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

Linear functions

Functions in general Linear functions

Linear (in)equalities

Functions in general

Functions: example

• Base price: 5 Euro• Per kilometer: 2 Euro

price 5 2 7 19

Price of a 7 km ride?

What does a taxi ride cost me with company A?

• Base price: 5 Euro• Per kilometer: 2 Euro

5 2y x

Price of an x km ride?

Functions: example

What does a taxi ride cost me with company A?

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

First way: Most concrete form!

Through a TABLE, e.g. for y = 2x + 5:

x y

0 5

1 7

2 9

… …

Functions : 3 representations

Functions : 3 representations

Second way: Most concentrated form!

Through the EQUATION, e.g. y = 2x + 5.

formula y = 5 + 2x:EQUATION OF THE FUNCTION

Third way:Most visual form!

In the example, the graph is a (part of a) STRAIGHT LINE!

1 5

2

14

x

y

Through the GRAPH,e.g. for y = 2x + 5:

Functions : 3 representations

Functions : Summary

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

y = 5 + 2x

FIXED PART + VARIABLE PART

FIXED PART + MULTIPLE OF INDEPENDENT VARIABLE

FIXED PART + PART PROPORTIONAL TO THE INDEPENDENT VARIABLE

Linear functions : 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

Caution: m and q FIXED (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 function : 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

• 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 q

• 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).

ym

x

Graphical significance of mTherefore:

Linear functions: equation

Significance of the parameter m

q in the example of taxi company A

q shows where the graph cuts the Y-axis: Y-INTERCEPT

In general:

Linear functions : graph

Graphical significance of the parameter q

graph of linear function is (part of) a STRAIGHT LINE!

• 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 function : 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

-2 2

-2

2

x

y

m < 0-2 2

-2

2

x

y

m = 0

-2 2

-2

2

x

y

m > 0

Linear functions : graph

Graphical significance of the parameter m

if x is increased by x units, y is increased by mx units

xmy Increase formula:

Parallel lines have same slope

Linear functions: graph

Graphical significance of the parameter m

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 : graph

Graphical significance of m and q

Linear functions : Exercises

• exercise 4• exercise 5 (only the indicated points are to be used!)

for E: parallel lines have the same slope!

Figure 5

Slope of a straight linegiven by two points:

2 1

2 1

vertical distance

horizontal distance

y y ym

x x x

Equations of straight lines

Linear functions : equation

• straight line through a given point and with a given slope:line through point (x0, y0) with slope m has equation 0 0y y m x x

Equations of straight lines

Linear funtions: 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 unitbond: 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

• 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:

BS B

10 000 250125 3.125

80q

q q

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:

B S

10 000 8040 0.32

250Sq

q q

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 : Implicitly

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

.3

6

4

5

xx

Linear equalities

GRAPHICALLY:equation:solution: 2.5function with equation y=2x-5

2.5 is a zero of the function

052 x

GRAPHICALLY:equation:solution: 2two corresponding functions

2 is x-coordinate of intersection point

1245.0 xx

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

63

x

Linear inequalities

GRAPHICALLY:inequality:solution: x>2.5function with equation y=2x-5

for x>2.5 graph is above horizontal axis

052 x

GRAPHICALLY:inequality:solution: x<2two corresponding functions

for x<2 green graph is higher than blue one

1245.0 xx

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

12

82

yx

yx

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

(3,9)

x

y

(0,7)

(0,3)

(6,6)

(2,0)

A

B

C

D E

F

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