Textbook Question
Evaluate each function at the given values of the independent variable and simplify. h(x) = x4 - x2 +1 b. h (-1)
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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 31
Write the standard form of the equation of the circle with the given center and radius. Center (0, 0), r = 7
Verified step by step guidance1
Recall that the standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is given by the formula:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Identify the center \((h, k)\) and radius \(r\) from the problem. Here, the center is \((0, 0)\) and the radius is \(7\).
Substitute the values of \(h = 0\), \(k = 0\), and \(r = 7\) into the standard form equation:
\[ (x - 0)^2 + (y - 0)^2 = 7^2 \]
Simplify the equation by removing the zeros inside the parentheses and squaring the radius:
\[ x^2 + y^2 = 49 \]
This equation represents the circle in standard form with the given center and radius.
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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. This form clearly shows the circle's center coordinates and radius, making it easy to graph or analyze.
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Circles in Standard Form
Center of the Circle
The center of a circle is the fixed point from which all points on the circle are equidistant. In the equation, the center is represented by (h, k). For this problem, the center is at the origin (0, 0), simplifying the equation.
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Radius of the Circle
The radius is the distance from the center to any point on the circle. It is a positive value denoted by r in the equation. Squaring the radius (r²) is essential in the standard form to represent the circle's size accurately.
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Related Practice
Textbook Question
In Exercises 31–32, the domain of each piecewise function is (-∞, ∞) (a) Graph each function. (b) Use the graph to determine the function's range.
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Textbook Question
Which graphs in Exercises 29–34 represent functions that have inverse functions?
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Textbook Question
Find ƒ+g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1
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Textbook Question
Evaluate each function at the given values of the independent variable and simplify. h(x) = x4 - x2+1 a. h (2)
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Textbook Question
Evaluate each function at the given values of the independent variable and simplify. g(x) = x² - 10x - 3 c. g(-x)
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