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

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

It’s most often utilized when talking about velocity, although it has numerous uses across different industries. Due to its value, this formula is a specific concept that learners should grasp.

This article will go over 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 denotes the change of one figure in relation to another. In every day terms, it's utilized to define the average speed of a variation over a specific period of time.

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

R = Δy / Δx

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

The change through the numerator and denominator is shown by the greek letter Δ, read as delta y and delta x. It is further denoted as the variation between 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 described as:

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

Average Rate of Change = Slope

Plotting out these numbers in a Cartesian plane, is beneficial when reviewing differences in value A in comparison with value B.

The straight line that links 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

To summarize, in a linear function, the average rate of change among two figures 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 passing through two random 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 have discussed the slope formula and what the values mean, finding the average rate of change of the function is possible.

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

Step 1: Determine Your Values

In these equations, math scenarios typically offer you two sets of values, from which you extract x and y values.

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

In this situation, next you have to search for the values on the x and y-axis. Coordinates are typically provided in an (x, y) format, like this:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Calculate the Δx and Δy values. As you may remember, 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 add the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our figures in place, all that we have to do is to simplify the equation by deducting all the numbers. Thus, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve stated before, the rate of change is applicable to numerous diverse situations. The aforementioned examples were applicable to the rate of change of a linear equation, but this formula can also be used in 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 scenario, the values given will have one f(x) equation and one X Y graph value.

Negative Slope

Previously if you remember, the average rate of change of any two values can be graphed. The R-value, then is, equal 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 axis.

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 tells us that the object is increasing in value, and the secant line is trending upward from left to right. In relation to our aforementioned example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

In this section, we will review 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 must do is a straightforward substitution because the delta values are already provided.

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 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 you can see, the average rate of change is equal to the slope of the line connecting 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 calculating 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, solve for the values of the functions in the equation. In this situation, we simply replace 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

Once we have all our values, all we have to 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

Grade Potential Can Help You Get a Grip on Math

Math can be a demanding topic to learn, but it doesn’t have to be.

With Grade Potential, you can get paired with an expert tutor that will give you personalized instruction based on your abilities. With the professionalism of our teaching services, getting a grip on equations is as simple as one-two-three.

Connect with us now!