# Introduction and straight-line functions

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## Introduction

The gradient of a straight line graph is calculated as:

$\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

for two points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ 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, $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ as:

$\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

This is the same as [link] .

## 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.

 ${x}_{1}$ ${x}_{2}$ ${y}_{1}$ ${y}_{2}$ $\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ A-B A-C B-C

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

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