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📊honors statistics review

Rise Over Run

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

Rise over run, also known as the slope of a line, is a way to quantify the steepness or inclination of a straight line. It represents the ratio of the change in the vertical (y) direction to the change in the horizontal (x) direction between two points on the line.

5 Must Know Facts For Your Next Test

  1. The rise over run formula is calculated as $\frac{\Delta y}{\Delta x}$, where $\Delta y$ represents the change in the y-coordinate and $\Delta x$ represents the change in the x-coordinate between two points on the line.
  2. The slope of a line can be positive, negative, zero, or undefined, depending on the relationship between the rise and the run.
  3. A positive slope indicates that the line is sloping upward from left to right, while a negative slope indicates that the line is sloping downward from left to right.
  4. A slope of zero means the line is horizontal, and a slope of undefined means the line is vertical.
  5. The slope of a line can be used to determine the angle of inclination of the line relative to the horizontal axis.

Review Questions

  • How can you calculate the rise over run of a line given two points on the line?
    • To calculate the rise over run of a line, you need to identify two points on the line, (x1, y1) and (x2, y2). The rise over run is then calculated as $\frac{y_2 - y_1}{x_2 - x_1}$. This gives you the slope of the line, which represents the steepness or inclination of the line in the coordinate plane.
  • Explain how the rise over run is related to the slope of a linear equation and its graphical representation.
    • The rise over run is directly equivalent to the slope of a linear equation. In the general linear equation $y = mx + b$, the slope 'm' represents the rise over run, which is the ratio of the change in the y-coordinate to the change in the x-coordinate between any two points on the line. This slope determines the steepness and direction of the line when graphed on the coordinate plane. A positive slope indicates an upward-sloping line, a negative slope indicates a downward-sloping line, a slope of zero indicates a horizontal line, and a slope of undefined indicates a vertical line.
  • Discuss how the rise over run can be used to analyze and compare the characteristics of different linear equations.
    • The rise over run, or slope, of a linear equation is a critical characteristic that can be used to analyze and compare the properties of different linear equations. By calculating the rise over run, you can determine the steepness and direction of the line, which provides insights into the relationship between the variables. For example, a steeper line with a higher absolute value of the slope will have a greater rate of change between the x and y variables compared to a line with a lower slope. Additionally, the sign of the slope can indicate whether the line is increasing or decreasing. This information can be used to make comparisons between linear equations and understand their underlying relationships.