Textbook Question
Find the value of the function for the given value of x. ƒ(x)=[[3-(x/2)]], for x=1
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Ch. 2 - Graphs and Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 3, Problem 48
Work each of the following. Find the equation of a circle with center at (-4, 3), passing through the point (5, 8).Write it in center-radius form.
Verified step by step guidance1
Recall that the center-radius form of a circle's equation is given by \(\left(x - h\right)^2 + \left(y - k\right)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
Identify the center \((h, k)\) from the problem: here, the center is \((-4, 3)\), so \(h = -4\) and \(k = 3\).
Calculate the radius \(r\) by finding the distance between the center \((-4, 3)\) and the point on the circle \((5, 8)\) using the distance formula: \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
Substitute the coordinates into the distance formula: \(r = \sqrt{(5 - (-4))^2 + (8 - 3)^2}\), then simplify inside the square root.
Write the equation of the circle by plugging \(h\), \(k\), and \(r^2\) into the center-radius form: \(\left(x - (-4)\right)^2 + \left(y - 3\right)^2 = r^2\).
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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 = r^2, where (h, k) is the center and r is the radius. This form directly relates the circle's geometric properties to its algebraic equation.
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Circles in Standard Form
Distance Formula
The distance formula calculates the distance between two points (x1, y1) and (x2, y2) as √[(x2 - x1)^2 + (y2 - y1)^2]. It is used here to find the radius by measuring the distance from the center to a point on the circle.
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06:36Solving Quadratic Equations Using The Quadratic Formula
Substitution into the Circle Equation
After finding the radius, substitute the center coordinates and radius squared into the center-radius form. This step finalizes the equation of the circle, ensuring it passes through the given point.
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5:18Circles in Standard Form
Related Practice
Textbook Question
Find the slope of each line, provided that it has a slope. through (2, -2) and (3, -4)
Textbook Question
Graph the line satisfying the given conditions. through (2, -4), m = 3/4
Textbook Question
For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=1/x
Textbook Question
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. 2x+3y=5
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Textbook Question
For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4.