Straight Lines 2 - Gradient

Straight LinesStraight Lines

Objectives:E Grade Plot the graphs of straight lines such as

x = 3 and y = 4 Complete a table of values for equationssuch as y = 3x + 1 and draw the graph

D Grade Solve problems involving graphs, such as findingwhere the line y = x +3 crosses the line y = 2

C Grade Recognise the equations of straight line graphssuch as y = -3x + 1Find the gradients of straight line graphs

Prior knowledge: Plot co-ordinates in all four quadrants

Straight LinesStraight Lines

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

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-1

-2

-3

-4

-5

-6

-7

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-10

Complete the table for the equation y = xx -3 -1 0 1 3y          

x -3 -1 0 1 3y          

x -3 -1 0 1 3y          

x -3 -1 0 1 3y          

Complete the table for the equation y = -x

Complete the table for the equation y = 3x

Complete the table for the equation y = 2x

-3 -1 0 1 3

3 1 0 -1 -3

-6 -2 0 2 6

-9 -3 0 3 9

xxx

x

x

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xx

The Gradient of a straight line

Straight LinesStraight Lines

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

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5

6

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-1

-2

-3

-4

-5

-6

-7

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-10

What do you notice about these straight lines?

y = x y = 2xy = 3xy = -x

They are not parallel- they have different gradients

xxx

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Straight LinesStraight Lines

We look at the gradient more closely:

y = x            

0 1 2 3 4 5 6

5

4

3

2

1

if x = 1 then y = 1

So we say :“every time we go across 1we go up 1” x

x

if x = 2 then y = 2

Straight LinesStraight Lines

y = 2x            

0 1 2 3 4 5 6

5

4

3

2

1

if x = 1 then y = 2

So we say :“every time we go across 1we go up 2”

x

x

if x = 2 then y = 4

if x = 0 then y = 0

x

Straight LinesStraight Lines

y = 3x            

0 1 2 3 4 5 6

5

4

3

2

1

if x = 1 then y = 3

So we say :“every time we go across 1we go up 3”

x

x

if x = 2 then y = 6

if x = 0 then y = 0

x

Straight LinesStraight Lines

y = -x            

0 1 2 3 4 5 6

2

1

0

-1

-2

if x = 1 then y = -1

So we say :“every time we go across 1we go down 1”

x

x

if x = 2 then y = -2

if x = 0 then y = 0

x

positive gradientA positive coefficient for x

negative gradientA negative coefficient for x

e.g. y = x, y = 2x, y = 3x e.g. y = -x, y = -2x, y = -3x

Straight LinesStraight Lines

To summarise for the gradient of a line:

The coefficient of x tells us the gradient of a straight line (how steep it is)

y = 2x

y = 3x

y = -x

y = x A gradient of 1 “every time we go across 1 we go up 1”

A gradient of 2 “every time we go across 1 we go up 2”Steeper than a gradient of 1

A gradient of 3 “every time we go across 1 we go up 3”Steeper than a gradient of 2

A gradient of -1 “every time we go across 1 we go down 1”A negative gradient

Straight LinesStraight Lines

To summarise for any straight line:For any equation in the form y = mx + c

The variable m can be + or -

m is the gradient

c is the y-intercept

Straight LinesStraight Lines

Finding the gradient for the line that passes through two pairs ofcoordinates:

The gradient is the same at every point on the straight line.

To find the gradient between two points find how much it hasgone up and compare this with how much it has gone across.

x

xHow much up (y-direction)

How much across (x-direction)

difference in y

difference in x

gradient =

Straight LinesStraight Lines

Find the gradient for the line that passes through (2,1) and (5,7)

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

7

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9

10

-1

-2

-3

-4

-5

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-10

x

x

difference in y 7 – 1 = 6

difference in x 5 – 2 = 3

gradient = 63

= 2

In general terms we can say (2,1) is (x1,y1)and (5,7) is (x2,y2)

Therefore: Gradient = y2- y1

x2 - x1

Now do these:

Straight LinesStraight Lines

12 - 04 - 0 = 34 - 012 - 0 = 1

30 - 55 - 0 = -17 - 57 - 3 = 1

2-4 - 5-2 - 4 =11

2

Straight LinesStraight Lines

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Complete the table for the equation y = xx -3 -1 0 1 3y          

x -3 -1 0 1 3y          

x -3 -1 0 1 3y          

x -3 -1 0 1 3y          

Complete the table for the equation y = -x

Complete the table for the equation y = 3x

Complete the table for the equation y = 2x

Worksheet 1