Question

Given the function f(x) = 3x - 1, explain how to find the average rate of change between x = 1 and x = 4.

Answer 1

3

Explanation:

The average rate of f(x) in the closed interval [ a, b ] is

`(f(b)-f(a))/(b-a)`

Here [ a, b ] = [ 1, 4 ] , then

f(b) = f(4) = 3(4) - 1 = 12 - 1 = 11

f(a) = f(1) = 3(1) - 1 = 3 - 1 = 2

average rate of change =
`(11-2)/(4-1)` =
`(9)/(3)` = 3

Answer 2

Explanation:

f(1) = 3×1 - 1 = 2

f(4) = 3×4 - 1 = 12-1 = 11

so, the functional value changes 11-2=9 units on an x interval of 4-1=3 units length.

the average change rate is the total change across the x interval relative to the interval length.

that is

9/3 = 3

which is the slope (= the factor of x) in the line equation.

for a line its change rate for any point is the same constant. and that is therefore automatically also the average change rate across an interval of x values.

if the change rate would be different for different parts of the function, it would not be a straight line.