the same as \(f(x)\). But we found symmetry of a different kind: it follows \(f(-x) = -f(x)\), suggesting specific behavior that hints at y-axis types of relationships or behaviors. This transitions to the next topic.
Symmetry about the origin
Symmetry about the origin involves a rotation around the center point of the graph (0,0). To test this symmetry, if you substitute both \(-x\) and \(-y\) into the function and it remains true, the graph is symmetric about the origin.
  • This type of symmetry can be described as: If \(f(-x) = -f(x)\).
  • Origin symmetry implies if you rotated the graph 180 degrees around the origin, it would appear unchanged.

Examining \(f(x) = x|x|\) further, we check for this by taking the function values with \(-x\). Doing the math, output doesn't match the origin symmetry condition since \(-f(-x)\) amounts to \(x|x|\), differing from \(f(x)\).
Hence, the function doesn't hold symmetry around the origin. You can visualize this with graphing tools, watching for lack of overlap upon 180-degree rotation.

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Most popular questions from this chapter

Use the following steps to prove that \(\log _{b}(x y)=\log _{b} x+\log _{b} y\). a. Let \(x=b^{p}\) and \(y=b^{q}\). Solve these expressions for \(p\) and \(q\) respectively. b. Use property El for exponents to express \(x y\) in terms of \(b, p\) and \(q\). c. Compute \(\log _{b}(x y)\) and simplify.

Use shifts and scalings to graph then given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. $$h(x)=-4 x^{2}-4 x+12$$

Use analytical methods to find the following points of intersection. Use a graphing utility only to check your work. Find the point(s) of intersection of the parabolas \(y=x^{2}\) and \(y=-x^{2}+8 x\)

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. $$x^{3}-y^{5}=0$$

Kelly has finished a picnic on an island that is \(200 \mathrm{m}\) off shore (see figure). She wants to return to a beach house that is \(600 \mathrm{m}\) from the point \(P\) on the shore closest to the island. She plans to row a boat to a point on shore \(x\) meters from \(P\) and then jog along the (straight) shore to the house. a. Let \(d(x)\) be the total length of her trip as a function of \(x .\) Graph this function. b. Suppose that Kelly can row at \(2 \mathrm{m} / \mathrm{s}\) and jog at \(4 \mathrm{m} / \mathrm{s}\). Let \(T(x)\) be the total time for her trip as a function of \(x\). Graph \(y=T(x)\) c. Based on your graph in part (b), estimate the point on the shore at which Kelly should land in order to minimize the total time of her trip. What is that minimum time?
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