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

# 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 throughout academics, particularly in physics, chemistry and finance.

It’s most frequently applied when discussing momentum, though it has many uses throughout different industries. Due to its value, this formula is something that students should grasp.

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 value when compared to another. In every day terms, it's used to identify the average speed of a variation over a certain period of time.

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

R = Δy / Δx

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

The change within the numerator and denominator is represented by the greek letter Δ, read 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 described as:

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

## Average Rate of Change = Slope

Plotting out these figures in a Cartesian plane, is useful when talking about dissimilarities in value A versus 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 figures is equivalent to the slope of the function.

This is mainly why average rate of change of a function is the slope of the secant line intersecting two arbitrary 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 achievable.

To make understanding this topic simpler, here are the steps you should obey to find the average rate of change.

### Step 1: Find Your Values

In these types of equations, math questions 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 situation, next you have to search for the values along 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

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 input the values as follows.

R = 4 - 2 / 3 - 1

### Step 3: Simplify

With all of our numbers in place, all that is left is to simplify the equation by subtracting all the numbers. Thus, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, by plugging in 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 earlier, the rate of change is relevant to multiple diverse scenarios. The aforementioned examples were applicable to the rate of change of a linear equation, but this formula can also be applied to 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 instance, the values provided will have one f(x) equation and one Cartesian plane value.

### Negative Slope

As you might remember, the average rate of change of any two values can be graphed. The R-value, is, identical to its slope.

Every so often, the equation results 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 diminishing in value. For example, rate of change can be negative, which means a decreasing position.

### Positive Slope

On the contrary, a positive slope denotes 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. In relation to our previous example, if an object has positive velocity and its position is ascending.

## Examples of Average Rate of Change

In this section, we will discuss the average rate of change formula through some examples.

### Example 1

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

In the given example, all we need to do is a simple substitution since 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 Cartesian plane.

For this example, we still have to look for 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 provided, 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 last 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 case, we simply substitute the values on the equation using 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 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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