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

Functions in general

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,…

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?

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

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

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

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!

Functions : Summary - Example - Definition - 3 representations : table, equation, graph

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!

Cost of a ride with company B,C,..? Linear function : equation Cost of a ride with company B,C,..? Examples : y = 4.50 + 2.10x; y = 5.20 + 1.90x; 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!

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 20 000 Euro and drive it for x km, if the costs amount to 0.8 Euro per km? y = 20 000 + 0.8x 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 = 3 + 0.2q hence y = mx + q! FK zijn de vaste kosten, de kosten die er zijn als nog niets geproduceerd wordt.

!! 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!

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

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:

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

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

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!

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

Linear functions : graph Graphical significance of m and q We can see this significance very clearly here … http://www.rfbarrow.btinternet.co.uk/htmks3/Linear1.htm Or here… http://standards.nctm.org/document/eexamples/chap7/7.5/index.htm

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

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

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

Linear function : Exercises

Linear functions : Implicitly Invest a capital of 10 000 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 = 10 000

Linear functions : Implicitly We have: 80qS + 250qB = 10 000 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.

Linear functions : Implicitly We have: 80qS + 250qB = 10 000 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!

Linear functions : Implicitly We have: 80qS + 250qB = 10 000 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!

Linear functions : Implicitly Connection, RELATION, between qS and qB: 80qS + 250qB = 10 000: 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

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:

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.

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

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

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

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

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).

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

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

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

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

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

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

Exercise 5 Back