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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 77
Chapter 3, Problem 77

If three distinct points A, B, and C in a plane are such that the slopes of nonvertical line segments AB, AC, and BC are equal, then A, B, and C are collinear. Otherwise, they are not. Use this fact to determine whether the three points given are collinear. (-1, -3), (-5, 12), (1, -11)

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Recall that three points A, B, and C are collinear if the slopes of the line segments AB, AC, and BC are all equal. This means the slope between any two pairs of points must be the same.
Label the points as A(-1, -3), B(-5, 12), and C(1, -11). We will calculate the slopes of AB, AC, and BC using the slope formula: \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\).
Calculate the slope of AB: \(m_{AB} = \frac{12 - (-3)}{-5 - (-1)} = \frac{12 + 3}{-5 + 1} = \frac{15}{-4}\).
Calculate the slope of AC: \(m_{AC} = \frac{-11 - (-3)}{1 - (-1)} = \frac{-11 + 3}{1 + 1} = \frac{-8}{2}\).
Calculate the slope of BC: \(m_{BC} = \frac{-11 - 12}{1 - (-5)} = \frac{-23}{6}\). Compare \(m_{AB}\), \(m_{AC}\), and \(m_{BC}\). If all three are equal, the points are collinear; otherwise, they are not.

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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 Segment

The slope measures the steepness of a line segment connecting two points and is calculated as the change in y-coordinates divided by the change in x-coordinates. It indicates how much y changes for a unit change in x. For points (x1, y1) and (x2, y2), slope = (y2 - y1) / (x2 - x1), provided the line is not vertical.
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The Slope of a Line

Collinearity of Points

Three points are collinear if they lie on the same straight line. This occurs when the slopes of the line segments connecting each pair of points are equal. If the slopes between AB, AC, and BC are the same, the points A, B, and C are collinear.
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Using Slopes to Test Collinearity

To determine if three points are collinear, calculate the slopes of the segments AB, AC, and BC. If all slopes are equal (and none are undefined due to vertical lines), the points lie on the same line. Otherwise, they are not collinear.
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Related Practice
Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.

ƒ(x)=x3,g(x)=x2+3x−1ƒ(x)=x^3, g(x)=x^2+3x-1

Textbook Question

Given functions f and g, (b)(g∘ƒ)(x)(g∘ƒ)(x) and its domain. See Examples 6 and 7.

ƒ(x)=x3,g(x)=x2+3x−1ƒ(x)=x^3, g(x)=x^2+3x-1

Textbook Question

Given functions f and g, find (b)(g∘ƒ)(x)(g∘ƒ)(x) and its domain. See Examples 6 and 7.

ƒ(x)=√x,g(x)=x+3ƒ(x)=√x, g(x)=x+3

Textbook Question

Graph each function. f(x)={∣x∣if x<36−xif x≥3f(x) =\(\begin{cases}\)|x| & \(\text{if }\) x < 3 \\6 - x & \(\text{if }\) x \(\geq\) 3\(\end{cases}\)

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Textbook Question

Consider the following nonlinear system. Work Exercises 75 –80 in order.

y=∣x−1∣y = | x - 1 |

y=x2−4y = x^2 - 4

How is the graph of y = x^2 - 4 obtained by transforming the graph of y=x2y = x^2?

Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=√x, g(x)=x-1