Rate of Change Formula - What Is the Rate of Change Formula? Examples

The rate of change formula is one of the most important math formulas across academics, especially in chemistry, physics and finance.

It’s most often applied when talking about thrust, although it has many applications across various industries. Because of its usefulness, this formula is something that learners should understand.

This article will discuss the rate of change formula and how you should solve them.

Average Rate of Change Formula

In mathematics, the average rate of change formula shows the variation of one figure in relation to another. In every day terms, it's utilized to evaluate the average speed of a variation over a specified period of time.

At its simplest, the rate of change formula is written as:

R = Δy / Δx

This computes the change of y in comparison to the change of x.

The change within the numerator and denominator is shown by the greek letter Δ, read as delta y and delta x. It is also denoted as the difference within the first point and the second point of the value, or:

Δy = y2 - y1

Δx = x2 - x1

As a result, the average rate of change equation can also be described as:

R = (y2 - y1) / (x2 - x1)

Average Rate of Change = Slope

Plotting out these figures in a X Y axis, is beneficial when working with differences in value A when compared to value B.

The straight line that joins these two points is also known as secant line, and the slope of this line is the average rate of change.

Here’s the formula for the slope of a line:

y = 2x + 1

In short, in a linear function, the average rate of change between two values is equal to the slope of the function.

This is the reason why the average rate of change of a function is the slope of the secant line intersecting two random endpoints on the graph of the function. Meanwhile, the instantaneous rate of change is the slope of the tangent line at any point on the graph.

How to Find Average Rate of Change

Now that we understand the slope formula and what the figures mean, finding the average rate of change of the function is possible.

To make understanding this topic less complex, here are the steps you should keep in mind to find the average rate of change.

Step 1: Find Your Values

In these sort of equations, mathematical scenarios usually offer you two sets of values, from which you will get x and y values.

For example, let’s assume the values (1, 2) and (3, 4).

In this situation, then you have to locate the values along the x and y-axis. Coordinates are generally given in an (x, y) format, like this:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Find the Δx and Δy values. As you can recollect, the formula for the rate of change is:

R = Δy / Δx

Which then translates to:

R = y2 - y1 / x2 - x1

Now that we have found all the values of x and y, we can add the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our values plugged in, all that is left is to simplify the equation by subtracting all the values. Thus, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, by replacing all our values and simplifying the equation, we get the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

As we’ve mentioned before, the rate of change is relevant to multiple different scenarios. The previous examples focused on the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function observes a similar principle but with a different formula because of the distinct values that functions have. This formula is:

R = (f(b) - f(a)) / b - a

In this situation, the values given will have one f(x) equation and one X Y graph value.

Negative Slope

As you might recall, the average rate of change of any two values can be graphed. The R-value, is, identical to its slope.

Sometimes, the equation concludes in a slope that is negative. This means that the line is trending downward from left to right in the X Y graph.

This means that the rate of change is decreasing in value. For example, velocity can be negative, which means a decreasing position.

Positive Slope

On the contrary, a positive slope indicates that the object’s rate of change is positive. This shows us that the object is increasing in value, and the secant line is trending upward from left to right. In relation to our last example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

Now, we will run through the average rate of change formula through some examples.

Example 1

Extract the rate of change of the values where Δy = 10 and Δx = 2.

In this example, all we have to do is a simple substitution because the delta values are already specified.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Find the rate of change of the values in points (1,6) and (3,14) of the Cartesian plane.

For this example, we still have to find the Δy and Δx values by utilizing the average rate of change formula.

R = y2 - y1 / x2 - x1

R = (14 - 6) / (3 - 1)

R = 8 / 2

R = 4

As you can see, the average rate of change is equivalent to the slope of the line linking two points.

Example 3

Find the rate of change of function f(x) = x2 + 5x - 3 on the interval [3, 5].

The final example will be finding the rate of change of a function with the formula:

R = (f(b) - f(a)) / b - a

When finding the rate of change of a function, determine the values of the functions in the equation. In this case, we simply substitute the values on the equation with the values given in the problem.

The interval given is [3, 5], which means that a = 3 and b = 5.

The function parts will be solved by inputting the values to the equation given, such as.

f(a) = (3)2 +5(3) - 3

f(a) = 9 + 15 - 3

f(a) = 24 - 3

f(a) = 21

f(b) = (5)2 +5(5) - 3

f(b) = 25 + 10 - 3

f(b) = 35 - 3

f(b) = 32

Now that we have all our values, all we must do is replace them into our rate of change equation, as follows.

R = (f(b) - f(a)) / b - a

R = 32 - 21 / 5 - 3

R = 11 / 2

R = 11/2 or 5.5

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