Algebra Rate of Change Calculator
Calculate the average rate of change (slope) between two points.
Point 1 (Initial)
Point 2 (Final)
How to Calculate Rate of Change in Algebra
In algebra, the Rate of Change refers to how much a quantity changes over time or in relation to another quantity. Graphically, this is represented by the slope of the line connecting two points. It measures the steepness and direction of a line.
When the relationship is linear (a straight line), the rate of change is constant. When the relationship is non-linear (a curve), this calculation provides the average rate of change between the two selected points.
The Rate of Change Formula
To calculate the rate of change between two coordinates $(x_1, y_1)$ and $(x_2, y_2)$, we use the slope formula:
Rate of Change (m) = (y₂ – y₁) / (x₂ – x₁)
This is often remembered as "Rise over Run":
- Rise (Δy): The vertical change, calculated as the difference in Y values.
- Run (Δx): The horizontal change, calculated as the difference in X values.
Step-by-Step Calculation Guide
Follow these simple steps to find the rate of change manually:
- Identify Point 1: Determine your starting x and y values $(x_1, y_1)$.
- Identify Point 2: Determine your ending x and y values $(x_2, y_2)$.
- Calculate Δy: Subtract $y_1$ from $y_2$.
- Calculate Δx: Subtract $x_1$ from $x_2$.
- Divide: Divide the result of Δy by Δx.
Example Calculation
Let's say you want to calculate the rate of change for a car that travels distance over time. At 2 hours ($x_1$), the car has traveled 100 miles ($y_1$). At 5 hours ($x_2$), the car has traveled 310 miles ($y_2$).
- $(x_1, y_1) = (2, 100)$
- $(x_2, y_2) = (5, 310)$
- $\Delta y = 310 – 100 = 210$
- $\Delta x = 5 – 2 = 3$
- Rate of Change = $210 / 3 = 70$ miles per hour.
Understanding the Results
- Positive Rate: The line goes up from left to right. The quantity is increasing.
- Negative Rate: The line goes down from left to right. The quantity is decreasing.
- Zero Rate: The line is horizontal. There is no change in Y relative to X.
- Undefined: The line is vertical. Division by zero occurs because $x_2 = x_1$.