Textbook Question
Use the graph of y = f(x) to graph each function g. g(x) = f(x + 1) − 2
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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 23
Find the midpoint of each line segment with the given endpoints. (-3, -4) and (6, −8)
Verified step by step guidance1
Recall that the midpoint \( M \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ M = \left( \frac{ x_1 + x_2 }{ 2 }, \frac{ y_1 + y_2 }{ 2 } \right) \]
Identify the coordinates of the given endpoints: \( (x_1, y_1) = (-3, -4) \) and \( (x_2, y_2) = (6, -8) \).
Substitute the values into the midpoint formula:
\[ M = \left( \frac{ -3 + 6 }{ 2 }, \frac{ -4 + (-8) }{ 2 } \right) \]
Simplify the expressions inside the parentheses separately:
\[ \frac{ -3 + 6 }{ 2 } = \frac{3}{2} \quad \text{and} \quad \frac{ -4 + (-8) }{ 2 } = \frac{ -12 }{ 2 } \]
Write the midpoint as the ordered pair using the simplified values:
\[ M = \left( \frac{3}{2}, -6 \right) \]
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Coordinate Plane and Points
The coordinate plane is a two-dimensional system where each point is defined by an ordered pair (x, y). Understanding how to plot and interpret points is essential for visualizing line segments and their properties.
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Graphs & the Rectangular Coordinate System
Line Segment
A line segment is the part of a line bounded by two endpoints. Knowing the endpoints allows us to analyze properties like length and midpoint, which are fundamental in geometry and algebra.
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Midpoint Formula
The midpoint formula calculates the point exactly halfway between two endpoints. It is given by ((x1 + x2)/2, (y1 + y2)/2), averaging the x-coordinates and y-coordinates separately to find the midpoint.
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Related Practice
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The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ-1 (x)) = = x and ƒ-1 (f(x)) = x. f(x) = √x
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Write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions. The graph of ƒ is perpendicular to the line whose equation is 3x - 2y - 4 = 0 and has the same y-intercept as this line.
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