Learn on PengiBig Ideas Math, Course 3Chapter 4: Graphing and Writing Linear Equations

Lesson 2: Slope of a Line

In this Grade 8 lesson from Big Ideas Math Course 3, students learn how to define and calculate the slope of a line using the formula m = (y₂ − y₁) / (x₂ − x₁), identifying rise and run between any two points on a line. Students practice finding slopes from graphs and tables, and explore how positive and negative slopes describe whether a line rises or falls from left to right. The lesson also uses similar triangles to show why slope is constant between any two points on the same line, addressing Common Core standard 8.EE.6.

Section 1

Slope Formula

Property

The slope of the line between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
This formula calculates the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run).

Examples

  • Find the slope between (2,3)(2, 3) and (7,9)(7, 9). Using the formula: m=9−37−2=65m = \frac{9 - 3}{7 - 2} = \frac{6}{5}.
  • Find the slope between (−1,5)(-1, 5) and (3,−3)(3, -3). Using the formula: m=−3−53−(−1)=−84=−2m = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2.
  • Find the slope between (−4,−2)(-4, -2) and (−6,8)(-6, 8). Using the formula: m=8−(−2)−6−(−4)=10−2=−5m = \frac{8 - (-2)}{-6 - (-4)} = \frac{10}{-2} = -5.

Explanation

The slope formula is a way to calculate rise over run without a graph. It finds the vertical distance between points (y2−y1)(y_2 - y_1) and divides it by the horizontal distance (x2−x1)(x_2 - x_1) to find the steepness.

Section 2

Slopes of Horizontal and Vertical Lines

Property

The slope of a horizontal line is zero.
The slope of a vertical line is undefined.

Examples

  • The slope of the horizontal line y=5y = 5 is 0. For any two points like (1,5)(1, 5) and (3,5)(3, 5), the slope calculation is m=5−53−1=02=0m = \frac{5 - 5}{3 - 1} = \frac{0}{2} = 0.
  • The slope of the vertical line x=−2x = -2 is undefined. For any two points like (−2,1)(-2, 1) and (−2,4)(-2, 4), the slope is m=4−1−2−(−2)=30m = \frac{4 - 1}{-2 - (-2)} = \frac{3}{0}, which is undefined.
  • A line parallel to the x-axis is horizontal, so its slope is always 0. A line parallel to the y-axis is vertical, and its slope is always undefined.

Explanation

Slope is rise over run. A horizontal line has zero rise since yy does not change, so its slope is 0. A vertical line has zero run since xx does not change, which means dividing by zero—an undefined operation.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Slope Formula

Property

The slope of the line between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
This formula calculates the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run).

Examples

  • Find the slope between (2,3)(2, 3) and (7,9)(7, 9). Using the formula: m=9−37−2=65m = \frac{9 - 3}{7 - 2} = \frac{6}{5}.
  • Find the slope between (−1,5)(-1, 5) and (3,−3)(3, -3). Using the formula: m=−3−53−(−1)=−84=−2m = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2.
  • Find the slope between (−4,−2)(-4, -2) and (−6,8)(-6, 8). Using the formula: m=8−(−2)−6−(−4)=10−2=−5m = \frac{8 - (-2)}{-6 - (-4)} = \frac{10}{-2} = -5.

Explanation

The slope formula is a way to calculate rise over run without a graph. It finds the vertical distance between points (y2−y1)(y_2 - y_1) and divides it by the horizontal distance (x2−x1)(x_2 - x_1) to find the steepness.

Section 2

Slopes of Horizontal and Vertical Lines

Property

The slope of a horizontal line is zero.
The slope of a vertical line is undefined.

Examples

  • The slope of the horizontal line y=5y = 5 is 0. For any two points like (1,5)(1, 5) and (3,5)(3, 5), the slope calculation is m=5−53−1=02=0m = \frac{5 - 5}{3 - 1} = \frac{0}{2} = 0.
  • The slope of the vertical line x=−2x = -2 is undefined. For any two points like (−2,1)(-2, 1) and (−2,4)(-2, 4), the slope is m=4−1−2−(−2)=30m = \frac{4 - 1}{-2 - (-2)} = \frac{3}{0}, which is undefined.
  • A line parallel to the x-axis is horizontal, so its slope is always 0. A line parallel to the y-axis is vertical, and its slope is always undefined.

Explanation

Slope is rise over run. A horizontal line has zero rise since yy does not change, so its slope is 0. A vertical line has zero run since xx does not change, which means dividing by zero—an undefined operation.