Introduction and straight-line functions

Introduction

The gradient of a straight line graph is calculated as:

for two points $(x_1,y_1)$ and $(x_2,y_2)$ on the graph.

We can now define the average gradient between two points even if they are defined by a function which is not a straight line, $(x_1,y_1)$ and $(x_2,y_2)$ as:

This is the same as [link] .

Straight-line functions

Investigation : average gradient - straight line function

Fill in the table by calculating the average gradient over the indicated intervals for the function $f\left(x\right)=2x-2$ . Note that ( $x_1$ ; $y_1$ ) is the co-ordinates of the first point and ( $x_2$ ; $y_2$ ) is the co-ordinates of the second point. So for AB, ( $x_1$ ; $y_1$ ) is the co-ordinates of point A and ( $x_2$ ; $y_2$ ) is the co-ordinates of point B.

What do you notice about the gradients over each interval?

The average gradient of a straight-line function is the same over any two intervals on the function.