key term - Rise Over Run

Definition

Rise over run, also known as the slope, is a way to quantify the steepness or incline of a line on a coordinate plane. It represents the change in the vertical position (rise) of a point on the line compared to the change in its horizontal position (run), providing a measure of the line's slope or rate of change.

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 vertical coordinate and $\Delta x$ represents the change in the horizontal coordinate.
2. The slope or rise over run of a line can be positive, negative, zero, or undefined, indicating the direction and steepness of the line.
3. A positive slope indicates that the line is sloping upward from left to right, while a negative slope indicates a downward slope.
4. A slope of zero indicates a horizontal line, and an undefined slope indicates a vertical line.
5. The rise over run or slope of a line is a key characteristic that determines the line's behavior and can be used to analyze and interpret linear functions.

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 finding the change in the vertical coordinate ($\Delta y$) divided by the change in the horizontal coordinate ($\Delta x$) between two points on the line. This ratio represents the steepness or incline of the line, with a positive slope indicating an upward trend and a negative slope indicating a downward trend. The slope calculated using the rise over run formula is a critical characteristic that defines the behavior and properties of a linear function.
• Describe how the rise over run relates to the graphs of linear functions.
  • The rise over run, or slope, of a line directly determines the shape and orientation of the graph of a linear function. A positive slope results in a line that slopes upward from left to right, while a negative slope produces a line that slopes downward. A slope of zero creates a horizontal line, and an undefined slope indicates a vertical line. The numerical value of the rise over run also provides information about the steepness of the line, with larger absolute values of the slope corresponding to lines that are more steeply inclined. Understanding the relationship between the rise over run and the graph of a linear function is crucial for analyzing and interpreting the behavior of linear models.
• Analyze how changes in the rise over run of a line impact the properties and behavior of a linear function.
  • Variations in the rise over run, or slope, of a line can significantly influence the properties and behavior of a linear function. A larger positive slope indicates a steeper upward trend, while a larger negative slope represents a steeper downward trend. Changes in the slope affect the rate of change, or how quickly the dependent variable changes in relation to the independent variable. Slope also determines the angle of the line on the coordinate plane, with steeper lines having slopes closer to positive or negative infinity. Understanding how modifications to the rise over run impact a linear function's graph, rate of change, and other characteristics is essential for effectively modeling and interpreting linear relationships.

Related terms

Slope:

The slope of a line is a measure of its steepness, calculated as the change in the vertical coordinate (rise) divided by the change in the horizontal coordinate (run).

Rate of Change:

The rate of change, or slope, represents how quickly a quantity is changing with respect to another quantity, such as the change in the y-coordinate per unit change in the x-coordinate.

Linear Function:

A linear function is a function where the dependent variable changes at a constant rate with respect to the independent variable, resulting in a straight-line graph.