New Zealand
Level 8 - NCEA Level 3

# Domain and Range of Circles

Lesson

The diagram below shows that the domain of a circle consists of all $x$x-values within the interval $a-r\le x\le a+r$arxa+r and the range of of a circle consists of all $y$y-values within the interval $b-r\le y\le b+r$bryb+r.

We can also write the domain in interval notation as $\left[a-r,a+r\right]$[ar,a+r], and the range in interval notation as $\left[b-r,b+r\right]$[br,b+r].

The circle itself is not a function, but can be split into two semicircles, each of which are functions. We can rearrange the equation to make this clear:

 $\left(x-a\right)^2+\left(y-b\right)^2$(x−a)2+(y−b)2 $=$= $r^2$r2 $\left(y-b\right)^2$(y−b)2 $=$= $r^2-\left(x-a\right)^2$r2−(x−a)2 $y-b$y−b $=$= $\pm\sqrt{r^2-\left(x-a\right)^2}$±√r2−(x−a)2 $y$y $=$= $y=b\pm\sqrt{r^2-\left(x-a\right)^2}$y=b±√r2−(x−a)2

So, for example, in the new form the circle whose centre is located at $\left(2,5\right)$(2,5) and has radius $r=3$r=3 has the equation given by $y=5\pm\sqrt{9-\left(x-2\right)^2}$y=5±9(x2)2. It can be split into the two functions, say $f$f and $g$g, where $f\left(x\right)=5+\sqrt{9-\left(x-2\right)^2}$f(x)=5+9(x2)2 and $g\left(x\right)=5-\sqrt{9-\left(x-2\right)^2}$g(x)=59(x2)2.

Each of these functions are semicircles. Both functions will have the same domain; that of the original circle, given by $-1\le x\le5$1x5. The range of $f$f becomes $5\le y\le8$5y8 and the range of $g$g becomes $2\le y\le5$2y5 as shown here.

#### Practice questions

##### Question 1

Consider the graph of the circle shown below.

1. State the domain of the graph in interval notation.

2. State the range of the graph in interval notation.

##### Question 2

Consider the equation $\left(x+5\right)^2+\left(y+3\right)^2=16$(x+5)2+(y+3)2=16.

1. Plot the graph described by the equation.

2. State the domain of the graph in interval notation.

3. State the range of the graph in interval notation.

##### Question 3

The top of a semicircle has a domain of $\left[-10,2\right]$[10,2] and a range of $\left[-2,4\right]$[2,4].

1. Plot the semicircle.

2. State the equation for the semicircle in the form $y=\pm\sqrt{r^2-\left(x-h\right)^2}+k$y=±r2(xh)2+k.

### Outcomes

#### M8-1

Apply the geometry of conic sections

#### 91573

Apply the geometry of conic sections in solving problems