preparing for the formative assessment

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

Good morning! 

Schedule: 

• Revise chapter 3 
• Practise (formative assessment)

What paragraph do you find the most difficult?

3.1 The graph of a linear function

3.2 Finding the equation of a line

3.3 linear relationships

3.4 The balance method

3.5 Solving equations

3.6 Applying equations

3.1 The graph of a linear function

How to draw a graph of a linear function in the coordinate plane

1. Make a table with at least 2 points. 
2. Plot these points in the coordinate plane. 
3. Draw a line through these points. 
4. Don't forget to write the formula next to your graph! 

Example

Draw the graph of y = -3x + 2. 

1. 

x	0	1
y	2	-1

For x = 0 

y = -3 · 0 + 2 = 0 + 2 = 2

y = 2

For x = 1

y = -3 · 1 + 2 = -3 + 2 = -1

y = -1

This gives us the points:  

(0,2) and (1,-1)

Example 

2. Plot (0,2) and (1,-1)

Example 

2. Plot (0,2) and (1,-1)

3. Draw a line through (0,2) and

(1,-1)

Example 

2. Plot (0,2) and (1,-1)

3. Draw a line through (0,2) and

(1,-1)

4. Write the formula next to the 

graph!!

3.1 The graph of a linear function 

How to check if a point is on a line

To check if point (x,y) in on a line you fill in the x-coordinate in the formula of the line and see if the answer matches the y-coordinate.

If the answer and the y-coordinate are the same, the point is on the line. 

Example

Calculate wether points A(9,15) and B(-2,-8) are on line y = 2x - 3

Check point A: 

Filling in x = 9 in y = 2x-3 gives: 

y = 2 · 9 - 3 = 18 - 3 = 15

Check point B:

Filling in x = -2 in y = 2x-3  gives:

y = 2 · -2 - 3 = -4 - 3 = -7  

Point A(9,15) is on the line

Point B(-2,-8) is not on the line

3.2 Finding the equation for a line

The formula for a linear relationship between x and y in an equation in the form 

y = ax+b

Going 1 to the right means going a up

The line has y-intercept (0,b)

y = 2x -3

Going 1 to the right means going 2 up

This line has y-intercept (0,-3)

how to find the equation of a line 

1. Assume y = ax + b 

2. Find the y-intercept. This is (0,b)

3. Pick two grid points on the line. 

4. Write down the equation. 

a=​run​​rise​​

a = .....

What is the
y-intercept?

1. (0,1)
2. (1,0)
3. (0,-1)
4. (-1,0)

What is the
equation of this line?

Answers from the previous questions:
a = -2 

y-intercept = (0,-1)

Equations of parallel lines

Lines with the same gradient are parallel 

Parallel lines have the same gradient

Line l and k both have gradient -2

Line l and k are parallel

Line m has gradient 3

Line m is not parallel to line l or k

3.3 linear relationships 

How to find the equation of a line where the axis have different lables than x and y.

The horizontal axis is labelled p and the vertical axis is labelled N

The equation is in the form: 

N = ap + b

Find the equation

1. The equation is in the form 

N = ap + b 

2. The line goes through (0, -80), so b = -80 

3.                                        

4.  N = ap + b, 

b = -80 and a = 5 so 

N = 5p - 80       

a=​run​​rise​​=​20​​100​​=5

The sum graph 

To draw the sum graph of two graphs you need two points of this graph. 

You can find these points in two way:
1. By adding the y-values of the two graphs with the same x-value 

2. By finding the points directly above the x-intercepts of the two graphs

You don't need to know this for the test!

1. Adding y-values

Given are the lines f and g with equations f: y = -2x + 6 and

g: y = 3x - 2  

f. 

g. 

f + g.  

Two points on the sum graph are : 

(0,4) and (1,5)

x	0	1
y	6	4

x	0	1
y	-2	1

x	0	1
y	6 + -2	4 + 1

You don't need to know this for the test!

You see the sum graph of line f and g. In what order would you do these steps to find the sum graph?

You don't need to know this for the test!

Step 1: find the x-intercept 

Step 2:

 Find the y-values directly above the y-intercepts 

Step 3: Draw a line through the new points 

1. substracting y-values

Given are the lines f and g with equations f: y = -2x + 6 and

g: y = 3x - 2  

f. 

g. 

f - g.  

Two points on the sum graph are : 

(0,8) and (1,3)

x	0	1
y	6	4

x	0	1
y	-2	1

x	0	1
y	6 - -2	4 - 1

The difference graph

You don't need to know this for the test!

1. finding the y-values with the x-intercepts

Step 1: Find the x-intercept with the same y-value as the intersection point

Step 2: Find the x-intercept of the line you are substracting

Step 3: Find the point with the same x-value as the point you found in step 2 on the other line (in this case line f)

Step 4: Draw a line through the two points from step 1 and step 3 

In this case we are trying to find lin f-g

So we are looking for the x-intercept of line g

You don't need to know this for the test!

When you have an equation and you want to find the solution for x:

You can Add or Substract the same number on both sides

You can multiply or divide by the same number on both sides

Example: 

5x + 7      = 32

         -7          -7

5x + 7 -7 = 32 -7

5x + 0      = 25

5x              = 25 

:5                   :5

5x:5           = 25:5

x                  = 5

3.4 The balance method

We always want the variables on the lefthand side, 

That's why we try to remove the 7 from the LHS first

Try it yourself! 

1. -7x + 3 = 24                                                       2. -5 + 6x = 37

Try to remove the 3 on the left hand side first. 

We only want x on the left hand side!

Hint

Try to remove the -5 on the left hand side first. 

We only want x on the left hand side!

Hint

Solutions 

1. -7x + 3        = 24                                             2. -5 + 6x         = 37 

            - 3     - 3                                                        +5              +5

    -7x + 3 - 3 = 24 - 3                                            -5 + 5 + 6x = 37 + 5

    -7x + 0       = 21                                                    0 + 6x           =  42 

    -7x               = 21                                                    6x                  = 42

     :-7                     :-7                                                 :6                        :6

    -7x : -7          = 21:-7                                             6x : 6            = 42 : 6 

     x                     = -3                                                   x                    = 7

    

3.5 Solving equations

In what order should you solve an equation (with brackets)? 

1. 

2. 

3. 

4. 

If there are brackets, multiply them out. 

Simplify the LHS and the RHS if needed.

Move all the terms with a variabel to the LHS and move all the terms without a variabel to the RHS.

Divide by the number in front of the variabel

3.5 Solving equations 

Equations with fractions 

When you want to change a fraction into a whole number, you multiply the fraction by it's denominator because: 

so what should

we do when      

​4​​3​​⋅​1​​4​​=​4​​12​​=3

​5​​3​​x=6

4/1 = 4 

?

How would you solve
?

​5​​3​​x=6

3.6 Applying equations 

You can solve many problems using an equation!! 

Problem: 

In 4 years I will be 5 times as old as I was 16 years ago. How old am i now? 

I want to know my age right now so we will call this x. Right now I am x years old.

How old am I in 4 years? 

How old was I 16 years ago?  

What is equal to my age 16 years ago times 5?

3.6 Applying equations 

You can solve many problems using an equation!! 

Problem: 

In 4 years I will be 5 times as old as I was 16 years ago. How old am i now? 

Right now my age is x

How old am I in 4 years? x + 4

How old was I 16 years ago?  x - 16

What is equal to my age 16 years ago times 5? (x - 16) · 5 = x + 4

Now solve for x to find my age! 

What can I work on right now? 

Mixed exercises: p. 122-123

Diagnostic test: p.126-127

Revision: p. 128-129