# 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 used mathematical formulas throughout academics, particularly in physics, chemistry and finance.

It’s most frequently used when talking about velocity, though it has many applications across different industries. Because of its utility, this formula is something that students should learn.

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 describes the change of one figure in relation to another. In every day terms, it's employed to evaluate the average speed of a change over a specific period of time.

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

R = Δy / Δx

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

The variation within the numerator and denominator is portrayed by the greek letter Δ, read as delta y and delta x. It is further denoted as the difference between 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 X Y axis, is beneficial when discussing differences in value A versus value B.

The straight line that joins these two points is also 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 summation, in a linear function, the average rate of change among two figures is the same as 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 random 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 achievable.

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

### Step 1: Find Your Values

In these sort of equations, math scenarios generally offer you 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 find the values via 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

Calculate 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 numbers plugged in, 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 stated, by plugging in 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 mentioned earlier, the rate of change is pertinent to multiple diverse situations. The previous 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 obeys the same principle but with a different formula because of 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 axis value.

### Negative Slope

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

Occasionally, the equation concludes in a slope that is negative. This means that the line is trending downward from left to right in the X Y axis.

This translates to 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 means that the object’s rate of change is positive. This tells 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 ascending.

## Examples of Average Rate of Change

Next, we will review 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 this example, all we have to do is a straightforward substitution due to the fact that the delta values are already specified.

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 search for the Δy and Δx values by employing 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 equal to the slope of the line joining two points.

### Example 3

Find the rate of change of function f(x) = x2 + 5x - 3 on the interval [3, 5].

The third example will be finding the rate of change of a function with the formula:

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

When finding the rate of change of a function, solve for the values of the functions in the equation. In this situation, we simply substitute 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

With all our values, all we must 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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