Textbook Question
Solve each equation in Exercises 15–34 by the square root property.
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Ch. 1 - Equations and Inequalities
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 2, Problem 23a
Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = |x| + 1
Verified step by step guidance1
Step 1: Understand the equation y = |x| + 1. This is an absolute value function, where |x| represents the absolute value of x. The graph of |x| is a V-shaped graph, and the '+1' shifts the graph vertically upward by 1 unit.
Step 2: Create a table of values for x and y. Use the given x-values: -3, -2, -1, 0, 1, 2, 3. For each x-value, calculate y by substituting x into the equation y = |x| + 1. Remember that the absolute value of a number is always non-negative.
Step 3: For example, when x = -3, |x| = 3, so y = 3 + 1 = 4. Similarly, calculate y for all other x-values: -2, -1, 0, 1, 2, and 3.
Step 4: Plot the points (x, y) on a coordinate plane using the calculated values from the table. For instance, plot points like (-3, 4), (-2, 3), (-1, 2), (0, 1), (1, 2), (2, 3), and (3, 4).
Step 5: Connect the points to form the graph. The graph will have a V-shape, with the vertex at (0, 1) and the arms of the V extending symmetrically upward. This is the graph of y = |x| + 1.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value Function
The absolute value function, denoted as |x|, measures the distance of a number x from zero on the number line, always yielding a non-negative result. For example, |3| = 3 and |-3| = 3. This function is crucial for understanding how the graph behaves, as it creates a V-shape, reflecting the positive and negative values of x symmetrically.
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Function Composition
Graphing Linear Equations
Graphing linear equations involves plotting points on a coordinate plane that satisfy the equation. In this case, the equation y = |x| + 1 can be graphed by calculating y for various x values. The resulting points are then connected to form the graph, which illustrates the relationship between x and y visually.
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Transformations of Functions
Transformations of functions involve shifting, reflecting, or stretching the graph of a function. In the equation y = |x| + 1, the '+1' indicates a vertical shift of the absolute value graph upwards by one unit. Understanding transformations helps in predicting how changes to the equation affect the graph's position and shape.
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Related Practice
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In Exercises 21–28, divide and express the result in standard form. 5i/(2 - i)
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Textbook Question
Solve each equation in Exercises 15–34 by the square root property.
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Divide and express the result in standard form. 8i/(4 - 3i)
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In Exercises 15–26, use graphs to find each set. [3, ∞) ∩ (6, ∞)
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In Exercises 1–26, solve and check each linear equation. 16 = 3(x - 1) - (x - 7)
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