1. Linear functions Functions in general Linear functions Linear (in)equalities

2. Functions in general

3. Functions: example What does a taxi ride cost me with company A?
Base price: 5 Euro
Per kilometer: 2 Euro
Price of a 7 km ride?
This is the most simple situation. In practice, there may be other factors that influence the price: number of passengers, waiting rate (e.g. in case of a traffic jam), within or outside a certain area,…

4. Functions: example What does a taxi ride cost me with company A?
Base price: 5 Euro
Per kilometer: 2 Euro
Price of an x km ride?

5. Functions : definition
x (length of ride) en y (price of ride):
VARIABLES
y depends on x:
y is FUNCTION of x, notation: y(x) or y=f(x)
y: DEPENDENT VARIABLE
x: INDEPENDENT VARIABLE
INPUT : x  OUTPUT y
De uitdrukking “is functie van” is in de natuurlijke taal niet zo gebruikelijk. Misschien in een zin zoals “het loon is functie van het aantal dienstjaren”?
Bij “vergelijking” hier NIET denken aan een onbekende die gezocht moet worden!
Function: rule that assigns to each input
exactly 1 output

6. Functions : 3 representations
First way: Most concrete form!
Through a TABLE, e.g. for y = 2x + 5: x y 5 1 7 2 9 …

7. Functions : 3 representations
Second way: Most concentrated form!
Through the EQUATION, e.g. y = 2x + 5.
formula y = 5 + 2x:
EQUATION OF THE FUNCTION

8. Functions : 3 representations
Third way:
Most visual form!
Through the GRAPH,
e.g. for y = 2x + 5:
In wiskunde zetten we bij een grafiek altijd de onafhankelijke veranderlijke op de horizontale as en de afhankelijke op de verticale as. In economie is het soms anders. In het taxivoorbeeld zijn alleen de positieve waarden van x en y zinvol! Bij eerstegraadsfuncties is de grafiek altijd een rechte!
In the example, the graph is a (part of a) STRAIGHT LINE!

9. Functions : Summary
- Example - Definition representations : table, equation, graph

10. Linear functions : equation
y = 5 + 2x
FIXED PART + VARIABLE PART
FIXED PART + MULTIPLE OF
INDEPENDENT VARIABLE
FIXED PART + PART PROPORTIONAL
TO THE INDEPENDENT VARIABLE
Wijzen op begrippen vaste kosten, variabele kosten en marginale kosten.
Betekenis evenredig benadrukken!

11. Cost of a ride with company B,C,..?
Linear function : equation
Cost of a ride with company B,C,..?
Examples : y = x; y = x; etc. …
In general:
y = base price + price per km  x
y = q + m x
y = m x + q
FIRST DEGREE FUNCTION!
LINEAR FUNCTION
Parameters zijn in feite ook wel “veranderlijken” …!
Caution: m and q FIXED (for each company): parameters
x and y: VARIABLES!

12. DIFFERENT SITUATIONS which give rise to first degree functions?
Linear function : equation
DIFFERENT SITUATIONS which give rise to first degree functions?
Cost y to purchase a car of Euro and drive it for x km, if the costs amount to 0.8 Euro per km?
y = x hence … y = mx + q!
Production cost c to produce q units, if the fixed cost is 3 and the production cost is 0.2 per unit?
c = q hence y = mx + q!
FK zijn de vaste kosten, de kosten die er zijn als nog niets geproduceerd wordt.

13. !! Situations where function is NOT a FIRST DEGREE FUNCTION?
Linear functions : equation
!! Situations where function is NOT a FIRST DEGREE FUNCTION?
To crash with a taxi at a speed of 100 km/h is MUCH more deadly than at 50 km/h, since the energy E is proportional to the SQUARE of the speed v.
For a taxi of 980 kg: E = 490v²
i.e. NOT of the form y = mx + q
Therefore NOT a linear function!

14. Linear functions : equation
Significance of the parameter q
Taxi company A: y = 2x + 5. Here q = 5: the base price.
q can be considered as THE VALUE OF y WHEN x = 0.
Graphical significance of q

15. Linear functions: equation
Significance of the parameter m
Taxi company A: y = 2x + 5, m = 2: the price per km.
m is CHANGE OF y WHEN x IS INCREASED BY 1.
If x is increased by e.g. 3 (the ride is 3 km longer), y will be increased by 2  3 = 6 (we have to pay 6 Euro more).
In mathematical notation: if x = 3 then y = 2  3 = m  x.
Always: y = mx (INCREASE FORMULA).
Graphical significance of m
Therefore:

16. Linear functions : graph
graph of linear function is (part of) a STRAIGHT LINE!
Graphical significance of the parameter q
q in the example of taxi company A
In general:
q shows where the graph cuts the Y-axis: Y-INTERCEPT

17. Linear function : graph
Graphical significance of the parameter m
m in the example of taxi company A
if x is increased by 1 unit, y is increased by m units
m is the SLOPE of the straight line

18. Linear functions : graph
Graphical significance of the parameter m
Sign of m determines
whether the line is going up / horizontal / down
whether linear function is increasing / constant(!!) / decreasing
Size of m determines how steep the line is
Wijzen op de rol van de keuze van de eenheden op de assen!

