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 formulas across academics, specifically in physics, chemistry and accounting.

It’s most often used when talking about velocity, though it has multiple uses throughout many industries. Because of its value, this formula is something 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 denotes the change of one figure when compared to another. In every day terms, it's utilized to evaluate the average speed of a variation over a specified period of time.

Simply put, the rate of change formula is expressed as:

R = Δy / Δx

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

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

Δy = y2 - y1

Δx = x2 - x1

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

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

Average Rate of Change = Slope

Plotting out these values in a X Y axis, is beneficial when talking about differences in value A in comparison with 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 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 intersecting 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 understand the slope formula and what the figures mean, finding the average rate of change of the function is achievable.

To make learning this topic less complex, here are the steps you need to obey to find the average rate of change.

Step 1: Determine Your Values

In these equations, math problems generally provide you with 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 case, then you have to search for the values via the x and y-axis. Coordinates are usually 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

Find 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 figures inputted, all that is left is to simplify the equation by deducting all the values. So, our equation then becomes the following.

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 before, the rate of change is relevant to numerous diverse scenarios. The previous 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 obeys a similar principle but with a unique formula due to the distinct 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 axis value.

Negative Slope

Previously if you 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.

Every so often, the equation concludes in a slope that is negative. This means that the line is trending downward from left to right in the Cartesian plane.

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

Positive Slope

On the contrary, a positive slope denotes 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 previous example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

Next, 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 need to do is a straightforward substitution since the delta values are already given.

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 you can see, the average rate of change is equivalent 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 final example will be extracting 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, calculate the values of the functions in the equation. In this situation, we simply replace the values on the equation with 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 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

Grade Potential Can Help You Get a Grip on Math

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

With Grade Potential, you can get set up with an expert teacher that will give you individualized support based on your abilities. With the quality of our tutoring services, getting a grip on equations is as simple as one-two-three.

Book now!