Textbook Question
Determine whether each equation defines y as a function of x. 2x + y^2 = 6
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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 6
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, -1) and (3, −1)
Verified step by step guidance1
Identify the coordinates of the two points: Point 1 is (4, -1) and Point 2 is (3, -1).
Recall the formula for the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\):
\(slope = \frac{y_2 - y_1}{x_2 - x_1}\)
Substitute the given points into the slope formula:
\(slope = \frac{-1 - (-1)}{3 - 4}\)
Simplify the numerator and denominator separately:
Numerator: \(-1 - (-1) = -1 + 1 = 0\)
Denominator: \(3 - 4 = -1\)
Calculate the slope using the simplified values:
\(slope = \frac{0}{-1}\) which equals 0, indicating the line is horizontal and does not rise or fall.
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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 of a line and is calculated as the change in y-values divided by the change in x-values between two points. It is given by the formula m = (y2 - y1) / (x2 - x1). This value indicates how much y changes for a unit change in x.
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Undefined Slope
A slope is undefined when the denominator in the slope formula (x2 - x1) equals zero, meaning the line is vertical. Vertical lines have no horizontal change, so their slope cannot be expressed as a real number.
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Interpreting Slope Direction
The sign of the slope indicates the line's direction: a positive slope means the line rises from left to right, a negative slope means it falls, zero slope means the line is horizontal, and an undefined slope means the line is vertical.
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Related Practice
Textbook Question
In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation. {(-3, -3), (-2, −2), (−1, −1), (0, 0)}
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Textbook Question
Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (2, −3) and perpendicular to the line whose equation is y = (1/5)x + 6
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Textbook Question
In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation. {(3, −2), (5, −2), (7, 1), (4, 9)}
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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. (-2, -6) and (3, −4)
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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. (0, 0) and (3,-4)
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