Question

How do I find the average rate of change for a function between two given values?

Answer 1

Here in this type of question we will have given a function which is $f(x)$. To find the average let the given value be ${x_1}$ and ${x_2}$. Now by using the average rate of change formula given by: Average rate of change $= \dfrac{\text{Change In Output}}{\text{Change In Input}}$.

Complete step by step answer:
Here in this question, they have asked to find the average rate of a function between two values. So let the function be $f(x)$ and the two values be ${x_1}$ and ${x_2}$which are nothing but the input.
As we know the two points are given that are ${x_1}$ and ${x_2}$ we can substitute this in the given function $f(x)$ to find the corresponding value which will be the output of the given function.
When we substitute ${x_1}$ in the given function $f(x)$ we get the corresponding function as $f({x_1})$.
In the similar way now substitute ${x_2}$ in the given function $f(x)$ to find the corresponding value. Hence we get $f({x_2})$.
Now we need to find the average rate of change for the given function $f(x)$ as they have asked for it. So in order to find the average rate of change of function we make use of the formula given by Average rate of change $= \dfrac{\text{Change In Output}}{\text{Change In Input}}$.
Hence the change in output is written $f({x_2}) - f({x_1})$ and the change in the input is written as ${x_2} - {x_1}$. Now substitute in the above average rate of change formula to get the required answer.
Therefore, we get

Average rate of change $= \dfrac{{f({x_2}) - f({x_1})}}{{{x_2} - {x_1}}}$

Hence we can make use of the above formula to find the average rate of change for function between two values.

Note:
The average rate of change is nothing but the slope formula itself. But they have asked in another way to confuse you. So try to analyze and write the answer accordingly. If you know the general average formula then we can easily arrive at the required answer.