📚 Topic Summary
Completing the square is a technique used to rewrite quadratic expressions in a more convenient form. When dealing with circles, this technique allows us to transform the equation of a circle from its general form to its standard form, $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. This makes it easy to identify the circle's center and radius directly from the equation. By applying algebraic manipulation, we can convert a complex equation into a simple and understandable representation of a circle.
This method involves manipulating the equation by adding and subtracting specific values to create perfect square trinomials for both $x$ and $y$ terms. Once the equation is in standard form, graphing and analyzing the circle becomes straightforward.
🧮 Part A: Vocabulary
Match the following terms with their definitions:
- Term: Radius
- Term: Center
- Term: Standard Form
- Term: Completing the Square
- Term: General Form
Definitions:
- The form $(x - h)^2 + (y - k)^2 = r^2$
- The distance from the center to any point on the circle
- A technique to rewrite quadratics
- The point (h, k) in the middle of the circle
- The form $Ax^2 + Ay^2 + Bx + Cy + D = 0$
| Term |
Definition |
| Radius |
The distance from the center to any point on the circle |
| Center |
The point (h, k) in the middle of the circle |
| Standard Form |
The form $(x - h)^2 + (y - k)^2 = r^2$ |
| Completing the Square |
A technique to rewrite quadratics |
| General Form |
The form $Ax^2 + Ay^2 + Bx + Cy + D = 0$ |
✍️ Part B: Fill in the Blanks
To convert the general form of a circle's equation to standard form, we use a technique called __________ ___________ ___________. This involves creating ___________ ___________ ___________ for both the $x$ and $y$ terms. The standard form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the ___________ of the circle and $r$ represents the ___________.
Answer: completing the square, perfect square trinomials, center, radius
🤔 Part C: Critical Thinking
Explain in your own words why completing the square is a useful technique when working with equations of circles. Give an example of a situation where using this technique would be particularly helpful.