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

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Linear functions Functions in general Linear
functionsLinear (in)equalities
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Functions in general
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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?
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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?
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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
  yf(x)
• y DEPENDENT VARIABLE
• x INDEPENDENT VARIABLE

INPUT x ? OUTPUT y
Function rule that assigns to each input
exactly 1 output
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Functions 3 representations

• First way Most concrete form!
• Through a TABLE, e.g. for y 2x 5

x y
0 5
1 7
2 9

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Functions 3 representations

• Second way Most concentrated form!
• Through the EQUATION, e.g. y 2x 5.

formula y 5 2x EQUATION OF THE FUNCTION
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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!
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Functions Summary
- Example - Definition - 3 representations
table, equation, graph
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Linear functions equation

• y 5 2x
• FIXED PART VARIABLE PART
• FIXED PART MULTIPLE OF
• INDEPENDENT VARIABLE
• FIXED PART PART PROPORTIONAL
• TO THE INDEPENDENT VARIABLE

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Cost of a ride with company B,C,..?
Linear function equation

• 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!
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DIFFERENT SITUATIONS which give rise to first
degree functions?
Linear function equation

• 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!

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!! Situations where function is NOT a FIRST
DEGREE FUNCTION?
Linear functions equation

• 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!

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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

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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
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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
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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
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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

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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
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Linear functions graph
Graphical significance of m and q

• We can see this significance very clearly here
• http//www.rfbarrow.btinternet.co.uk/htmks3/Linear
  1.htm
• Or here
• http//standards.nctm.org/document/eexamples/chap7
  /7.5/index.htm

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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!
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Equations of straight lines
Linear functions equation

• Slope of a straight line
• given by two points

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Equations of straight lines
Linear funtions equation

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

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Linear function Exercises

• exercise 5
• exercise 6
• exercise 8

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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

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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.

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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!
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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!
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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

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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

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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.

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Linear functions Summary
- equation first degree function ymxq,
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
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Linear equalities

• Exercise 9
• A LINEAR EQUATION in the unknown x is an
  equation that can be written in the form a x
  b0 , with a and b numbers and a ? 0.
• Exercise 3

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Linear equalities

• TWO TYPICAL EXAMPLES
• Example 5x-83x-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

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Linear equalities

• GRAPHICALLY
• equation
• solution 2.5
• function with equation y2x-5
• 2.5 is a zero of the function

GRAPHICALLY equation solution 2 two
corresponding functions 2 is x-coordinate
of intersection point
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Linear inequalities

• Exercise 12
• A LINEAR INEQUALITY in the unknown x is an
  inequality that can be written in the form axblt0
  or axb0 or axbgt0 or axb0, with a and b
  numbers (a ? 0).

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Linear inequalities

• STRATEGY to solve
• Example 5x-8gt3x-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

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Linear inequalities

• GRAPHICALLY
• inequality
• solution xgt2.5
• function with equation y2x-5
• for xgt2.5 graph is above horizontal axis

GRAPHICALLY inequality solution xlt2 two
corresponding functions for xlt2 green graph
is higher than blue one
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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
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Systems of 2 linear equations

• Exercise 10 (a)
• Supplementary exercises
• Exercise 10 (b, c)
• Exercises 11

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Linear (in)equalities Summary
- linear equation - linear inequalities
- system of two linear equations
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Exercises !
TO LEARN MATHEMATICS TO DO A LOT OF EXERCISES
YOURSELF, UNDERSTAND MISTAKES AND DO THE
EXERCISES AGAIN CORRECTLY
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Exercise 5
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