2TH/vwo §1.5 Line through two given points.

• Wiskunde
• Middelbare school
• havo, vwo
• Leerjaar 2

§1.5

Learning goal:

how to make a formula for a line that passes through two (given) points

Let's do a useful exercise together!

This subject is found pretty difficult, sometimes.

That's why we do this together, step by step!

Task: make a linear formula.

Given are two points:

P(10, 6) and Q(18, 30)

You will do similar tasks later. 

Now we tackle this one together!

First: it helps to make a Sketch!

Without it the task stays pretty obscure...

The given points

P(10, 6)  and  Q(18, 30) 

are drawn 

in this coordinate system.

TASK: calculate the GRADIENT.

ANS (answer it in next slide)

Use the given points,

P(10, 6)  and  Q(18, 30) !

Gradient is ...

1. 5
2. -3
3. 1/3
4. 3

Solution:

Going from P(10, 6)  to  Q(18, 30) means:

-  going up 30 - 6 = 24 (= vertical increase) 

-  going to the right 18 - 10 = 8 (= horizontal increase)

If we divide the vertical increase by the horizontal increase we get the 

gradient:    24 : 8 = 3 = gradient !

TASK 

Find the y - intercept!

There's one problem: you cannot read 

it off! 

So a CALCULATION is needed here!

THERE ARE 2 METHODS FOR FINDING THE Y-INTERCEPT.

Here comes the 1st one.

                    

Finding the y-intercept following 

Method 1: 

-  we use the general shape of a linear formula: y = ax + b

- we fill in the gradient, that we just calculated:

   y = 3x + b

- to find the y-intercept, we fill in one of the given points;

  see next slide!

In the formula:   y = 3x + b

we fill in P(10, 6) as follows:

6 = 3    10 + b            Note: the dot means 'times'!

6 = 30 + b

b = -24       We found the value of b, so we found the y-intercept!

⋅

⋅

Now we can make the formula complete:

We had y = 3x + b  already.

We just calculated that b = -24

So the formula is:

y = 3x -24                                                     

2nd Method to find the y-intercept!

Another way to calculate the Gradient and the

y-intercept !                 With TABLES.

The points P(10, 6) and Q(18, 30) are placed in this table:

Adding 8  to  x    and

adding  24  to  y,

means the same as:

adding   1  to  x  and

adding  3  to  y !                               That's because  24 : 8 = 3

Adding 1 to x means adding 3 to y.
Then the gradient is ..

1. 1
2. -3
3. 3
4. 1/3

Solution: the GRADIENT is:  

                                                      3,

because going one place to the right in the graph,

means going 3 places up!

To find the y-intercept with the table,

we have to find the y coordinate to 0  as x-coordinate:

Try to find the y-intercept!

 So: what has to be filled in 

on the dots?

ANS

1. -30
2. -24
3. 24
4. 0

To find the y-intercept with the table,

we have to find the y coordinate to 0  as x-coordinate:

The numbers  1  and  3

have the same ratio as 

-10  and  -30.

So we do 6 -30 = -24

-24 is the y-intercept!

FORMULA:

y = 3x - 24

we do 33a together:

So open your Notebook, Then 

write '33' in the margin.

take a pencil and a geo and make the sketch!

Solution:

By answering questions 

33b, c, d, e, f and g

you will again find the 

linear formula for the graph

through two given points!

Useful advice:

On p.27 you see the METHOD, how to make a linear formula for a line through two points.

USE THIS METHOD all the time, when you have to make a formula!

So for instance for questions:   

35a, b, c      36a      and  37d         

Homework:

§1.5, not 32, 34  and  38.

By the way, later on in this lesson I will give you a little more advice about the exercises.