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Introduction

The gradient of a straight line graph is calculated as:

y 2 - y 1 x 2 - x 1

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:

y 2 - y 1 x 2 - x 1

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 ( x ) = 2 x - 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 y 2 - y 1 x 2 - x 1
A-B
A-C
B-C

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.

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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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