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

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

It’s most frequently used when talking about momentum, though it has numerous uses across different industries. Due to its utility, this formula is something that students should understand.

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

Average Rate of Change Formula

In math, the average rate of change formula denotes the change of one figure in relation to another. In practice, it's utilized to define the average speed of a change over a certain period of time.

To put it simply, the rate of change formula is expressed as:

R = Δy / Δx

This measures the variation of y compared to the variation of x.

The variation through the numerator and denominator is represented by the greek letter Δ, read as delta y and delta x. It is further denoted as the variation 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 helpful when talking about dissimilarities in value A when compared to value B.

The straight line that connects these two points is 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 figures is the same as the slope of the function.

This is mainly why average rate of change of a function is the slope of the secant line passing through two arbitrary endpoints on the graph of the function. In the meantime, 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 have discussed the slope formula and what the values mean, finding the average rate of change of the function is feasible.

To make studying this principle simpler, here are the steps you should keep in mind to find the average rate of change.

Step 1: Understand Your Values

In these sort of equations, math problems usually provide you with two sets of values, from which you extract x and y values.

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

In this scenario, then you have to search for the values via the x and y-axis. Coordinates are typically given in an (x, y) format, as you see in the example below:

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 figures in place, all that is left is to simplify the equation by subtracting all the numbers. Thus, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve mentioned earlier, the rate of change is relevant to numerous diverse situations. 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 an identical principle but with a distinct formula because of the unique values that functions have. This formula is:

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

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

Negative Slope

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

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

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

Positive Slope

At the same time, a positive slope means 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 last example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

In this section, we will discuss the average rate of change formula via some examples.

Example 1

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

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

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Calculate 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 search for the Δy and Δx values by employing the average rate of change formula.

R = y2 - y1 / x2 - x1

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

R = 8 / 2

R = 4

As given, the average rate of change is equal to the slope of the line linking two points.

Example 3

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

The final example will be calculating 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, determine the values of the functions in the equation. In this case, we simply substitute 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

Once 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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