Textbook Question
In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² – 10x – 6y – 30 = 0
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3. Functions
Intro to Functions & Their Graphs
2:38 minutesProblem 47
Textbook Question
In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x + 2)² + (y - 2)² = 4
Verified step by step guidance1
Identify the standard form of a circle's equation: \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
Compare the given equation \((x + 2)^2 + (y - 2)^2 = 4\) with the standard form.
Rewrite \((x + 2)^2\) as \((x - (-2))^2\) to identify \(h = -2\).
Rewrite \((y - 2)^2\) as \((y - 2)^2\) to identify \(k = 2\).
Recognize that \(r^2 = 4\), so \(r = \sqrt{4}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Equation of a Circle
The standard form of the equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. In the given equation, (x + 2)² + (y - 2)² = 4, we can identify the center as (-2, 2) and the radius as 2, since r² = 4 implies r = 2.
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Circles in Standard Form
Graphing Circles
To graph a circle, plot the center point on the coordinate plane and use the radius to mark points in all directions (up, down, left, right) from the center. The resulting shape is a circle, and understanding how to accurately represent this visually is crucial for analyzing the relation's domain and range.
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Domain and Range
The domain of a relation refers to all possible x-values, while the range refers to all possible y-values. For the circle described, the domain is the interval [-4, 0] and the range is [0, 4], as these values encompass the horizontal and vertical extents of the circle based on its center and radius.
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