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

The rate of change formula is one of the most used mathematical formulas throughout academics, especially in physics, chemistry and finance.

It’s most often utilized when discussing velocity, though it has numerous uses throughout various industries. Because of its usefulness, this formula is something that learners should understand.

This article will share the rate of change formula and how you can work with them.

Average Rate of Change Formula

In mathematics, the average rate of change formula shows the variation of one figure in relation to another. In every day terms, it's used to identify the average speed of a variation over a specified period of time.

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

R = Δy / Δx

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

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

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

Average Rate of Change = Slope

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

The straight line that links 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 short, in a linear function, the average rate of change among two figures is equal to the slope of the function.

This is the reason why the 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. Meanwhile, 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 understanding this topic simpler, here are the steps you should obey to find the average rate of change.

Step 1: Determine Your Values

In these equations, math scenarios generally provide you with two sets of values, from which you solve to find x and y values.

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

In this case, next you have to locate the values along the x and y-axis. Coordinates are generally 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 found 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 values in place, all that is left is to simplify the equation by subtracting all the numbers. Therefore, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, just by replacing all our values and simplifying the equation, we achieve the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

As we’ve stated previously, the rate of change is relevant to multiple different scenarios. The aforementioned examples were applicable 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 principle but with a distinct formula due to the unique 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 Cartesian plane value.

Negative Slope

If you can recall, the average rate of change of any two values can be graphed. The R-value, then is, equivalent to its slope.

Occasionally, the equation concludes in a slope that is negative. This means that the line is descending from left to right in the Cartesian plane.

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

Positive Slope

On the contrary, a positive slope indicates that the object’s rate of change is positive. This shows us that the object is gaining 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 increasing.

Examples of Average Rate of Change

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

Example 1

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

In the given example, all we need to do is a plain 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 axis.

For this example, we still have to find 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 the same as 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 last example will be finding 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, solve for the values of the functions in the equation. In this case, 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 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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