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

The rate of change formula is one of the most widely used mathematical concepts throughout academics, most notably in physics, chemistry and accounting.

It’s most frequently utilized when talking about momentum, although it has multiple uses throughout various industries. Because of its utility, this formula is a specific concept that students should grasp.

This article will discuss the rate of change formula and how you can work with them.

Average Rate of Change Formula

In math, the average rate of change formula describes the change of one figure in relation to another. In practice, 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 variation of y compared to the variation of x.

The variation within the numerator and denominator is shown by the greek letter Δ, read as delta y and delta x. It is additionally expressed as the variation within the first point and the second point of the value, or:

Δy = y2 - y1

Δx = x2 - x1

Because of this, the average rate of change equation can also be expressed as:

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

Average Rate of Change = Slope

Plotting out these figures in a X Y graph, is useful when working with dissimilarities in value A versus 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 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 know the slope formula and what the figures mean, finding the average rate of change of the function is achievable.

To make understanding this concept simpler, here are the steps you must obey to find the average rate of change.

Step 1: Understand Your Values

In these types of equations, math scenarios usually give you two sets of values, from which you solve to find x and y values.

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

In this situation, next you have to locate the values via the x and y-axis. Coordinates are generally given in an (x, y) format, as in this example:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Calculate the Δx and Δy values. As you may recall, the formula for the rate of change is:

R = Δy / Δx

Which then translates to:

R = y2 - y1 / x2 - x1

Now that we have obtained 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 in place, all that we have to do is to simplify the equation by subtracting all the values. Thus, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve stated previously, the rate of change is pertinent to numerous different situations. The aforementioned examples were applicable to the rate of change of a linear equation, but this formula can also be relevant for functions.

The rate of change of function obeys the same rule but with a different formula because of the different values that functions have. This formula is:

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

In this instance, the values given will have one f(x) equation and one Cartesian plane value.

Negative Slope

If you can remember, the average rate of change of any two values can be plotted on a graph. The R-value, therefore is, equivalent to its slope.

Occasionally, the equation results in a slope that is negative. This denotes that the line is descending from left to right in the X Y graph.

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

Positive Slope

At the same time, a positive slope indicates that the object’s rate of change is positive. This means that the object is increasing in value, and the secant line is trending upward from left to right. In terms of our aforementioned 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 with some examples.

Example 1

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

In the given example, all we need to do is a plain substitution due to the fact that 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 look for 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 provided, the average rate of change is identical to the slope of the line connecting two points.

Example 3

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

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

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

When calculating the rate of change of a function, calculate the values of the functions in the equation. In this case, we simply replace the values on the equation with the values provided 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 plug in 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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