Textbook Question
In Exercises 64–66, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. r(x) = 2√(x + 2)
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 3, Problem 66
A line segment through the center of each circle intersects the circle at the points shown. a. Find the coordinates of the circle's center. b. Find the radius of the circle. c. Use your answers from parts (a) and (b) to write the standard form of the circle's equation.
Verified step by step guidance1
Identify the two points on the circle that the line segment passes through. For the first circle, these points are (-5, 9) and (-3, 5). For the second circle, the points are (3, 6) and (5, 4).
Find the center of the circle by calculating the midpoint of the line segment connecting the two points. Use the midpoint formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).
Calculate the radius of the circle by finding the distance from the center to either of the two points on the circle. Use the distance formula: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), then divide by 2 since the segment is a diameter.
Write the standard form of the circle's equation using the center \((h, k)\) and radius \(r\): \[(x - h)^2 + (y - k)^2 = r^2\].
Substitute the values of the center coordinates and the radius squared into the standard form equation to express the equation of the circle.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Midpoint Formula
The midpoint formula finds the center point between two given points by averaging their x-coordinates and y-coordinates. It is essential here to find the circle's center since the line segment passes through the center and connects two points on the circle.
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Distance Formula
The distance formula calculates the length between two points in the coordinate plane. It helps determine the radius of the circle by finding the distance from the center to one of the points on the circle.
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Standard Form of a Circle's Equation
The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Using the center and radius found, this formula expresses the circle algebraically.
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Related Practice
Textbook Question
In Exercises 59-66, a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Use the slope and y-intercept to graph the linear function. 4y+ 28 = 0
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Textbook Question
Find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2).
f(x) = 1/x, g(x)= 1/x
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Textbook Question
Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = (1/2) (x − 1)² – 1
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Textbook Question
Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = -2(x+2)²+1
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Textbook Question
Use the graph of f to find each indicated function value.
f(-2)
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