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

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

It’s most often applied when discussing momentum, although it has numerous uses across different industries. Due to its value, this formula is something that learners should understand.

This article will share the rate of change formula and how you can solve it.

Average Rate of Change Formula

In math, the average rate of change formula shows the variation of one value when compared to another. In practice, it's utilized to define the average speed of a change 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 also denoted as the variation between 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 portrayed as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y graph, is useful when working with dissimilarities in value A in comparison with value B.

The straight line that connects 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 summation, in a linear function, the average rate of change among two values is equivalent to the slope of the function.

This is why the 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. Simultaneously, 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 values mean, finding the average rate of change of the function is achievable.

To make learning this principle simpler, here are the steps you must follow to find the average rate of change.

Step 1: Determine Your Values

In these equations, mathematical problems typically provide you with 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 instance, next you have to search for the values on the x and y-axis. Coordinates are typically provided 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

Calculate 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 all the values of x and y, we can input the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our numbers plugged in, all that we have to do is to simplify the equation by deducting all the numbers. Therefore, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, by simply plugging in all our values and simplifying the equation, we obtain the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

As we’ve shared previously, the rate of change is pertinent to many diverse situations. The aforementioned examples were more relevant to the rate of change of a linear equation, but this formula can also be applied to 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 situation, the values given will have one f(x) equation and one X Y graph value.

Negative Slope

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

Occasionally, the equation concludes in a slope that is negative. This means that the line is descending from left to right in the X Y axis.

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

Positive Slope

On the contrary, a positive slope means that the object’s rate of change is positive. This tells us 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 increasing.

Examples of Average Rate of Change

Now, we will run through the average rate of change formula through 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 plain substitution because the delta values are already given.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Extract the rate of change of the values in points (1,6) and (3,14) of the X Y axis.

For this example, we still have to look 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 identical to the slope of the line joining two points.

Example 3

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

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

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

When extracting 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 using the values specified 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 have to do is substitute 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

Grade Potential Can Help You Improve Your Grasp of Math Concepts

Mathematical can be a demanding topic to study, but it doesn’t have to be.

With Grade Potential, you can get paired with an expert teacher that will give you individualized instruction based on your current level of proficiency. With the quality of our teaching services, comprehending equations is as simple as one-two-three.

Book now!