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

The rate of change formula is one of the most used mathematical principles throughout academics, specifically in chemistry, physics and accounting.

It’s most often applied when talking about velocity, however it has multiple applications across many industries. Because of its utility, this formula is a specific concept that students should grasp.

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 denotes the variation of one value in relation to another. In practice, it's used to determine the average speed of a change over a specific period of time.

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

R = Δy / Δx

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

The variation through the numerator and denominator is represented by the greek letter Δ, expressed as delta y and delta x. It is also expressed 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 Cartesian plane, is useful when discussing dissimilarities in value A versus 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 among two values is the same as 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 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 feasible.

To make studying this concept less complex, here are the steps you must follow to find the average rate of change.

Step 1: Find Your Values

In these equations, mathematical questions usually 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 scenario, then you have to locate the values on the x and y-axis. Coordinates are usually given in an (x, y) format, as in this example:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Calculate the Δx and Δy values. As you can recollect, 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 numbers plugged in, all that we have to do is to simplify the equation by subtracting all the values. Thus, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve shared before, the rate of change is applicable to many diverse 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 observes the same rule 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 case, the values given will have one f(x) equation and one X Y axis value.

Negative Slope

If you can remember, the average rate of change of any two values can be graphed. The R-value, 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 decreasing in value. For example, rate of change can be negative, which means a decreasing position.

Positive Slope

On the other hand, a positive slope shows 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 aforementioned example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

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

Example 1

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

In the given example, all we must 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 X Y graph.

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 you can see, the average rate of change is identical 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 extracting the rate of change of a function, calculate the values of the functions in the equation. In this case, we simply substitute 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 have to 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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