圆：根据平移书写方程式

Equations From Translations

This topic has already been explained pretty well here and here. Just as recap though, there are two types of translation. These are translation horizontally and vertically. You can combine these. To translate a circle 3 units to the right and 4 units up, we take our equation, $x^2 + y^2 = r^2$ and subtract $3$ from the $x$ term and $4$ from the $y$ term. The equation will now look like this $(x - 3)^2 + (y - 4)^2 = r^2$. As we can see, we now have a general formula. We subtract $h$ units from the $x^2$ term to move the circle right $h$ units, and subtract $k$ units from the $y^2$ term. Our new formula is now:

$(x - h)^2 + (y - k)^2 = r^2$

Wait a second - your first thought might be that we should add $3$ and $4$ to the $x$ and $y$! If this is confusing to you, use the link above. Now, some review.

(Video) Equation of a Translated Circle

by Eddie Woo

In this video by Eddie Woo, you will learn about the equation of a translated circle.

Summary

To find the equation of the circle, we use the distance formula. We can use our translated center to find the equation of a translated circle.

We can plug in the center of the circle into the distance formula to find the new equation. We know that the center of the circle is at $(-2,4)$. We know that the radius is $3$.

$$\begin{align} d &= \sqrt{(x_1-x_2)^2+(y_1-y_2)^2} \\ 3 &= \sqrt{(x +2 )^2+(y-4)^2} \\ 9 &= (x +2 )^2+(y-4)^2 \\ \end{align}$$

Writing an Equation of a Circle for a Translated Circle

We know that a circle is a set of points that are all the same distance from a center point. The standard equation of a circle is:

$$(x-h)^2 + (y-k)^2 = r^2$$

$(h,k)$ is the center, and $r$ is the radius.

When a circle is translated, we can use the distance formula to find the equation of the new circle. For example, if a circle is centered at $(3,2)$ and the radius is $5$, the equation is:

$$(x-3)^2 + (y+2)^2 = 25$$

Notice here that the signs inside the parentheses are opposite of the center's coordinates. This is because the equation subtracts the center values to keep the distance formula correct.

Translated circles can shift up, down, left, or right. However - their shape will always stay the same. Once you look at a graph to find the center and radius, you can plug them into the formula. You will always be able to use the center and radius to graph.

In the next lesson, you will learn about how to complete the square to convert the equations of a circle into different forms.