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 throughout academics, most notably in chemistry, physics and finance.

It’s most often applied when discussing velocity, though it has numerous applications throughout many industries. Due to its utility, this formula is something that students should learn.

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

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

R = Δy / Δx

This calculates the change of y compared to the variation of x.

The change within the numerator and denominator is shown by the greek letter Δ, read as delta y and delta x. It is also portrayed 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 described as:

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

Average Rate of Change = Slope

Plotting out these figures in a X Y graph, is helpful when working with differences in value A in comparison with value B.

The straight line that links 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 values is equivalent to 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 random 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 grasping this concept easier, here are the steps you should keep in mind to find the average rate of change.

Step 1: Understand Your Values

In these types of equations, math questions generally give you two sets of values, from which you will get x and y values.

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

In this instance, next you have to find 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 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 figures inputted, all that we have to do is to simplify the equation by subtracting all the values. Therefore, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, 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 mentioned before, the rate of change is pertinent to many different situations. The aforementioned examples were applicable to the rate of change of a linear equation, but this formula can also be applied to functions.

The rate of change of function follows an identical principle but with a distinct formula due to the different values that functions have. This formula is:

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

In this case, the values provided will have one f(x) equation and one Cartesian plane value.

Negative Slope

If you can remember, the average rate of change of any two values can be plotted. The R-value, then is, equal to its slope.

Every so often, the equation concludes in a slope that is negative. This indicates 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, velocity can be negative, which results in a declining position.

Positive Slope

At the same time, 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. With regards to our previous example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

Now, we will review the average rate of change formula with some examples.

Example 1

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

In this example, all we must do is a simple substitution because 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 search for the Δy and Δx values by utilizing the average rate of change formula.

R = y2 - y1 / x2 - x1

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

R = 8 / 2

R = 4

As provided, the average rate of change is identical to the slope of the line joining two points.

Example 3

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

The third example will be finding 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 using the values given 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

With all our values, all we need 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

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