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    This module is included inLens: Siyavula: Mathematics (Gr. 10-12)
    By: Siyavula

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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.

Parabolic Functions

Investigation : Average Gradient - Parabolic Function

Fill in the table by calculating the average gradient over the indicated intervals for the function f(x)=2x-2f(x)=2x-2:

Table 2
  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          
B-C          
C-D          
D-E          
E-F          
F-G          

What do you notice about the average gradient over each interval? What can you say about the average gradients between A and D compared to the average gradients between D and G?

Figure 2
Figure 2 (MG10C12_002.png)

The average gradient of a parabolic function depends on the interval and is the gradient of a straight line that passes through the points on the interval.

For example, in Figure 3 the various points have been joined by straight-lines. The average gradients between the joined points are then the gradients of the straight lines that pass through the points.

Figure 3: The average gradient between two points on a curve is the gradient of the straight line that passes through the points.
Figure 3 (MG10C12_003.png)

Method: Average Gradient

Given the equation of a curve and two points (x1x1, x2x2):

  1. Write the equation of the curve in the form y=...y=....
  2. Calculate y1y1 by substituting x1x1 into the equation for the curve.
  3. Calculate y2y2 by substituting x2x2 into the equation for the curve.
  4. Calculate the average gradient using:
    y2-y1x2-x1y2-y1x2-x1
    (3)

Exercise 1: Average Gradient

Find the average gradient of the curve y=5x2-4y=5x2-4 between the points x=-3x=-3 and x=3x=3

Solution
  1. Step 1. Label points :

    Label the points as follows:

    x 1 = - 3 x 1 = - 3
    (4)
    x 2 = 3 x 2 = 3
    (5)

    to make it easier to calculate the gradient.

  2. Step 2. Calculate the yy coordinates :

    We use the equation for the curve to calculate the yy-value at x1x1 and x2x2.

    y 1 = 5 x 1 2 - 4 = 5 ( - 3 ) 2 - 4 = 5 ( 9 ) - 4 = 41 y 1 = 5 x 1 2 - 4 = 5 ( - 3 ) 2 - 4 = 5 ( 9 ) - 4 = 41
    (6)
    y 2 = 5 x 2 2 - 4 = 5 ( 3 ) 2 - 4 = 5 ( 9 ) - 4 = 41 y 2 = 5 x 2 2 - 4 = 5 ( 3 ) 2 - 4 = 5 ( 9 ) - 4 = 41
    (7)
  3. Step 3. Calculate the average gradient :
    y 2 - y 1 x 2 - x 1 = 41 - 41 3 - ( - 3 ) = 0 3 + 3 = 0 6 = 0 y 2 - y 1 x 2 - x 1 = 41 - 41 3 - ( - 3 ) = 0 3 + 3 = 0 6 = 0
    (8)
  4. Step 4. Write the final answer :

    The average gradient between x=-3x=-3 and x=3x=3 on the curve y=5x2-4y=5x2-4 is 0.

End of Chapter Exercises

  1. An object moves according to the function d=2t2+1d=2t2+1 , where dd is the distance in metres and tt the time in seconds. Calculate the average speed of the object between 2 and 3 seconds. The speed is the gradient of the function dd Click here for the solution
  2. Given: f(x)=x3-6xf(x)=x3-6x. Determine the average gradient between the points where x=1x=1 and x=4x=4. Click here for the solution

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