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Average gradient: Straight line functions

Module by: Free High School Science Texts Project. E-mail the author

Introduction

The gradient of a straight line graph is calculated as:

y 2 - y 1 x 2 - x 1 y 2 - y 1 x 2 - x 1
(1)

for two points (x1,y1)(x1,y1) and (x2,y2)(x2,y2) 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, (x1,y1)(x1,y1) and (x2,y2)(x2,y2) as:

y 2 - y 1 x 2 - x 1 . y 2 - y 1 x 2 - x 1 .
(2)

This is the same as Equation 1.

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)=2x-2f(x)=2x-2. Note that (x1x1;y1y1) is the co-ordinates of the first point and (x2x2;y2y2) is the co-ordinates of the second point. So for AB, (x1x1;y1y1) is the co-ordinates of point A and (x2x2;y2y2) is the co-ordinates of point B.

Table 1
  x 1 x 1 x 2 x 2 y 1 y 1 y 2 y 2 y 2 - y 1 x 2 - x 1 y 2 - y 1 x 2 - x 1
A-B          
A-C          
B-C          
Figure 1
Figure 1 (MG10C12_001.png)

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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Lenses

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