Textbook Question
Evaluate each expression. See Example 5. |-8|
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Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem 53
In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 5 and 6.
center (2, 0), radius 6
Verified step by step guidance1
Recall that the center-radius form of a circle's equation is given by \[(x - h)^2 + (y - k)^2 = r^2\] where \[(h, k)\] is the center and \[r\] is the radius.
Identify the center coordinates and radius from the problem: the center is \[(2, 0)\] and the radius is \[6\].
Substitute the center coordinates into the formula: replace \[h\] with \[2\] and \[k\] with \[0\], so the equation becomes \[(x - 2)^2 + (y - 0)^2 = r^2\].
Substitute the radius value into the equation: replace \[r\] with \[6\], so the equation becomes \[(x - 2)^2 + y^2 = 6^2\].
Simplify the right side by squaring the radius: the equation is \[(x - 2)^2 + y^2 = 36\]. This is the center-radius form of the circle's equation.
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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 in Center-Radius Form
The center-radius 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 directly shows the circle's location and size, making it easy to write the equation when the center and radius are known.
Recommended video:
06:03
Equations of Circles & Ellipses
Coordinate Geometry and Plotting Points
Understanding how to plot points on the Cartesian plane is essential for graphing the circle. The center (h, k) is the reference point, and the radius determines the distance from the center to any point on the circle, guiding the drawing of the circle's boundary.
Recommended video:
3:00Determining Different Coordinates for the Same Point Example 2
Radius and Distance in the Plane
The radius is the fixed distance from the center to any point on the circle. Knowing how to interpret and use this distance helps in both writing the equation and graphing the circle accurately, ensuring all points satisfy the distance condition.
Recommended video:
5:04Converting between Degrees & Radians
Related Practice
Textbook Question
Let f(x) = -3x + 4 and g(x) = -x² + 4x + 1. Find each of the following. Simplify if necessary. See Example 6. ƒ(p)
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Textbook Question
In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 5 and 6.
center (0, 4), radius 4
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Textbook Question
Add or subtract, as indicated. See Example 4. (5/12x²y) - (11/6xy)
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Textbook Question
Find each product. See Example 5. (4m + 2n)²
Textbook Question
Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = x³ - x + 9
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