Textbook Question
Begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. r(x) = (x − 2)³ +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 103
Exercises 103–105 will help you prepare for the material covered in the next section. Let (x1, y₁) = (7, 2) and (x2, y2) = (1, −1). Find √[(x2 − x1)² + (y2 − y₁)²]. Express the - answer in simplified radical form.
Verified step by step guidance1
Identify the formula for the distance between two points in a coordinate plane: d = √[(x₂ − x₁)² + (y₂ − y₁)²].
Substitute the given coordinates (x₁, y₁) = (7, 2) and (x₂, y₂) = (1, −1) into the formula: d = √[(1 − 7)² + (−1 − 2)²].
Simplify the expressions inside the parentheses: (1 − 7) becomes −6, and (−1 − 2) becomes −3. The formula now becomes d = √[(-6)² + (-3)²].
Square the values: (-6)² = 36 and (-3)² = 9. The formula now becomes d = √[36 + 9].
Add the squared values: 36 + 9 = 45. The distance formula simplifies to d = √45. Simplify the radical by factoring: √45 = √(9 × 5) = 3√5.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Distance Formula
The distance formula is a mathematical equation used to determine the distance between two points in a Cartesian coordinate system. It is derived from the Pythagorean theorem and is expressed as √[(x2 - x1)² + (y2 - y1)²]. This formula calculates the straight-line distance between points (x1, y1) and (x2, y2), which is essential for solving problems involving spatial relationships.
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Simplified Radical Form
Simplified radical form refers to the expression of a square root in its simplest terms, where no perfect square factors remain under the radical sign. For example, √8 can be simplified to 2√2. Understanding how to simplify radicals is crucial for presenting answers in a clear and concise manner, especially in algebraic contexts.
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Coordinate System
A coordinate system is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point in this system is represented by an ordered pair (x, y), which indicates its position relative to the axes. Familiarity with the coordinate system is vital for interpreting and solving problems involving points, distances, and geometric figures.
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Related Practice
Textbook Question
Solve and graph the solution set on a number line: 3|2x-1| ≥ 21
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Textbook Question
Solve: 5x3/4- 15 = 0.
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Textbook Question
Begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. h(x) = x³/2
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Textbook Question
In Exercises 105–106, find the midpoint of each line segment with the given endpoints. (2, 6) and (-12, 4)
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Textbook Question
Exercises 103–105 will help you prepare for the material covered in the next section. Use a rectangular coordinate system to graph the circle with center (1, -1) and radius 1.
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