19. Linear functions: graph
Graphical significance of the parameter m
if x is increased by x units, y is increased by mx units
Increase formula:
Parallel lines have same slope

20. Linear functions : graph
Graphical significance of m and q
We can see this significance very clearly here …
Or here…

21. Linear functions : Exercises
exercise 5 (only the indicated points are to be used!)
Figure 5
for E: parallel lines have the same slope!

22. Equations of straight lines
Linear functions : equation
Equations of straight lines
Slope of a straight line
given by two points:

23. Equations of straight lines
Linear funtions: equation
Equations of straight lines
straight line through a given point and with a given slope:
line through point (x0, y0) with slope m has equation

24. Linear function : Exercises

25. Linear functions : Implicitly
Invest a capital of Euro in a certain share and a certain bond
share: 80 Euro per unit
bond: 250 Euro per unit
How much of each is possible with the given capital?
Let qS be the number of units of the share and qB the number of units of the bond.
We must have: 80qS + 250qB =

26. Linear functions : Implicitly
We have: 80qS + 250qB =
There are infinitely many possibilities for qS en qB
e.g.: qS = 0, qB = 40;
qS = 125, qB = 0;
qS = 100, qB = 8
etc. …
Not all combinations are possible!
There is a connection, A RELATION, between qS and qB.

27. Linear functions : Implicitly
We have: 80qS + 250qB =
We can represent the connection, THE RELATION, between qS and qB more clearly, EXPLICITLY, as follows:
qS is dependent, qB independent variable,
connection is of the form y = mx + q
hence LINEAR FUNCTION!

28. Linear functions : Implicitly
We have: 80qS + 250qB =
We can represent the connection, THE RELATION, between qS and qB more clearly, EXPLICITLY, as follows:
Now qB is dependent, qS is independent variable,
connection is again of the form y = mx + q
hence LINEAR FUNCTION!

29. Linear functions : Implicitly
Connection, RELATION, between qS and qB:
80qS + 250qB = : IMPLICIT equation
both variables on the same side, form ax + by + c = 0
qB = 40  0.32qS: EXPLICIT equation
dependent variable isolated in left hand side, right hand side contains only the independent variable, form y = mx + q
qS = 125  3.125qB: EXPLICIT equation

30. Linear functions : Implicitly
THE RELATION between qS and qB corresponds in this case to LINEAR FUNTION (two possibilities!) and can therefore be presented graphically (in two ways!) as A PART OF A STRAIGHT LINE:

31. Linear functions : Implicitly
The graph of a first degree function with equation y = mx + q is A STRAIGHT LINE.
An equation of the form ax + by + c = 0 with b  0 determines a first degree function and thus is also the equation of a straight line. (In order to isolate y we have to DIVIDE by b, hence we need b  0!)
Every equation of the form ax + by + c = 0 WHERE a AND b ARE NOT BOTH 0 determines a straight line! See exercise 7.

32. Linear functions: Summary
- equation: first degree function y=mx+q, interpretation m,q - graph : straight line interpretation m,q - setting up equations of straight line based on two points slope and point - implicit linear function

33. Linear equalities Exercise 9
A LINEAR EQUATION in the unknown x is an equation that can be written in the form a x + b=0 , with a and b numbers and a ≠ 0.
Exercise 3

34. Linear equalities TWO TYPICAL EXAMPLES: Example: 5x-8=3x-2
Terms involving x on 1 side, rest on the other.
Example:
Write the equation in a form that is free of fractions, by multiplying by the (least common) multiple of all denominators.
Exercise 1

35. Linear equalities GRAPHICALLY: GRAPHICALLY: equation: solution: 2.5
function with equation y=2x-5
2.5 is a zero of the function
GRAPHICALLY:
equation:
solution: 2
two corresponding functions
2 is x-coordinate of intersection point

36. Linear inequalities Exercise 12
A LINEAR INEQUALITY in the unknown x is an inequality that can be written in the form ax+b<0 or ax+b≤0 or ax+b>0 or ax+b≥0, with a and b numbers (a ≠ 0).

37. Linear inequalities STRATEGY to solve: Example: 5x-8>3x-2
Terms involving x on one side, rest on the other side gives. If you divide by positive (negative) number sense of inequality remains (changes)
Exercise 2
Exercise 13

38. Linear inequalities GRAPHICALLY: GRAPHICALLY: inequality:
solution: x>2.5
function with equation y=2x-5
for x>2.5 graph is above horizontal axis
GRAPHICALLY:
inequality:
solution: x<2
two corresponding functions
for x<2 green graph is higher than blue one

39. System of linear equalities
Example:
STRATEGY to solve:
- Elimination-by-combination method
- Elimination-by-substitution method
- Elimination-by-setting equal mehod
GRAPHICALLY:
Intersection of two lines

40. Systems of 2 linear equations
Exercise 10 (a)
Supplementary exercises:
Exercise 10 (b, c)
Exercises 11

41. Linear (in)equalities : Summary
- linear equation linear inequalities system of two linear equations

42. Exercises !
TO LEARN MATHEMATICS = TO DO A LOT OF EXERCISES YOURSELF, UNDERSTAND MISTAKES AND DO THE EXERCISES AGAIN CORRECTLY

43. Exercise 5
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