Functions and Models Reference
Circles
A circle is the set of all points a fixed distance from a center. In coordinate form, that fixed distance is the radius.
Fact Table
| Form | What It Shows | Graphing Move |
|---|---|---|
| $(x-h)^2+(y-k)^2=r^2$ | Center $(h,k)$ and radius $r$. | Plot center, then move $r$ units up, down, left, and right. |
| $x^2+y^2=r^2$ | Center at the origin. | Use radius $\sqrt{r^2}$. |
| General form | Center is hidden. | Complete the square in $x$ and $y$. |
Content Formulas
Circle
$$(x-h)^2+(y-k)^2=r^2$$
Distance
$$r=\sqrt{(x-h)^2+(y-k)^2}$$
The signs inside the parentheses are opposite the center coordinates: $(x-3)^2+(y+2)^2=25$ has center $(3,-2)$.
Classic Examples
Read Center and Radius
Find the center and radius of $(x-4)^2+(y+1)^2=36$.
$$h=4,\quad k=-1$$
$$r=\sqrt{36}=6$$
$$\text{center: }(4,-1)$$
Complete the Square
Write $x^2+y^2-6x+4y-12=0$ in center-radius form.
$$(x^2-6x)+(y^2+4y)=12$$
$$(x^2-6x+9)+(y^2+4y+4)=12+9+4$$
$$(x-3)^2+(y+2)^2=25$$