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College Algebra Chapter 2 Functions and Graphs
Section 2.2 Circles
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1. Write an Equation of a Circle in Standard Form
2. Write the General Form of an Equation of a Circle
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Write an Equation of a Circle in Standard Form
A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The fixed distance from any point on the circle to the center is called the radius.
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Write an Equation of a Circle in Standard Form
Standard Form of Equation of a Circle: Center: ( h, k ) Radius: r (r > 0)
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Write an Equation of a Circle in Standard Form
Example: Graph a circle with center ( –3, 2 ) and radius 4. Equation: Mark the center with an open circle. Plot points 4 units up, down, left, and right from the center. Draw the circle through the points.
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Example 1: Write an equation of the circle in standard form and graph the circle. Center: (0, –3) Radius: 2
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Example 2: Write an equation of the circle in standard form and graph the circle. Center: (4, 3) Radius: 3
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Example 3: Write an equation of the circle in standard form and graph the circle. The center is (3, –2) and the circle is tangent to the y-axis.
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1. Write an Equation of a Circle in Standard Form
2. Write the General Form of an Equation of a Circle
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Write the General Form of an Equation of a Circle
General Form of Equation of a Circle:
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Example 4: Write the equation of the circle in general form:
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Write the General Form of an Equation of a Circle
To write the equation of a circle from general form into standard form, we must complete the square twice.
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Example: Write the equation of the circle in standard form. Identify the center and radius. Group the x terms, group the y terms. Move the constant term to the right.
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Example continued: Complete the squares. Identify center and radius from the equation.
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Example 5: Write the equation of the circle in standard form. Identify the center and radius.
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Example 6: Write the equation of the circle in standard form. Identify the center and radius.
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Determining if an Equation Represents the Graph of a Circle
The equation represents a circle of radius r only if If , then represents the single point (h, k). The solution set is { (h, k) }. If , then the solution set is the empty set { }.
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Example 7: Determine if the equation represents the equation of a circle.
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Graphing a Circle on a Calculator
A basic graphing calculator usually requires equations in the form y = ____________ Given: Solve for y to obtain two equations
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Graphing a Circle on a Calculator, continued
Enter each equation. Graph on the standard viewing window. The graph appears distorted because of the rectangular screen on the calculator.
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Graphing a Circle on a Calculator, continued
To eliminate the distortion, go to the ZOOM menu and select ZSQUARE.
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Graphing a Circle on a Calculator, continued
ZDECIMAL will also produce a "square" window, but it is considerably smaller.
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Graphing a Circle on a Calculator, continued
We can zoom out once to see the entire graph.
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Graphing a Circle on a Calculator, continued
We can also make our own custom window.
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