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# Average Rate of Change Formula

The average rate of change describes the average rate at which one quantity is changing with respect to another. It gives an idea of how much the function changed per unit in the given interval. In simple terms, in the rate of change, the amount of change in one item is divided by the corresponding amount of change in another. Let's look into the average rate of change formula in detail.

## What is the Average Rate of Change?

The **average rate of change** of a function f(x) over an interval [a, b] is defined as the ratio of "change in the function values" to the "change in the endpoints of the interval". i.e., the average rate of change can be calculated using [f(b) - f(a)] / (b - a). In other words, the average rate of change (which is denoted by A(x)) is the "ratio of change in outputs to change in inputs". i.e.,

A(x) = (change in outputs) / (change in inputs)

= Δy / Δx

= [f(b) - f(a)] / (b - a)

Here, Δy is the change in y-values (or) change in the function values and Δx is the change in x-values (or) the change in the endpoints of the interval. Some examples of the average rate of change are:

- A bus travels at a speed of 80 km per hour.
- The number of fish in a lake increases at the rate of 100 per week.
- The current in an electrical circuit decreases 0.2 amperes for a decrease of 1-volt voltage.

## Average Rate of Change Formula

The average rate of change function describes the average rate at which one quantity is changing with respect to something another quantity. The **average rate of change formula** is given as,

**A(x) = [f(b) - f(a)] / (b - a)**

where,

- A(x) = Average rate of change
- f(a) = Value of function f(x) at a
- f(b) = Value of function f(x) at b

## Examples Using Average Rate of Change Formula

**Example 1: **Calculate the average rate of change of a function, f(x) = 2x + 10 as x changes from 3 to 7.

**Solution:**

Given:

f(x) = 2x + 10, a = 3, b = 7.

- f(3) = 2(3) + 10 = 6 + 10 = 16
- f(7) = 2(7) + 10 = 14 + 10 = 24

Using the average rate of change formula,

A(x) = [f(b)−f(a)] / (b−a)

A(x) = [f(7)−f(3)]/ (7−3)

A(x) = (24 - 16) / 4

A(x) = 8/4

A(x) = 2

Therefore, the rate of change is 2.

**Example 2: **Evaluate the average rate of change of the function f(x) = x^{2} – 5x in the interval 4 ≤ x ≤ 8.

**Solution:**

Given: f(x) = x^{2} – 5x, a = 4, b = 8.

- f(4) = f(4) = (4)
^{2}– 5(4) = 16 – 20 = -4 - f(8) = f(8) = (8)
^{2}– 5(8) = 64 – 40 = 24

Using the average rate of change formula,

A(x) = [f(b)−f(a)] / (b−a)

= (24−(−4)) / (8−4)

= 28/4

= 7

Therefore, rate of change A(x) = 7.

**Example 3: **Find the rate of change of a function, f(x) = 25x + 18 as x changes from 5 to 8

**Solution: ** Given,

f(x) = 25x + 18, a = 5 and b = 8

- f(5) = 25×5 + 18 = 143
- f(8) = 25×8 + 18 = 218

Using the average rate of change formula,

A(x) = [f(b)−f(a)] / (b−a)

= [218 - 143] / (8 - 5)

= 75 / 3

= 25

Therefore, the average rate of change A(x) = 25

## FAQs on Average Rate of Change Formula

### What is Meant by Average Rate of Change Formula?

The **average rate of change **is the change one quantity with respect to the change in another. It is a measure of how much the function changed per unit in a particular interval. If f(x) is the function and [a, b] is the interval, then the formula is A(x) = [f(b) - f(a)] / (b - a)

### What is the Formula to Find the Average Rate of Change?

The average rate of change formula is given as,

A(x) = [f(b) - f(a)] / (b - a)

where,

- A(x) = Average rate of change
- f(a) = Value of function f(x) at a
- f(b) = Value of function f(x) at b

### How to Find Average Rate of Change Over an Interval?

To find the average rate of change of a function f(x) over an interval [a, b]:

- Find f(a) and f(b).
- Substitute the values in the formula [f(b) - f(a)] / (b - a) and simplify.

Note that it can be calculated using the formula [f(a) - f(b)] / (a - b) as well. But make sure to follow the same order both in the numerator and the denominator.

### What is the Formula to Find the Rate of Change of a Linear Function?

For a linear function, the rate of change is represented by the parameter (m) in the slope-intercept form for a line: y=mx+b, and is visible in a table or on a graph.

### Is the Average Rate of Change the Same as Slope?

The slope is considered as the average rate of change of a point where the average is taken and is reduced to zero. The slope is the rise over the run which is defined as the average rate of change in y coordinates over the change in x coordinates.

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