Textbook Question
Graph each equation in a rectangular coordinate system. y = -2
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 48
In Exercises 46–49, give the slope and y-intercept of each line whose equation is given. Then graph the line. 2x + 3y + 6 = 0
Verified step by step guidance1
Rewrite the given equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. Start by isolating the term with y. Subtract 2x and 6 from both sides: 3y = -2x - 6.
Divide every term in the equation by 3 to solve for y: y = (-2/3)x - 2. Now the equation is in slope-intercept form.
Identify the slope (m) and the y-intercept (b) from the equation y = (-2/3)x - 2. The slope is m = -2/3, and the y-intercept is b = -2.
To graph the line, start by plotting the y-intercept (0, -2) on the coordinate plane. This is the point where the line crosses the y-axis.
Use the slope m = -2/3 to find another point on the line. From the y-intercept, move down 2 units (negative rise) and right 3 units (positive run). Plot this second point, then draw a straight line through the two points to complete the graph.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Slope-Intercept Form
The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope and b represents the y-intercept. This form is useful for quickly identifying the slope and y-intercept of a line, making it easier to graph the equation. Understanding this format allows students to convert standard forms of equations into slope-intercept form for analysis.
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Slope-Intercept Form
Finding the Slope
The slope of a line indicates its steepness and direction, calculated as the change in y over the change in x (rise over run). In the context of the equation provided, rearranging it into slope-intercept form will reveal the slope directly. A positive slope indicates the line rises from left to right, while a negative slope indicates it falls.
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06:49The Slope of a Line
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis, represented by the value of y when x equals zero. In the slope-intercept form, this value is directly given as b. Identifying the y-intercept is crucial for graphing the line accurately, as it provides a starting point on the y-axis.
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Related Practice
Textbook Question
In Exercises 31–50, find fg and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)
Textbook Question
Use the graph of y = f(x) to graph each function g. g(x) = -f(x + 1) − 1
Textbook Question
In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. x² + (y − 1)² = 1
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Textbook Question
In Exercises 31–50, find f/g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)
Textbook Question
Find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)