Linear functionsFunctions in general Linear functionsLinear (in)equalities
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?
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 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:
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 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
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 + mx y = mx + q FIRST DEGREE FUNCTION! LINEAR FUNCTION Caution: m and qFIXED (for each company): parameters x and y: VARIABLES!
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!
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= mx (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
Linear functions: graph Graphical significance of the parameter m if x is increased by x units, y is increased by mx 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 4 • exercise 5 (only the indicated points are to be used!) Figure 5 for E: parallel lines have the same slope!
Linear functions : equation Equations of straight lines Slope of a straight line given by two points:
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 • exercise 5 • exercise 6 • exercise 8
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, qBindependent 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, qSis 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, hencewe 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: equation: solution: 2 twocorrespondingfunctions 2 is x-coordinate of intersection point GRAPHICALLY: equation: solution: 2.5 function with equation y=2x-5 2.5 is a zero of the function
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: inequality: solution: x<2 twocorrespondingfunctions for x<2 green graph is higherthan blue one GRAPHICALLY: inequality: solution: x>2.5 functionwithequation y=2x-5 for x>2.5 graph is above horizontal axis
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