Textbook Question
In Exercises 11–26, determine whether each equation defines y as a function of x. 4x = y²
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 19
Find the midpoint of each line segment with the given endpoints. (6, 8) and (2, 4)
Verified step by step guidance1
Recall the midpoint formula for a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\):
\[\text{Midpoint} = \left( \frac{\,x_1 + x_2}{2}, \frac{\,y_1 + y_2}{2} \right)\]
Identify the coordinates of the given endpoints:
\( (x_1, y_1) = (6, 8) \) and \( (x_2, y_2) = (2, 4) \).
Substitute the values of \(x_1\), \(x_2\), \(y_1\), and \(y_2\) into the midpoint formula:
\[\left( \frac{6 + 2}{2}, \frac{8 + 4}{2} \right)\]
Simplify the expressions inside the parentheses by adding the coordinates:
\[\left( \frac{8}{2}, \frac{12}{2} \right)\]
Finally, divide each sum by 2 to find the midpoint coordinates:
\[\left( 4, 6 \right)\]
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Midpoint Formula
The midpoint formula calculates the point exactly halfway between two given points in a coordinate plane. It is found by averaging the x-coordinates and the y-coordinates of the endpoints separately: Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2). This formula helps locate the center of a line segment.
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Coordinate Plane
The coordinate plane is a two-dimensional surface defined by a horizontal x-axis and a vertical y-axis. Points are represented as ordered pairs (x, y), where x indicates horizontal position and y indicates vertical position. Understanding this system is essential for plotting points and calculating distances or midpoints.
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Line Segment
A line segment is a part of a line bounded by two distinct endpoints. Unlike a line, it has a fixed length. Knowing the endpoints of a line segment allows for calculations such as length, midpoint, and slope, which are fundamental in coordinate geometry.
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Related Practice
Textbook Question
use the graph of y = f(x) to graph each function g.
g(x) = f(x-1)
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Textbook Question
Write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions. The graph of ƒ passes through (−1, 5) and is perpendicular to the line whose equation is x = 6.
Textbook Question
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+2)³
Textbook Question
Determine whether each equation defines y as a function of x. y = √x +4
Textbook Question
Find the domain of each function. f(x) = 1/√(x - 3)
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