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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 47
Chapter 3, Problem 47

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

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Identify the standard form of the circle equation, which is \(\left(x - h\right)^2 + \left(y - k\right)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
Compare the given equation \(\left(x + 2\right)^2 + \left(y - 2\right)^2 = 4\) to the standard form. Note that \(x + 2\) can be rewritten as \(x - (-2)\), so the center coordinates are \((-2, 2)\).
Determine the radius by taking the square root of the right side of the equation: \(r = \sqrt{4}\).
Use the center and radius to sketch the circle on a coordinate plane, plotting the center at \((-2, 2)\) and marking points \(r\) units away in all directions.
Identify the domain and range of the circle from the graph or equation: the domain is all \(x\) values within \(r\) units of the center's \(x\)-coordinate, and the range is all \(y\) values within \(r\) units of the center's \(y\)-coordinate.

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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 a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. By comparing the given equation to this form, you can identify the circle's center and radius directly.
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Circles in Standard Form

Graphing Circles

Graphing a circle involves plotting its center and using the radius to mark points in all directions. This visual representation helps in understanding the shape and position of the circle on the coordinate plane.
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Circles in Standard Form

Domain and Range of a Circle

The domain of a circle is the set of all possible x-values, and the range is the set of all possible y-values covered by the circle. These can be found by considering the center coordinates and radius, determining the intervals for x and y.
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Circles in Standard Form
Related Practice
Textbook Question

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = x³, g(x) = x³ +2

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Textbook Question

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x)= |x|, g(x) = |x| +1

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Textbook Question

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