Textbook Question
Determine whether each relation is a funciton, Give the domain and range for each relation. (1, 10), (2, 500), (13, π)
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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 3
Write an equation for line L in point-slope form and slope-intercept form.
Verified step by step guidance1
Identify the slope of the given blue line. The equation is given as \(y = \frac{x}{2} - 2\), so the slope \(m_1\) is \(\frac{1}{2}\).
Since line L (red line) is perpendicular to the blue line, find the slope of line L. The slope of a line perpendicular to another is the negative reciprocal of the original slope. So, \(m_2 = -\frac{1}{m_1} = -2\).
Determine a point on line L from the graph. From the image, the red line passes through the point \((2, -2)\).
Write the equation of line L in point-slope form using the point \((2, -2)\) and slope \(-2\): \(y - y_1 = m(x - x_1)\), which becomes \(y - (-2) = -2(x - 2)\).
Convert the point-slope form to slope-intercept form by simplifying and solving for \(y\): \(y = -2x + b\), where \(b\) is the y-intercept found by substituting the point into the equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Slope of a Line
The slope of a line measures its steepness and is calculated as the ratio of the vertical change to the horizontal change between two points. It is often represented as 'm' in the equation y = mx + b. Understanding slope is essential for writing equations of lines and analyzing their relationships.
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06:49
The Slope of a Line
Perpendicular Lines and Their Slopes
Two lines are perpendicular if the product of their slopes is -1. This means the slope of one line is the negative reciprocal of the other. Recognizing this relationship helps in finding the slope of a line perpendicular to a given line, which is crucial for writing its equation.
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07:52Parallel & Perpendicular Lines
Point-Slope and Slope-Intercept Forms of a Line
The point-slope form, y - y1 = m(x - x1), uses a known point and slope to write a line's equation. The slope-intercept form, y = mx + b, expresses the line using slope and y-intercept. Both forms are fundamental for representing lines and converting between different equation formats.
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05:46Point-Slope Form
Related Practice
Textbook Question
Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. (-2, 1) and (2, 2)
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Textbook Question
Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (4, -1) and (-6, 3)
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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
Find the domain of each function. g(x) = 3/(x-4)
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Textbook Question
Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x)=3x+8 and g(x) = (x-8)/3
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