LINES

gradient

The gradient or gradient of a line is a number that tells us how “steep” the line is and which direction it goes.

If you move along the line from left to right and are climbing, it is a positive gradient.

These are all positive gradients. The “steeper” the line, the larger the gradient value.

This one has the greatest gradient

This one has

the smallest

gradient

We compute the gradient by taking the ratio of how much the line rises (goes up) and how much the line runs (goes over)

rise

run

If we took two points on the line, we could compute the rise by looking at the difference between the y values.

(x1, y1)

(x2, y2)

y2 - y1

We could compute the run by looking at the difference between the x values.

x2 - x1

So the gradient or gradient is the rise over the run 12

12

xx

yym

gradient is designated with an m

This is the gradient formula12

12

xx

yym

These are all negative gradients. The “steeper” the line, the larger the absolute value of the gradient. (basically this means if you ignore the negative, the larger the number, the steeper the line---but in the negative gradient direction).

Gradient

If you move along the line from left to right and are descending, it is a negative gradient.

This one has the greatest absolute value gradient

This one has the smallest absolute value gradient

Let’s figure out the gradient of this line. We

know it should be a positive

number.

x

y

Choose two points on the line.

(0, 0)

(1, 2)2

1

The rise over the run is 2 over 1 which is 2. Let’s compute it with the gradient formula.

201

02

mWhat if we'd chosen two different points on the line?

(-2, -4)

(2, 4)

2

4

8

22

44

mIt doesn't matter which two points we pick, we'll always get a constant ratio of 2 for this line.

x

y

If we look at any points on this line we see that they all have a y coordinate of 3 and the x coordinate varies.

(-4, 3) (-1, 3) (2, 3)Let's choose the points (-4, 3) and (2, 3) and compute the gradient.

06

0

42

33

m

This makes sense because as you go from left to right on the line, you are not rising or falling (so zero gradient).

The equation of this line is y = 3 since y is 3 everywhere along the line.

In general, the equation of a horizontal line is y = b, where b is the y coordinate of any point on the line.In general, the equation of a horizontal line is y = b, where b is the y coordinate of any point on the line.

x

y

If we look at any points on this line we see that they all have a x coordinate of - 2 and the y coordinate varies.

(-2, 3)

(-2, 0)

(-2, -2)

Let's choose the points (-2, 3) and (-2, - 2) and compute the gradient.

undefined0

5

22

32

m

Dividing by 0 is undefined so we say the gradient is undefined. You can't go from left to right on the line since there isn't a left and right.

The equation of this line is x = - 2 since x is - 2 everywhere along the line.

In general, the equation of a vertical line is x = a, where a is the x coordinate of any point on the line.In general, the equation of a vertical line is x = a, where a is the x coordinate of any point on the line.

positive gradient

negative gradient

zero gradient (or no gradient)

und

efin

ed

gra

die

ntIt is easy to remember 0 gradient

because the line does not slope at all (it is horizontal)

It is easy to remember undefined gradient because you can’t move along from left to right (it is vertical)

x

y

2

2

2

3

x

y

2

3

4

6

22

42

m

We often have points on a line but want to find an equation of the line. We'll see how to do this by looking at an example. Find the equation of a line the contains the points (- 2, 4) and (2, - 2).

First let's plot the points and graph the line.

Now let's compute the gradient---we know it will be negative by looking at the line.

Pick a general point on the line, (x, y).

(x, y)

Use the point (2, -2) and this general point in the gradient formula subbing in the gradient we found.

(2, -2)

222

3 yx

Let's get it in a neater form. If we get rid of brackets and fractions and get the x and y on one side (with positive x term) and constants on the other side we'll have standard form.

This is an equation for the line.

222

3 yx

If we get rid of brackets and fractions and get the x and y on one side (with positive x term) and constants on the other side we'll have standard form.

get rid of brackets 232

3 yx

get rid of fractions by multiplying by - 2 4263 yx

get the x and y terms on one side (with positive x term)

4623 yx

constants on the other side 223 yx 223 yx

x

y

Choose any x and sub it in this equation and solve for y and you will get a point (x, y) that lies on the line.

x = 0 2203 y1y

(0, 1) is on the line

general or standard

form

Let's generalize what we did to get a formula for finding the equation of a line. Let's call the specific point we know on the line (x1, y2).

Multiply both sides by x - x1

rearranging a bit we have:

1

1

xx

yym

(x - x1)(x - x1)

11 xxmyy

This is called the point-gradient formula because it will find the equation of a line when you have a specific point (x1, y2) on the line and the gradient.

11 xxmyy

We can also use it when we know two points on the line because we could find the gradient first and then use one of the points for the specific point.

Example when you have a point and the gradient

A point on a line and the gradient of the line are given. Find two additional points on the line.

25,1 m

-1 0

Remember that gradient is the change in y over the change in x. The gradient is 2 which can be made into the fraction

1

2

(0,3)

So this point is on the line also. You can see that this point is found by changing (adding) 2 to the y value of the given point and changing (adding) 1 to the x value.

+2+1To find another point on the line, repeat this process with your new point

(0,3)+1 +2(1,1) (1,5)

-1 0

A point on a line and the gradient of the line are given. Find two additional points on the line.

-1 0

(1,5)

-1 0

If we find the equation of the line using the point-gradient formula, we can easily find additional points on the line by subbing in various x values and finding the y values.

11 xxmyy

25,1 m

125 xy

125 xy 225 xy

32 yx

x = 0

302 y

3y

A way to do the last problem using the equation of the line

Let's take this equation and solve for y.

-1 0

-1 0

This form of the equation is called gradient-intercept form because it contains the gradient and the y intercept of the line.

32 yx32 xy

bmxy

gradient-intercept form

bmxy

y in

terc

ept

grad

ient

Example of given an equation, find the gradient and y intercept

Find the gradient and y intercept of the given equation and graph it.

0443 yxFirst let's get this in gradient-intercept form by solving for y. -3x +4 -3x +4

434 xy-4 -4

14

3 xy

Now plot the y

intercept

From the y intercept, count the gradient

Change in y

Change in x

Now that you have 2 points you can draw

the line

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au