### About The Circle:

When working with circles, we want to be able to place the equation of the circle in center-radius form. This allows us to quickly determine the center and radius of the circle from its equation. We can use our equation to quickly graph the circle. This can easily be done by plotting the center and four additional points based on the radius. A smooth curve then connects the points.

Test Objectives

- Demonstrate the ability to place the equation of a circle in center-radius form
- Demonstrate the ability to find the center and radius from the equation of a circle
- Demonstrate the ability to sketch the graph of a circle

#1:

Instructions: Identify the center and radius of each, and then sketch the graph.

a) $$(x + 1)^2 + y^2 = 9$$

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#2:

Instructions: Identify the center and radius of each, and then sketch the graph.

a) $$(x - 2)^2 + (y + 3)^2 = 16$$

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#3:

Instructions: Identify the center and radius of each, and then sketch the graph.

a) $$x^2 + y^2 + 6x + 8 = 0$$

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#4:

Instructions: Identify the center and radius of each, and then sketch the graph.

a) $$2x^2 + 2y^2 + 6x + 14y + 27 = 0$$

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#5:

Instructions: Identify the center and radius of each, and then sketch the graph.

a) $$4x^2 + 4y^2 + 4x + 24y + 21 = 0$$

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Written Solutions:

#1:

Solutions:

a) $$center\hspace{.25em}radius-form:$$ $$(x-(-1))^2 + (y-0)^2 = 3^2$$ $$Center: (-1,0)$$ $$Radius: 3$$

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#2:

Solutions:

a) $$center\hspace{.25em}radius-form:$$ $$(x - 2)^2 + (y - (-3))^2 = 4^2$$ $$Center: (2,-3)$$ $$Radius: 4$$

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#3:

Solutions:

a) $$center\hspace{.25em}radius-form:$$ $$(x - (-3))^2 + (y - 0)^2 = 1^2$$ $$Center: (-3,0)$$ $$Radius: 1$$

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#4:

Solutions:

a) $$center\hspace{.25em}radius-form:$$ $$\left(x - \left(\frac{3}{2}\right)\right)^2 + \left(y - \left(-\frac{7}{2}\right)\right)^2 = 1^2$$ $$Center: \left(-\frac{3}{2},-\frac{7}{2}\right)$$ $$Radius: 1$$

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#5:

Solutions:

a) $$center\hspace{.25em}radius-form:$$ $$\left(x - \left(-\frac{1}{2}\right)\right)^2 + (y - (-3))^2 = 2^2$$ $$Center: \left(-\frac{1}{2},-3\right)$$ $$Radius: 2$$