Middle Years

Lesson

Graphs of equations of the form $\left(x-h\right)^2+\left(y-k\right)^2=r^2$(`x`−`h`)2+(`y`−`k`)2=`r`2 (where $h$`h`, $k$`k`, and $r$`r` are any number and $r\ne0$`r`≠0) are called circles.

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

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

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

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

Summary

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

Circles have a centre at $\left(h,k\right)$(`h`,`k`) and a radius of $r$`r`

Circles can be transformed in the following ways (starting with the circle defined by $x^2+y^2=1$`x`2+`y`2=1):

- Vertically translated by $k$
`k`units: $x^2+\left(y-k\right)^2=1$`x`2+(`y`−`k`)2=1 - Horizontally translated by $h$
`h`units: $\left(x-h\right)^2+y^2=1$(`x`−`h`)2+`y`2=1 - Scaled (both vertically and horizontally) by a scale factor of $r$
`r`: $x^2+y^2=r^2$`x`2+`y`2=`r`2

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

Plot the graph for the given circle.

Loading Graph...Write the equation of the circle.

Consider the circle $x^2+y^2=25$`x`2+`y`2=25.

State the coordinates of the centre of the circle.

What is the radius of the circle?

Find the $x$

`x`values of the $x$`x`-intercepts. Write all solutions on the same line separated by a comma.Find the $y$

`y`values of the $y$`y`-intercepts. Write all solutions on the same line separated by a comma.Graph the circle.

Loading Graph...

The circle $x^2+y^2=4^2$`x`2+`y`2=42 is translated $4$4 units down. Which of the following diagrams shows the new location of the circle?

- Loading Graph...ALoading Graph...BLoading Graph...ALoading Graph...B