Textbook Question
Use the graph of y = f(x) to graph each function g. g(x)=2f(x-1)
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 51
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 + 1)² + y² = 25
Verified step by step guidance1
Recognize that the given equation is in the standard form of a circle: \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.
Rewrite the equation \( (x + 1)^2 + y^2 = 25 \) as \( (x - (-1))^2 + (y - 0)^2 = 5^2 \) to identify the center and radius clearly.
From the rewritten form, identify the center of the circle as \( (-1, 0) \) and the radius as \( 5 \).
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 \(-1 - 5\) and \(-1 + 5\), or \([-6, 4]\).
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 \(0 - 5\) and \(0 + 5\), or \([-5, 5]\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of a Circle's Equation
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. Recognizing this form allows you to identify the circle's center and radius directly from the equation.
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Circles in Standard Form
Graphing Circles
Graphing a circle involves plotting its center and using the radius to mark points equidistant from the center in all directions. This visual representation helps in understanding the shape and position of the circle on the coordinate plane.
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5:18Circles 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. 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].
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5:18Circles in Standard Form
Related Practice
Textbook Question
Graph using intercepts: 2x - 5y - 10 = 0
Textbook Question
In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = √x (x = 0, 1, 4, 9) and g(x) = √x −1 (x = 0, 1, 4, 9)
Textbook Question
Use the graph of y = f(x) to graph each function g. g(x) =(1/2) f(2x)
Textbook Question
Find
a. (fog) (x)
b. (go f) (x)
c. (fog) (2)
d. (go f) (2).
f(x) = 2x, g(x) = x+7
Textbook Question
In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = ∛x + 1