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5 Equations and graphs
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AustraliaVIC
Level 9

5.09 Graphs of circles

Lesson
Worksheet
Practice
Lesson

Graphs of circles

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Graphs of equations of the form \\ \left(x-h \right)^{2} + \left(y-k \right)^{2} = r^{2} (where h,\, k, and r are any number and r \neq 0) are called circles.

The circle defined by x^{2} + y^{2} = 1. It has a centre at (0,0) and a radius of 1 unit.

A circle can be scaled both vertically and horizontally by changing the value of r. In fact, r is the radius of the circle.

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This graph shows x^{2} + y^{2} = 1 expanded by a scale factor of 2 to get x^{2} + y^{2} = 4, and compressed by 2 to get x^{2} + y^{2} = \dfrac{1}{4}.

Examples

Example 1

A circle has its centre at the origin and a radius of 9 units.

a

Plot the graph for the given circle.

Worked Solution
Create a strategy

Plot the centre of the circle.

Count 9 units in each direction from the centre: left, right, up, and down and draw a round curve passing through each these points.

Apply the idea
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The centre of the circle is at the origin. So plot the point (0,0).

Now we count 9 units in all four directions to get the points (-9,0),\, (9,0),\, (0,9) and (0,-9).

To graph the circle, connect the points using a round curve passing through the 4 points.

b

Write the equation of the circle.

Worked Solution
Create a strategy

Use the form x^{2} + y^{2} = r^{2}.

Apply the idea
\displaystyle x^{2} + y^{2}\displaystyle =\displaystyle r^{2}
\displaystyle x^{2} + y^{2}\displaystyle =\displaystyle 9^{2}Substitute r=9
\displaystyle x^{2} + y^{2}\displaystyle =\displaystyle 81Evaluate 9^2

Example 2

Consider the circle x^{2} + y^{2} = 64.

a

Find the x values of the x-intercepts.

Worked Solution
Create a strategy

Substitute y=0 into the equation.

Apply the idea
\displaystyle x^{2} + y^{2}\displaystyle =\displaystyle 64Write the equation
\displaystyle x^{2} + (0)^{2}\displaystyle =\displaystyle 64Subsitute y=0
\displaystyle x^{2}\displaystyle =\displaystyle 64Evaluate
\displaystyle \sqrt{x^{2}}\displaystyle =\displaystyle \pm \sqrt{64}Square root both sides
\displaystyle x\displaystyle =\displaystyle \pm 8Evaluate
b

Find the y values of the y-intercepts.

Worked Solution
Create a strategy

Let x=0 and substitute into the given equation.

Apply the idea
\displaystyle x^{2} + y^{2}\displaystyle =\displaystyle 64Write the equation
\displaystyle (0)^{2} + y^{2}\displaystyle =\displaystyle 64Subsitute x=0
\displaystyle y^{2}\displaystyle =\displaystyle 64Evaluate
\displaystyle \sqrt{y^{2}}\displaystyle =\displaystyle \pm \sqrt{64}Square root both sides
\displaystyle y\displaystyle =\displaystyle \pm 8Evaluate
c

Plot the graph for the given circle.

Worked Solution
Create a strategy

Draw a round curve passing through the x\text{-} and y-intercepts.

Apply the idea
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Reflect and check

From the equation, the coordinates of the centre are (0,0), and the radius is r=\sqrt{64}=8. We can see that this agrees with the graph above.

Idea summary

The graph of an equation of the form \left(x-h \right)^{2} + \left(y-k \right)^{2} = r^{2} is a circle.

Circle have a centre at (h,k) and a radius of r.

Circles of the form x^{2} + y^{2} = 1 can be scaled by a scale factor of r to get the equation x^{2} + y^{2} = r^{2}.

Horizontal translations

A circle can be horizontally translated by increasing or decreasing the x-values by a constant number. However, the x-value together with the translation must be squared together. That is, to translate \\ x^{2} + y^{2} = 1 to the left by h units we get \left(x+h \right)^{2}+ y^{2} = 1.

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This graph shows x^{2} + y^{2} = 1 translated horizontally left by 2 to get \left(x+2 \right)^{2} + y^{2}=1, and right by 2 to get \left(x-2 \right)^{2} + y^{2} = 1.

Exploration

Let's try some horizontal translations. For this applet, move the slider for h to horizontally translate the circle and move the slider for r to adjust the radius.

Loading interactive...

When we subtract h from x then the circle is translated right h units. When we add h to x then the circle is translated left h units. As the radius increases the size of the circle also increases.

Idea summary

Circles of the form x^{2} + y^{2} = r^2 can be horizontally translated by h units to the left to get: \left(x+h \right)^{2}+ y^{2} = r^2 or h units to the right to get: \left(x-h \right)^{2}+ y^{2} = r^2.

Vertical translations

A circle can be vertically translated by increasing or decreasing the y-values by a constant number. However, the y-value together with the translation must be squared together. So to translate \\ x^{2} + y^{2} = 1 up by k units gives us x^{2} + \left(y-k \right)^{2} = 1.

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This graph shows x^{2} + y^{2} = 1 translated vertically up by 2 to get x^{2} + \left(y-2 \right)^{2} = 1, and down by 2 to get x^{2} + \left(y+2 \right)^{2} = 1.

Exploration

In the applet below, move the slider for k to vertically translate the circle and move the slider for r to adjust the radius.

Loading interactive...

When we subtract k from y then the circle is translated up k units. When we add k to y then the circle is translated down k units. As the radius increases the size of the circle also increases.

Notice that the centre of the circle x^{2} + y^{2} = 1 is at (0,0). Translating the circle will also translate the centre by the same amount. So the centre of \left(x-h \right)^{2} + \left(y-k \right)^{2} = r^{2} is at (h,k).

Examples

Example 3

The circle x^{2} + y^{2}=4^{2} is translated 4 units down. Which of the following diagrams shows the new location of the circle?

A
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B
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Worked Solution
Create a strategy

Sketch the original circle then translate it.

Apply the idea
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The circle with equation x^{2} + y^{2}=4^{2} is centred at the origin and with radius 4. The circle is graphed here.

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To translate the circle 4 units down, we move the centre from (0,0) to (0,-4), along with the rest of the points on the circle.

The translated circle is the lower circle shown here, labelled x^2+(y+4)^2=4^2.

The translated circle corresponds to option A which is the correct answer.

Idea summary

Circles of the form x^{2} + y^{2} = r^2 can be vertically translated by k units up get: \\ x^{2} + \left(y-k \right)^{2} = r^2 or k units down to get: x^{2} + \left(y-k \right)^{2} = r^2.

The general equation of a circle is given by:

\displaystyle (x-h)^2+(y-k)^2=r^2
\bm{(h,k)}
are the coordinates of the centre of the circle
\bm{r}
is the radius of the circle

Outcomes

VCMNA311

Graph simple non-linear relations with and without the use of digital technologies and solve simple related equations

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