Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Check for symmetry with respect to both axes and the origin. \(y=\sqrt{16-x^{2}}\)

Step-by-Step Solution

Answer

The given function \(y=\sqrt{16-x^{2}}\) is symmetric about the y-axis, but not symmetric about the origin.

Step 1: Checking if the function is even

We should replace \(x\) with \(-x\) and simplify to check if it equals the original function \(y\). For the function \(y=\sqrt{16-x^{2}}\), the modified function becomes \(\sqrt{16-(-x)^{2}} = \sqrt{16-x^{2}}\). This matches the original function exactly, so the function is even and symmetric about the y-axis.

Step 2: Checking if the function is odd

Now replace \(x\) with \(-x\) and \(y\) with \(-y\) to check if the equation remains valid. Doing this for the function \(y=\sqrt{16-x^{2}}\), we have \(-y=\sqrt{16-(-x)^{2}}\). Squaring both sides to remove the square root gives us \(y^{2}=16-x^{2}\), which is not equivalent to the original function. Hence, the function is not odd meaning it is not symmetric about the origin.

Key Concepts

Even Function

An even function is a type of function that exhibits a particular symmetry. When you replace each occurrence of \(x\) in the function with \(-x\), and if the function remains unchanged, it's an even function. In formulaic terms, a function \(f(x)\) is considered even if \(f(x) = f(-x)\) for every value of x in the function's domain.
This property tells us that the graph of the function is symmetric about the y-axis. Imagine folding the graph along the y-axis; both sides would align perfectly.

• This symmetry means the behavior on one side of the y-axis mirrors the other side.


• A common example of an even function is \(y = x^2\).

In the solved example, when \(x\) was replaced by \(-x\) in the function \(y=\sqrt{16-x^{2}}\), the expression remained unchanged. Hence, this function is even and symmetric about the y-axis.

Odd Function

Odd functions have a different kind of symmetry. For these functions, when every \(x\) is replaced with \(-x\), and the function's result also switches signs (i.e., if \(f(x) = -f(-x)\)), it indicates the function is odd. An intuitive way to picture this is that an odd function is symmetric about the origin.

• Imagine rotating the graph 180 degrees around the origin; the graph would look the same.


• An example of an odd function is \(y = x^3\).

When we applied this test to the function \(y=\sqrt{16-x^{2}}\), replacing \(x\) with \(-x\) led to modifying \(-y\) to check symmetry. It didn't satisfy the condition of an odd function because squaring both sides transformed it to \(y^2 = 16-x^2\), which wasn't the same as before; hence, confirming the function is not odd.

Symmetric about y-axis

Symmetry about the y-axis is a visual and functional characteristic of a graph. It means that if you were to fold the graph along the y-axis, each side would match up perfectly. For a function, this means when you graph the function and replace \(x\) with \(-x\), the graph looks unchanged.
The mathematical condition \(f(x) = f(-x)\) ensures this kind of symmetry. Many commonly encountered functions can exhibit this property if they are even.

• For instance, the function \(y = x^2\) demonstrates symmetry about the y-axis, because \(x^2 = (-x)^2\).


• In the solution, the function \(y=\sqrt{16-x^{2}}\) being unchanged upon replacing \(x\) with \(-x\) directly indicates its symmetry about the y-axis.