Question

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

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

Step-by-step solution

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.

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.

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.

Key concepts

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.