# OpenStax_CNX

You are here: Home » Content » Average Gradient

### Lenses

What is a lens?

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

#### Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• FETMaths

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

Review Status: In Review

Click the "FETMaths" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

### Recently Viewed

This feature requires Javascript to be enabled.

### Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

## 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.

 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

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:

 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?

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.

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)

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

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks