Question

How is the average rate of change related to linear function?

Answer 1

First we have to understand the basic definitions of average rate of change and linear function. We also have to consider a linear function for example to understand the solution more accurately.

Complete step by step solution:
Let us first understand what average rate of change is. Average rate of change is the value of the ratio of total change of any measurement by the total time taken for chat change to happen.
Now, let us understand what linear function is. A linear function is just a function in which one variable is related linearly to any constant. It is in the form of $f(x) = ax + b$, where a and b are constant and x is the variable.
Now, let us take a linear function to understand this question, so let it be
$\Rightarrow f(x) = 2x + 6$, and let it be from $x = 4$ to $x = 8$
Now to find the average rate of change of it we have to find the ratio of total change in y and total change is x
So, total change in $f(x)$ is
$\Rightarrow f(8) - f(4)$
$\Rightarrow (16 + 6) - (8 + 6)$
$\Rightarrow 8$
Now, we have to find total change in x
$\Rightarrow 8 - 4$
$\Rightarrow 4$
So, the ratio will be the average rate of change, so
$\Rightarrow \dfrac{{f(8) - f(4)}}{{8 - 4}}$
$\Rightarrow \dfrac{8}{4} = 2$

Hence, the average rate of this function is 2.

Note:
We can clearly observe that the coefficient of x in any linear function is its average rate of change. So in our example we have 2 as the coefficient of x, which is our solved average rate of change. We can calculate the average rate of change of any function by the method that we used in the solution.