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

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

It’s most often applied when talking about thrust, although it has numerous applications across various industries. Due to its usefulness, this formula is a specific concept that students should learn.

This article will share the rate of change formula and how you can solve it.

Average Rate of Change Formula

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

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

R = Δy / Δx

This calculates the change of y in comparison to the change of x.

The variation through the numerator and denominator is shown by the greek letter Δ, read as delta y and delta x. It is additionally denoted 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 expressed as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y axis, is beneficial when discussing differences in value A in comparison with value B.

The straight line that connects 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 between 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 going through 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 understand the slope formula and what the values mean, finding the average rate of change of the function is feasible.

To make learning this topic less complex, here are the steps you need to keep in mind to find the average rate of change.

Step 1: Understand Your Values

In these types of equations, math problems usually provide you with two sets of values, from which you extract x and y values.

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

In this case, then you have to find the values on 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 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 input 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 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 get the average rate of change for the two coordinates that we were provided.

Average Rate of Change of a Function

As we’ve mentioned previously, the rate of change is pertinent to numerous diverse scenarios. The previous 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 an identical rule but with a distinct formula due to the distinct values that functions have. This formula is:

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

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

Negative Slope

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

Sometimes, the equation concludes in a slope that is negative. This denotes that the line is trending downward from left to right in the X Y graph.

This means that the rate of change is diminishing in value. For example, velocity can be negative, which results in a decreasing position.

Positive Slope

In contrast, a positive slope denotes that the object’s rate of change is positive. This means that the object is increasing in value, and the secant line is trending upward from left to right. In terms of our previous example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

In this section, we will talk about the average rate of change formula via some examples.

Example 1

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

In this example, all we have to do is a straightforward substitution since the delta values are already provided.

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 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

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 calculating the rate of change of a function, calculate the values of the functions in the equation. In this case, 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

Once we have 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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