Rise-Over-Run

Definition

Rise-over-run is a way to express the slope of a line, which is a measure of the steepness or incline of the line. It represents the change in the vertical (y) direction compared to the change in the horizontal (x) direction as you move along the line.

5 Must Know Facts For Your Next Test

1. The rise-over-run formula is $\frac{\Delta y}{\Delta x}$, 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 interpreted as the rise-over-run ratio, indicating how much the line rises (or falls) vertically for every unit it moves horizontally.
3. A positive slope indicates that the line is rising from left to right, while a negative slope indicates that the line is falling from left to right.
4. A slope of 0 indicates a horizontal line, where there is no vertical change, and a slope of $\infty$ (undefined) indicates a vertical line, where there is no horizontal change.
5. The rise-over-run ratio can be used to determine the equation of a line in slope-intercept form, $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Review Questions

• Explain how the rise-over-run ratio is used to determine the slope of a line.
  • The rise-over-run ratio, $\frac{\Delta y}{\Delta x}$, is used to calculate the slope of a line. The rise represents the change in the vertical (y) direction, and the run represents the change in the horizontal (x) direction between two points on the line. By dividing the rise by the run, you obtain the slope, which is a measure of the steepness or incline of the line. This ratio allows you to quantify the rate of change between the x and y variables, providing important information about the line's behavior.
• Describe how the rise-over-run ratio can be used to write the equation of a line in slope-intercept form.
  • The rise-over-run ratio, $\frac{\Delta y}{\Delta x}$, can be used to determine the slope of a line, which is the coefficient $m$ in the slope-intercept form of a linear equation, $y = mx + b$. Once the slope is known, you can use the coordinates of a point on the line to solve for the y-intercept $b$, completing the equation. This allows you to fully describe the line's behavior using the rise-over-run ratio to find the slope and a point on the line to find the y-intercept.
• Analyze how the sign of the rise-over-run ratio affects the direction and steepness of the line.
  • The sign of the rise-over-run ratio, $\frac{\Delta y}{\Delta x}$, determines the direction of the line. A positive ratio indicates a line that is rising from left to right, while a negative ratio indicates a line that is falling from left to right. The magnitude of the ratio determines the steepness of the line, with a larger absolute value corresponding to a steeper line. A ratio of 0 indicates a horizontal line, while an undefined ratio (division by 0) indicates a vertical line. Understanding the relationship between the rise-over-run ratio and the line's direction and steepness is crucial for analyzing and interpreting the behavior of linear functions.

Related terms

Slope: Slope is a measure of the steepness or incline of a line, representing the ratio of the vertical change to the horizontal change between two points on the line.

Coordinate Plane: A coordinate plane is a two-dimensional grid used to represent and plot points, lines, and other geometric shapes, with the horizontal axis (x-axis) and vertical axis (y-axis).

Linear Equation: A linear equation is an equation that represents a straight line, where the variables are raised to the power of 1 and are connected by addition, subtraction, or equality.