$$ \frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1} $$

where Δy is the change in y, Δx is the change in x, y1 is the initial value of y, y2 is the final value of y, x1 is the initial value of x, and x2 is the final value of x.

1. In Math:

The rate of change formula is commonly used to find the slope of a line in coordinate geometry. Here's how to use it:

- Calculate the change in y (Δy) by subtracting the initial y-coordinate (y1) from the final y-coordinate (y2): Δy = y2 - y1.

- Calculate the change in x (Δx) by subtracting the initial x-coordinate (x1) from the final x-coordinate (x2): Δx = x2 - x1.

- Divide Δy by Δx to obtain the slope of the line: Slope = (Δy)/(Δx).

Example: Find the slope of the line passing through the points (-2, 3) and (4, 7).

Solution:

- Calculate Δy = 7 - 3 = 4.

- Calculate Δx = 4 - (-2) = 6.

- Slope = (Δy)/(Δx) = 4/6 = 2/3.

2. In Physics:

- Speed and Velocity: In Physics, particularly kinematics, the rate of change formula is employed to calculate speed or velocity.

Speed: Speed is the rate of change of distance with respect to time, so v (speed) = (Δd)/(Δt).

Velocity: Velocity considers direction as well, so it's the rate of change of displacement (a vector quantity) with respect to time. Here, v (velocity) = (Δx_2 - x_1)/(Δt_2 - t_1).

- Acceleration: Acceleration measures the rate at which velocity changes with respect to time. It can be calculated as a = (Δv)/(Δt).

Example: A cyclist travels 15 km in 30 minutes. Calculate the cyclist's average speed.

Solution:

First, convert time to hours for uniformity. 30 minutes = 0.5 hours.

- Distance (d) = 15 km.

- Time (t) = 0.5 h.

- Speed = (Δd)/(Δt) = 15 km/0.5 h = 30 km/h.