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

The rate of change formula is one of the most important math formulas throughout academics, specifically in chemistry, physics and accounting.

It’s most frequently used when talking about velocity, though it has many uses across different industries. Due to its usefulness, this formula is something that students should understand.

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

To put it simply, the rate of change formula is expressed as:

R = Δy / Δx

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

The variation within the numerator and denominator is represented by the greek letter Δ, read as delta y and delta x. It is additionally expressed as the variation within 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 portrayed as:

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

Average Rate of Change = Slope

Plotting out these figures in a Cartesian plane, is beneficial when talking about differences in value A versus value B.

The straight line that joins these two points is called the 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 figures is equal 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 have discussed the slope formula and what the figures mean, finding the average rate of change of the function is possible.

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

Step 1: Determine Your Values

In these types of equations, mathematical problems generally give you two sets of values, from which you extract x and y values.

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

In this instance, then you have to find the values on the x and y-axis. Coordinates are usually given in an (x, y) format, like this:

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 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 figures plugged in, all that remains is to simplify the equation by deducting all the numbers. Thus, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, by replacing 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 shared previously, the rate of change is relevant to multiple different scenarios. The previous examples focused on the rate of change of a linear equation, but this formula can also be applied to functions.

The rate of change of function obeys an identical principle but with a unique formula because of the unique values that functions have. This formula is:

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

In this instance, the values provided will have one f(x) equation and one X Y graph value.

Negative Slope

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

Occasionally, the equation results in a slope that is negative. This indicates that the line is trending downward from left to right in the Cartesian plane.

This translates to the rate of change is decreasing in value. For example, velocity can be negative, which results in 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 relation to our last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

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

Example 1

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

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

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Extract 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 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 given, the average rate of change is identical to the slope of the line linking 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 extracting 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, solve for the values of the functions in the equation. In this case, we simply replace the values on the equation with 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

Now that we have all our values, all we need 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

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