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Circles and semi-circle functions

In this topic, we check if circles and semi-circle functions are indeed functions.

A graph of a circle is formed when an arc is drawn from a fixed point (called the centre of the circle) in such a way that any point on the curve is the same distance from the centre. This distance from the centre of the circle to the end of the circle is called the radius of the circle.

The graph of a circle is represented by the equation x2 + y2 + 2gx + 2fy + c = 0, where

(-f, -g) is the centre of the circle and

r = sqrt{g^2~+~f^2~-~c} is the radius of the circle.

A special type of circle is x2 + y2 = r2 where (0, 0) is the origin or centre, and r is the radius of the circle.

Domain of a circle x2 + y2 = r2 is {x: −r ≤ x ≤ r}

and the range is {y: −r ≤ y ≤ r}

The general equation of a circle with centre (a, b) and radius ‘r’ is given by

(x – a)2 + (y – b)2 = r2


Is a circle a function?

We’ll do the ‘vertical line test’ to find out if a circle is a function or not.

vertical line test for circle

A vertical line cuts the circle at two points, therefore it fails the test for a function.

But we can make a circle into a function by breaking it into two functions (i.e. 2 semi-circles).

Rearranging the equation x2 + y2 = r2 we get

y = {pm}sqrt{r^2~-~x^2}.

y = sqrt{r^2~-~x^2} represents the top semi-circle, and

y = – sqrt{r^2~-~x^2} represents the bottom semi-circle.

So the equation of the semi-circle above x-axis with centre (0, 0) and radius r is given by

y = sqrt{r^2~-~x^2},

while the equation of the semi-circle below x-axis with centre (0, 0) and radius r is given by

y = – sqrt{r^2~-~x^2}

Domain: {x: −r ≤ x ≤ r}

Range: {y: 0 ≤ y ≤ r}  for top semi-circle, and

Domain: {x: −r ≤ x ≤ r}

Range: {y: −r ≤ y ≤ 0}  for bottom semi-circle


Is a semi-circle a function?

vertical test for positive semi-circle     vertical test for negative semi-circle


In this graph of a semi-circle, the vertical line meets only at one point.

Hence y = sqrt{r^2~-~x^2}

or y = – sqrt{r^2~-~x^2} is a function.


We now look at some examples of circles and semi-circle functions