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, especially in physics, chemistry and accounting.

It’s most frequently applied when discussing velocity, though it has many applications throughout different industries. Because of its usefulness, this formula is a specific concept that learners should understand.

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

Average Rate of Change Formula

In mathematics, the average rate of change formula denotes the variation of one figure in relation to another. In every day terms, it's utilized to determine the average speed of a change over a certain period of time.

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

R = Δy / Δx

This computes the variation of y in comparison to the variation of x.

The variation through the numerator and denominator is shown by the greek letter Δ, expressed as delta y and delta x. It is further expressed as the difference within 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 shown as:

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

Average Rate of Change = Slope

Plotting out these figures in a X Y graph, is beneficial when talking about dissimilarities in value A when compared to 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

In short, in a linear function, the average rate of change among two figures 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. 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 possible.

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

Step 1: Determine Your Values

In these equations, math questions generally 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 case, next you have to locate the values along the x and y-axis. Coordinates are generally 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 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 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 values inputted, all that remains is to simplify the equation by deducting all the numbers. Thus, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve mentioned previously, the rate of change is pertinent to multiple different scenarios. The aforementioned examples were more relevant 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 principle but with a different formula due to the distinct 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 remember, the average rate of change of any two values can be plotted on a graph. The R-value, therefore is, identical to its slope.

Occasionally, 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 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 talk about the average rate of change formula through some examples.

Example 1

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

In this example, all we must do is a simple substitution due to the fact that the delta values are already given.

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 find the Δy and Δx values by using 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 identical to the slope of the line joining 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 finding the rate of change of a function, determine the values of the functions in the equation. In this instance, 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 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 Improve Your Math Skills

Mathematical can be a difficult topic to study, 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 quality of our teaching services, getting a grip on equations is as easy as one-two-three.

Connect with us now!