key term - Rise-Over-Run

Definition

Rise-over-run, also known as slope, is a way to quantify the steepness or incline of a line on a graph. It represents the change in the vertical (y) direction compared to the change in the horizontal (x) direction, providing a numerical value that describes the slope of the line.

5 Must Know Facts For Your Next Test

1. The rise-over-run formula is expressed 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 changes in the y-coordinate and x-coordinate.
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 an undefined slope means the line is vertical.
5. The slope of a line can be used to determine the equation of the line, which is typically written in the form $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.

Review Questions

• Explain how the rise-over-run formula is used to calculate the slope of a line.
  • The rise-over-run formula, $\frac{\Delta y}{\Delta x}$, is used to calculate the slope of a line by determining the change in the y-coordinate ($\Delta y$) and the change in the x-coordinate ($\Delta x$) between any two points on the line. The ratio of these changes provides the numerical value that represents the steepness or incline of the line. This slope information can then be used to write the equation of the line in the standard form of $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• Describe how the sign of the slope (positive, negative, zero, or undefined) affects the orientation and characteristics of a line on a coordinate plane.
  • The sign of the slope determined by the rise-over-run formula provides important information about the orientation and characteristics of a line on a coordinate plane. 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. A slope of zero means the line is horizontal, and an undefined slope means the line is vertical. These different slope characteristics can be used to identify the type of linear relationship represented by the line and make predictions about its behavior on the coordinate plane.
• Analyze how the rise-over-run formula and the slope of a line are related to the equation of the line in the standard form $y = mx + b$.
  • The rise-over-run formula, $\frac{\Delta y}{\Delta x}$, is directly connected to the slope $(m)$ in the standard equation of a line, $y = mx + b$. The ratio of the change in the y-coordinate to the change in the x-coordinate, as calculated by the rise-over-run formula, provides the numerical value for the slope $(m)$ that appears in the linear equation. This relationship allows the slope to be determined from the rise-over-run formula and then used to write the equation of the line in the standard form, which includes both the slope $(m)$ and the y-intercept $(b)$. Understanding this connection is crucial for analyzing and working with linear equations.

Related terms

Slope:

The slope of a line is a measure of its steepness, calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate between any two points on the line.

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 or subtraction.

Coordinate Plane:

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