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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 2 - Functions and GraphsProblema 43
Capitolo 3, Problema 43

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 − 3)² + (y + 1)² = 36

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1
Recognize that the given equation is in the standard form of a circle: \(\left(x - h\right)^2 + \left(y - k\right)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
Identify the center \((h, k)\) by comparing the given equation \(\left(x - 3\right)^2 + \left(y + 1\right)^2 = 36\) to the standard form. Note that \(y + 1\) can be rewritten as \(y - (-1)\), so the center is at \((3, -1)\).
Determine the radius \(r\) by taking the square root of the right side of the equation: \(r = \sqrt{36}\).
To find the domain of the circle, consider the horizontal distance from the center. The domain is all \(x\) values such that \(x\) is between \(h - r\) and \(h + r\), or \(3 - r \leq x \leq 3 + r\).
To find the range of the circle, consider the vertical distance from the center. The range is all \(y\) values such that \(y\) is between \(k - r\) and \(k + r\), or \(-1 - r \leq y \leq -1 + r\).

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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. This form helps identify the circle's position and size directly from the equation.
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Circles in Standard Form

Domain and Range of a Circle

The domain and range of a circle are the sets of possible x and y values, respectively. For a circle centered at (h, k) with radius r, the domain is [h - r, h + r] and the range is [k - r, k + r], representing all points covered by the circle horizontally and vertically.
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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 confirm the circle's shape and aids in identifying the domain and range by observing the extent of the graph along the x- and y-axes.
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Circles in Standard Form
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