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 - 2x + y² – 15 = 0
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3. Functions
Intro to Functions & Their Graphs
2:55 minutesProblem 51
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 + 1)² + y² = 25
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.
Rewrite the given equation \((x + 1)^2 + y^2 = 25\) in the standard form by recognizing that \((x + 1)\) can be written as \((x - (-1))\).
From the rewritten equation \((x - (-1))^2 + (y - 0)^2 = 25\), identify the center \((h, k)\) of the circle as \((-1, 0)\).
Determine the radius \(r\) by taking the square root of 25, which is \(r = 5\).
To find the domain and range from the graph, note that the circle is centered at \((-1, 0)\) with radius 5, so the domain is \([-6, 4]\) and the range is \([-5, 5]\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Circle Equation
The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. In the given equation, (x + 1)² + y² = 25, we can identify the center as (-1, 0) and the radius as 5, since r² = 25 implies r = 5.
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Circles in Standard Form
Graphing Circles
To graph a circle, plot the center point and then 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 domain and range.
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Domain and Range
The domain of a circle is the set of all x-values that the circle can take, while the range is the set of all y-values. For the circle described, the domain is [-6, 4] and the range is [-5, 5], reflecting the horizontal and vertical extents of the circle based on its center and radius.
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