Slope

MATH 018

Combined Algebra

S. Rook

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Overview

• Section 3.4 in the textbook– Definition and properties of slope– Slope-intercept form of a line– Slopes of horizontal and vertical lines– Slopes of parallel and perpendicular lines

Definition and Properties of Slope

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Definition and Properties of Slope

• Slope (m): the ratio of the change in y (Δ y) and the change in x (Δ x)– Quantifies (puts a numerical value on) the

“steepness” of a line

• Given 2 points on a line, we can find its slope:

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12

xx

yy

x

ym

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Sign of the Slope of a Line

• To determine the sign of the slope, examine the graph of the line from left to right– Positive if the line rises– Negative if the line drops

Calculating Slope (Example)

Ex 1: Find the slope between the two points and then graph the line:

a) (-4, -9), (4, 3)

b) (1, -4), (-2, 8)

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Slope-Intercept Form

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Slope-Intercept Form

• Slope-intercept form: a linear equation in the form y = mx + b where

m is the slope

b is the y-coordinate of the y-intercept (0, b)

• To utilize the slope-intercept form of a line, y MUST be ISOLATED

Slope-Intercept Form (Example)

Ex 2: Find the slope AND y-intercept of each given line:

a) y = 7x – 3

b) 3x – 6y = 6

c) -15x – 5y = -10

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Slopes of Horizontal and Vertical Lines

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Slopes of Horizontal and Vertical Lines

• Suppose we have a horizontal line y = 2– What are two points that lie on this line?– What happens when we use the slope formula?– Thus, we can say ALL horizontal lines have a slope

of zero

• Suppose we have a vertical line x = -1– What are two points that lie on this line?– What happens when we use the slope formula?– Thus, we can say ALL vertical lines have an

undefined slope

Slopes of Horizontal and Vertical Lines (Example)

Ex 3: Find the slope of each given line:

a) -3 = y – 3

b) 5x = 10

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Slopes of Parallel and Perpendicular Lines

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Slopes of Parallel and Perpendicular Lines

• Two special relationships exist between pairs of lines

• Determined using the slope of the lines

• Parallel lines: two lines that have the SAME slope

• Perpendicular lines: two lines that have OPPOSITE RECIPROCAL slopes– In other words, the product of the slopes is -1

Slopes of Parallel and Perpendicular Lines (Example)

Ex 4: Given the line 4x – y = 2, find:

a) The value of the slope that is parallel to the line

b) The value of the slope that is perpendicular to the line

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Slopes of Parallel and Perpendicular Lines (Example)

Ex 5: Determine whether the given lines are parallel, perpendicular, or neither:

a) -2x + 4y = -8

8x + 4y = 16

b) y = 2x – 3

2y + 4x = 516

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Summary

• After studying these slides, you should know how to do the following:– Understand the definition of slope and be able to apply the slope

formula when given 2 points on a line– Apply the definition of the slope-intercept form of a line to extract

the slope and the y-intercept– Be able to give the slope of a horizontal or vertical line– Determine whether pairs of lines are parallel, perpendicular or

neither

• Additional Practice– See the list of suggested problems for 3.4

• Next lesson– Equations of Lines (Section 3.5)