Comprehensive graphic organizer helping students master the standard equation of a circle through guided notes and practice problems.

Here are the solutions to the problems on the worksheet.

Part 1: Fill in the Blanks

Equation Definitions:
* (h, k) = Center of the circle
* (x, y) = Any point on the circle
* r = Radius

Sentence Completion:
*The equation of a circle is used to graph a circle or find parts of the circle such as the center, radius, or a point on the circle.*

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Part 2: Find the center and radius

To solve these, compare the given equation to the standard form: $(x-h)^2 + (y-k)^2 = r^2$.
* The center is $(h, k)$. Note: If it says $(x+4)$, then $h$ is $-4$.
* The radius is the square root of the number on the right side ($\sqrt{r^2}$).

1. $x^2 + y^2 = 25$
* This is the same as $(x-0)^2 + (y-0)^2 = 25$.
* Center: $(0, 0)$
* Radius: $\sqrt{25} = 5$

2. $(x+8)^2 + (y-3)^2 = 100$
* Change signs for the center: $x+8$ becomes $-8$, $y-3$ stays $3$.
* Center: $(-8, 3)$
* Radius: $\sqrt{100} = 10$

3. $(x-4)^2 + y^2 = 1$
* Center: $(4, 0)$ (since there is no number with $y$, $k=0$)
* Radius: $\sqrt{1} = 1$

4. $(x-5)^2 + (y+1)^2 = 144$
* Change signs for the center: $x-5$ stays $5$, $y+1$ becomes $-1$.
* Center: $(5, -1)$
* Radius: $\sqrt{144} = 12$

5. $x^2 + (y+2)^2 = 18$
* Center: $(0, -2)$
* Radius: $\sqrt{18}$ (This simplifies to $3\sqrt{2}$, but leaving it as $\sqrt{18}$ is often accepted depending on your class rules). Let's use the simplified version: $3\sqrt{2} \approx 4.24$.

6. $x^2 + y^2 = 16$
* Center: $(0, 0)$
* Radius: $\sqrt{16} = 4$

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Part 3: Write the equation of the circle

Plug the center $(h, k)$ and radius $r$ into the formula: $(x-h)^2 + (y-k)^2 = r^2$. Remember to flip the signs for the center coordinates inside the parentheses.

1. Center: $(0, 0)$ Radius: $4$
* Equation: $x^2 + y^2 = 16$

2. Center: $(2, -3)$ Radius: $15$
* Flip signs: $(x-2)$ and $(y+3)$. Square the radius: $15^2 = 225$.
* Equation: $(x-2)^2 + (y+3)^2 = 225$

3. Center: $(-7, 0)$ Radius: $10$
* Flip signs: $(x+7)$ and $y$. Square the radius: $10^2 = 100$.
* Equation: $(x+7)^2 + y^2 = 100$

4. Center: $(-5, -1)$ Radius: $2$
* Flip signs: $(x+5)$ and $(y+1)$. Square the radius: $2^2 = 4$.
* Equation: $(x+5)^2 + (y+1)^2 = 4$

5. Center: $(0, 0)$ Radius: $1$
* Equation: $x^2 + y^2 = 1$

6. Center: $(4, 8)$ Radius: $22$
* Keep signs: $(x-4)$ and $(y-8)$. Square the radius: $22^2 = 484$.
* Equation: $(x-4)^2 + (y-8)^2 = 484$

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Part 4: Graphing and Writing from Graphs

Example: Graph the circle $(x-1)^2 + (y-1)^2 = 9$
1. Find the Center: The equation has $(x-1)$ and $(y-1)$, so the center is at $(1, 1)$. Put a dot there.
2. Find the Radius: The number on the right is $9$. The radius is $\sqrt{9} = \mathbf{3}$.
3. Plot Points: From the center $(1,1)$, count 3 units Up, Down, Left, and Right.
* Up: $(1, 4)$
* Down: $(1, -2)$
* Right: $(4, 1)$
* Left: $(-2, 1)$
4. Draw: Connect these dots with a smooth curve to make a circle.

Example: Use the graph to write the equation
*(Looking at the graph in the bottom right corner)*
1. Find the Center: The black dot is at the origin. Center = $(0, 0)$.
2. Find the Radius: Count the grid squares from the center to the edge of the circle. It goes 2 units out. Radius = $2$.
3. Write Equation:
* Formula: $(x-0)^2 + (y-0)^2 = 2^2$
* Simplify: $x^2 + y^2 = 4$

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Final Answer:

Fill in the Blanks:
(h, k) = Center
(x, y) = Point on the circle
r = Radius
Sentence: graph, center, radius, point

Find Center and Radius:
1. C: (0, 0), r: 5
2. C: (-8, 3), r: 10
3. C: (4, 0), r: 1
4. C: (5, -1), r: 12
5. C: (0, -2), r: $\sqrt{18}$ or $3\sqrt{2}$
6. C: (0, 0), r: 4

Write the Equation:
1. $x^2 + y^2 = 16$
2. $(x-2)^2 + (y+3)^2 = 225$
3. $(x+7)^2 + y^2 = 100$
4. $(x+5)^2 + (y+1)^2 = 4$
5. $x^2 + y^2 = 1$
6. $(x-4)^2 + (y-8)^2 = 484$

Graph/Write from Graph:
Graph: Plot center at (1,1) with radius 3.
Equation from graph: $x^2 + y^2 = 4$