LINEAR LINEAR GRAPHSGRAPHS

2D Graphs2D Graphs- Show how two quantities relate- Have labelled axes, usually with scales showing units

Height (m)

Age (years)

Bob

Jane

Tom

Mary

e.g.

a) Who is the tallest?

b) Who is the same age?

c) Who is the oldest?

Tom

Bob and Mary

Jane

Distance/Time Distance/Time GraphsGraphs- Are line graphs with time on the horizontal and distance on

the vertical axis.- If the line is horizontal the object is not moving- The steeper the line, the faster the movement

9 am 11 am 1 pm 3 pm

time of day

dis

tance

fro

m h

arb

our

(km

)10

8

6

4

2

0

Distance of yacht from harboure.g.

a) How far out from the harbour did the yacht travel?b) What happened while the graph was horizontal?c) Which part of the journey was quickest?

5 km

The yacht was stationary

Steepest

The return journey

Co-ordinatesCo-ordinates- The first number relates to the horizontal axis (x)- The second number relates to the vertical axis (y)

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

e.g. Plot the following pointsA = (4,2) and B = (ˉ 3,ˉ 4)

A

B

Remember to always label your x and y axis!

Plotting PointsPlotting Points- To draw straight line graphs we can use a rule to find and plot co-ordinatese.g. Complete the tables below to find co-ordinates in order to plot the following straight lines:a) y = 2x b) y = ½x – 1 c) y = -3x + 2 x y = 2x y = ½x –

1

-2

-1

0

1

2x y = -3x +

2

-2

-1

0

1

2

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

2 x -1

2 x -2

0

2

-4

4

-2 ½ x -1 – 1

½ x -2 – 1

-1

-½

-2

0

-1 ½

-3 x -1 + 2

-3 x -2 + 2

2

-1

8

-4

5

Gradients of LinesGradients of Lines- The gradient is a number that tells us how steep a line is.- The formula for gradient is:Gradient = rise

run

1st point

2nd point

rise

run

 e.g. Write the gradients of lines A and B

A

B

A =

B =

4

6 8

4

4 = 18 2

6 = 34 2

 e.g. Draw lines with the following gradientsa) 1 b) 3 c) 2 2 5

To draw, write gradients as fractions

= 3 1

When calculating gradients it is best to write as simplest fraction

y = mxy = mx- This is a rule for a straight line, where the gradient (m) is the number directly in front of the x- When drawing graphs of the form y = mx, the line always goes through the origin i.e. (0,0)e.g. Draw the following lines:a) y = 2x b) y = 4x c) y = 3x 5 4

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

1. Step off the gradient from the origin (0,0) 2. Join the plotted point back to the origin

= 4x 1

To draw, always write gradients as fractions

gradient

Negative Negative GradientsGradients

 e.g. Write the gradients of lines A and B

A =

B =

-3

2

10

-5

-5 = -110 2

-3 2

A

B

When calculating gradients it is best to write as simplest fraction

e.g. Draw the following lines:a) y = -2x b) y = -4x c) y = -3x 5 4

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

1. Step off the gradient from the origin (0,0) 2. Join the plotted point back to the origin

= -4x 1

To draw, always write gradients as fractions

gradient

InterceptsIntercepts- Is a number telling us where a line crosses the y-axis (vertical axis)i.e. The line y = mx + c has m as the gradient and c as the intercept e.g. Write the intercepts of the lines A, B and C

x-10 -8 -6 -4 -2 2 4 6 8 10

y

-10

-8

-6

-4

-2

2

4

6

8

10

A

B

C

A =

B =

C =

8

4

-3

Drawing Lines: Gradient and Intercept Drawing Lines: Gradient and Intercept MethodMethod- A straight line can be expressed using the rule y = mx + c

e.g. Draw the following lines:a) y = 1x + 2 b) y = -3x – 2 c) y = -4x + 8 2 7

x-10 -8 -6 -4 -2 2 4 6 8 10

y

-10

-8

-6

-4

-2

2

4

6

8

10To draw:1. Mark in intercept2. Step off gradient3. Join up points

= -3x – 2 1

Note: Any rule with no number in front of x has a gradient of 1 1e.g. y = x – 1

Writing Equations of LinesWriting Equations of Lines- A straight line can be expressed using the rule y = mx + c

e.g. Write equations for the following lines

x-10 -8 -6 -4 -2 2 4 6 8 10

y

-10

-8

-6

-4

-2

2

4

6

8

10

A

B

C

A: B: C:m = c = m = c = m = c =

y = 3x – 6 4

y = -2x + 1 3

y = 4x + 4 1

34

-2 3

41 -6 +1 +4

Horizontal and Vertical LinesHorizontal and Vertical Lines- Horizontal lines have a gradient of:

0Rule: y = c (c is the y-axis intercept)- Vertical lines have a

gradient that is:undefined

Rule: x = c (c is the x-axis intercept)e.g. Draw or write equations for the following lines:

a) y = 2 b) c) x = 4 d)

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

x = -1y = -3

b)

d)

Writing Equations Cont.Writing Equations Cont.When you are given two points and are expected to write an equation:- One method is set up a set of axes and plot the two points.

e.g. Write an equation for the line joining the points A=(1, 3) and B = (3, -1)

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345 m = -2

1c = 5

y = -2x + 5

Sometimes when plotting the points, you may need to extend the axes to find the intercept.

- Or, substitute the gradient and a point into y = mx + c to find ‘c’, the intercept

m = -2 1

using point

(1, 3)

y = mx + c 3 = -2 x 1 + c 3 = -2 + c 5 = c

+2 +2

y = -2x + 5

Equations in the Form ‘ax + by = c’Equations in the Form ‘ax + by = c’- Can use the cover up rule to find the two intercepts:

e.g. Draw the following lines:a) 2y – x = 4 b) 4x – 3y =12

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5

-4

-3

-2

-1

1

2

3

4

51. Cover up ‘y’ term to find x intercept

- x = 4÷ -1 ÷ -1

x = -4

2. Cover up ‘x’ term to find y intercept

2y = 4÷ 2 ÷ 2

y = 2

3. Join up intercepts with a straight line

4x = 12÷ 4 ÷ 4

x = 3

-3y = 12÷ -3 ÷ -3

y = -4

It is also possible to rearrange equations into the form y = mx + ce.g. Rearrange 2x – y = 6

-2x -2x- y = 6 – 2x÷ -1 ÷ -1

y = -6 + 2xy = 2x – 6

x1 2 3 4 5 6 7 8 9 10

y

102030405060708090

100110120130140

ApplicationsApplicationse.g. A pizzeria specializes in selling large size pizzas. The relationship between x, number of pizzas sold daily, and y, the daily costs is given by the equation, y = 10x + 50

1. Draw a graph of the equation

2. What are the costs if they sell 8 pizzas?$1303a. What is the cost per pizza?$103b. How is this shown by the graph?

The gradient of the line4a. What are the costs

if they sell no pizzas?

$504b. How is this shown by the graph?

Where the line crosses the y-axis