A function we will use over and over is one whose graph is a semi-circle (there can't be a single function whose graph is a circle because it would violate the vertical line test, so we focus on semi-circles). Maybe you remember from precalculus that the graph of the function

is a semi-circle, the center of the semi-circle is at the origin, and the radius is \(a\) where \(a\) is the positive square root of \(a^2\text{.}\)

Look closely at this function. As mentioned above we will use it often and if you can convince yourself its graph is a semi-circle you will be in a place to solve problems involving circles.

Domain: All values satisfying \(-a \le x \le a\text{.}\)

Range: All values satisfying \(0 \le y \le a\text{.}\)

Graph: A semi-circle, center at the origin, radius \(a\text{.}\)

Here is the graph for \(a=4\text{:}\)

You may think that graph doesn't really look like semi-circular and you'd be correct. It was drawn that way for a reason - to emphasize that if the units on the two axes are not identical in size (and there is no reason they need to be) then, even though the graph IS a semi-circle, and we KNOW the graph is a semi-circle, it may not look like one. You can see for this graph the \(x-\)axis contains 10 units from left-to-right but the \(y-\)axis only contains 6 units from bottom-to-top. Thus the semicircle is skewed. If we know the algebraic form that a semi-circular function must take then we won't be misled by a graph drawn with unequal units on the axes.

With a little care we can ensure the units are the same and the graph does indeed look like the semi-circle that we know it must be:

In this example we draw the graph of two functions on the same axes, each semi-circles but with different radii.

Solution:First we note that both functions have graphs that are semi-circles, fitting the form given above. The graph of \(f(x)\) has a radius of 2 while the graph of \(g(x)\) has a radius of 6. We will use the same unit size on both axes so the graphs will look like semi-circles.

Finally, we show that we can use two functions to get a graph of a complete circle. The graph of the function

is the bottom half of a semi-circle (due to the minus sign in front - more on this in the next section).

Solution:Since \(f(x)\) is the top half of a circle of radius 2 and \(g(x)\) is the bottom half of a circle of radius 2, together they should graph a complete circle. We will also use the same unit length on both axes so it will look like a circle: