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

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

It’s most frequently utilized when talking about momentum, though it has many uses throughout various industries. Because of its usefulness, this formula is something that learners should understand.

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

Average Rate of Change Formula

In math, the average rate of change formula shows the variation of one value when compared to another. In practical terms, it's employed to evaluate the average speed of a change over a certain period of time.

At its simplest, the rate of change formula is expressed as:

R = Δy / Δx

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

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

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

Average Rate of Change = Slope

Plotting out these figures in a X Y axis, is helpful when talking about differences in value A when compared to 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 summation, in a linear function, the average rate of change between two figures is the same as the slope of the function.

This is mainly why average rate of change of a function is the slope of the secant line going 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 know the slope formula and what the values mean, finding the average rate of change of the function is feasible.

To make learning this concept easier, here are the steps you should obey to find the average rate of change.

Step 1: Find Your Values

In these equations, mathematical scenarios generally provide you with 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 situation, next you have to find the values on the x and y-axis. Coordinates are usually provided in an (x, y) format, like this:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Find 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 plug-in the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our figures in place, all that is left is to simplify the equation by deducting all the values. Therefore, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, by simply replacing 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 applicable to many different situations. The aforementioned examples focused on the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function follows an identical rule but with a unique formula due to 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 Cartesian plane value.

Negative Slope

Previously if you remember, the average rate of change of any two values can be graphed. The R-value, therefore is, equivalent to its slope.

Sometimes, the equation results 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, velocity can be negative, which means a declining position.

Positive Slope

At the same time, a positive slope indicates that the object’s rate of change is positive. This means that the object is gaining 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

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

Example 1

Extract 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

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 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 provided, the average rate of change is equal to the slope of the line linking 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 finding the rate of change of a function with the formula:

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

When calculating 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 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 must 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

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