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 21
Write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions. The graph of ƒ passes through (−6, 4) and is perpendicular to the line that has an x intercept of 2 and a y-intercept of -4.
Verified step by step guidance1
Identify the slope of the given line using its intercepts. The line passes through the points (2, 0) and (0, -4). Use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
Calculate the slope of the given line by substituting the points: \(m = \frac{0 - (-4)}{2 - 0} = \frac{4}{2}\).
Determine the slope of the line perpendicular to the given line. Recall that perpendicular slopes are negative reciprocals, so if the original slope is \(m\), the perpendicular slope is \(-\frac{1}{m}\).
Use the point-slope form of a line equation with the perpendicular slope and the given point \((-6, 4)\): \(y - y_1 = m(x - x_1)\).
Rewrite the equation from the previous step into slope-intercept form \(y = mx + b\) by solving for \(y\) and simplifying.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Slope-Intercept Form of a Linear Equation
The slope-intercept form is y = mx + b, where m represents the slope and b is the y-intercept. This form makes it easy to identify the slope and where the line crosses the y-axis, which is essential for graphing and writing linear equations.
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03:56
Slope-Intercept Form
Finding the Slope from Intercepts
A line's slope can be found using two points, such as the x- and y-intercepts. The slope m is calculated as the change in y divided by the change in x (m = (y2 - y1) / (x2 - x1)). Knowing the slope of the given line is necessary to find the slope of the perpendicular line.
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03:56Slope-Intercept Form
Perpendicular Lines and Their Slopes
Two lines are perpendicular if their slopes are negative reciprocals, meaning m1 * m2 = -1. If one line has slope m, the perpendicular line's slope is -1/m. This relationship helps determine the slope of the required line passing through a given point.
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Related Practice
Textbook Question
Find the midpoint of each line segment with the given endpoints. (-2, -8) and (−6, −2)
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Textbook Question
Determine whether each equation defines y as a function of x. x+y³ = 8
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
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Textbook Question
Use the graph of y = f(x) to graph each function g.
g(x) = f(x-1)+2
1
views
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) = 1/x