Textbook Question
Use the graph of y = f(x) to graph each function g.
g(x) = f(x)+1
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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 1
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. (4, 7) and (8, 10)
Verified step by step guidance1
Identify the coordinates of the two points given: Point 1 is \((4, 7)\) and Point 2 is \((8, 10)\).
Recall the formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substitute the coordinates into the slope formula:
\[m = \frac{10 - 7}{8 - 4}\]
Simplify the numerator and denominator separately to find the slope fraction:
\[m = \frac{3}{4}\]
Interpret the slope: since \(m\) is positive and a fraction, the line rises as it moves from left to right.
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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 measures the steepness and direction of a line, calculated as the ratio of the change in y-values to the change in x-values between two points. It is found using the formula m = (y2 - y1) / (x2 - x1). A positive slope indicates the line rises, while a negative slope means it falls.
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Undefined Slope
A slope is undefined when the line is vertical, meaning the x-values of both points are the same. Since division by zero is undefined, the slope formula cannot be applied, and the line does not rise or fall but goes straight up and down.
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Interpreting Slope Direction
The sign and value of the slope determine the line's direction: positive slope means the line rises from left to right, negative slope means it falls, zero slope indicates a horizontal line, and undefined slope corresponds to a vertical line.
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Related Practice
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Find the domain of each function. f(x)=3(x-4)
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Write an equation for line L in point-slope form and slope-intercept form.
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Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (2, 3) and (14, 8)
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