Unit 28Straight Lines

Presentation 1 Positive and Negative Gradient

Presentation 2 Gradients of Perpendicular Lines

Presentation 3 Application of Graphs

Presentation 4 Equation of Straight Line

Unit 2828.1 Positive and Negative

Gradient

A

B

Example

Find the gradient of the line shown opposite.

Solution

-4 -3 -2 -1 0 1 2 3

5

4

3

2

1

-1

-2

-3

-4

-5

y

x

?

?

?

?

Vertical change = 10

Horizontal change = 6

x

y

A

B

Example

Find the gradient of the line shown opposite.

Solution

x

y

-2 -1 0 1 2 3 4

5

4

3

2

1

A

B

(-2, 4)

(4, 1)

??

?

?

?

x

y

Unit 2828.2 Gradient of Perpendicular

Lines

?

If two lines are perpendicular to one another, then the product of the two gradients is equal to -1.

So if is the gradient of one line , the other line has a

gradient of

Example

Show that the line segment joining the points A(3, 2) and B(5, 7) is perpendicular to the line segment joining the points P(2, 5) and Q(7, 3).

Solution?

??

?

?

??

?

??

Unit 2828.3 Application of Graphs

Distance-time graph

Distanc

e

Time

The gradient gives the velocity.If the gradient is zero, the object is not moving.

2000

1500

1000

500

0 50 100 150 200 250 300 350 400

Example

The graph shows the distance travelled by a girl on bicycle.

Find the speed she is travelling on each stage of the journey

Solution?

?

??

?

??

?

?

A

B

C

D

E

Time (s)

Distance (m

)

Solution

The distance is given by the area under the graph, which can be split into 3 sections A, B and C

Velocit

y

Time

velocity-time graph

The gradient gives the acceleration.If the gradient is zero, the object is moving at a constant velocity.The area under this graph is the distance travelled.

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14

Example

The graph shows how the speed of a bird varies as it flies between two trees. How far apart are the two trees?A B C18m 36m 6m

?? ? ???? ???

Time (s)

Velocity (m

/s)

Unit 2828.4 Equation of a Straight Line

The equation a of a straight line is usually written in the form

Where m is the gradient and c is the intercept.

Example

(a)

(b) Equation of line AB:

As it passes through (3, 1)

and

10

8

6

4

2

-3 -2 -1 0 1 2 3 4 5 6

A

B

G

x

y

?

??

?

?(-1, 9)

(3, 1)

?

??

???

? ??

O

10

8

6

4

2

-3 -2 -1 0 1 2 3 4 5 6

A

B

G

x

y

(-1, 9)

(3, 1)

(c) Coordinates of G: so

(d) Equation of line throughO, the origin, perpendicularto AB:

Equation:

i.e.

(e) Equation of line through O, parallel to AB

Equation:

i.e.

?

?

?

O

??

?

??

?

??