A circle can be scaled both vertically and horizontally by changing the value of r. In fact, r is the radius of the circle.
A circle has its centre at the origin and a radius of 9 units.
Plot the graph for the given circle.
Write the equation of the circle.
Consider the circle x^{2} + y^{2} = 64.
Find the x values of the x-intercepts.
Find the y values of the y-intercepts.
Plot the graph for the given circle.
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}.
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.
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.
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.
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.
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.
In the applet below, move the slider for k to vertically translate the circle and move the slider for r to adjust the radius.
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).
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?
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: