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Topic 1

Topic 1 : Elementary functions

Reading: JacquesSection 1.1 - Graphs of linear equations

Section 2.1 – Quadratic functions Section 2.2 – Revenue, Cost and Profit

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Linear Functions

• The function f is a rule that assigns an incoming number x, a uniquely defined outgoing number y.

y = f(x)• The Variable x takes on different values…... • The function f maps out how different values

of x affect the outgoing number y. • A Constant remains fixed when we study a

relationship between the incoming and outgoing variables

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Simplest Linear Relationship:

y = a+bx ← independent dependent ↵ ↑ variable variable intercept

This represents a straight line on a graph i.e. a linear function has a constant slope

• b = slope of the line = change in the dependent variable y, given a change in the independent variable x.

• Slope of a line = ∆y / ∆x= (y2-y1) / (x2-x1)

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Example: Student grades

• Example: • y = a + bx• y : is the final grade, • x : is number of hours

studied,• a%: guaranteed

• Consider the function: • y = 5+ 0x • What does this tell

us?• Assume different

values of x ………

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Example Continued: What grade if you study 0 hours? 5 hours?

• y=5+0x Output = constant slope Input

y a b X

5 5 0 0

5 5 0 1

5 5 0 2

5 5 0 3

5 5 0 4

5 5 0 5

Linear Functions

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10

20

30

40

50

60

0 1 2 3 4Independent X Variable

Dep

end

ent

Y V

aria

ble

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Example Continued….

• y=5+15x

output= constant slope input y a b X 5 5 15 0

20 5 15 1 35 5 15 2 50 5 15 3 65 5 15 4

Linear Functions

-5

5

15

25

35

45

55

65

0 1 2 3 4Independent X Variable

Dep

end

ent

Y V

aria

ble If x = 4, what grade

will you get?Y = 5 + (4 * 15) = 65

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Demand functions: The relationship between price and quantity

Demand Function: D=a-bP D= 10 -2P

D a - b P 10 10 -2 0 8 10 -2 1 6 10 -2 2 4 10 -2 3

Demand Function

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2

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6

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0 1 2 3 4 5

Price

Q D

eman

d

If p =5, how much will be demanded?

D = 10 - (2 * 5) = 0

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Inverse Functions:

• Definition• If y = f(x) • then x = g(y)• f and g are inverse functions• Example• Let y = 5 + 15x • If y is 80, how many hours per week did they

study?

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Example continued…..

• If y is 80, how many hours per week did they study?

• Express x as a function of y: 15x = y – 5....

• So the Inverse Function is: x = (y-5)/15

• Solving for value of y = 80x = (80-5 / 15)

x = 5 hours per week

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An inverse demand function

• If D = a – bP then the inverse demand curve is given by P = (a/b) – (1/b)D

• E.g. to find the inverse demand curve of the function D= 10 -2P ……

First, re-write P as a function of D

2P = 10 – D

Then, simplify

So P= 5 – 0.5D is the inverse function

More Variables:

• Student grades again:y = a + bx + cz

• y : is the final grade, • x : is number of hours

studied,• z: number of

questions completed• a%: guaranteed

• Example: • If y = 5+ 15x + 3z, and a

student studies 4 hours per week and completes 5 questions per week, what is the final grade?

• Answer:• y = 5 + 15x + 3z

• y = 5 + (15*4) + (3*5)

• y = 5+60+15 = 80

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Another example: Guinness Demand.

• The demand for a pint of Guinness in the Student bar on a Friday evening is a linear function of price. When the price per pint is €2, the

demand ‘is €6 pints.

When the price is €3, the demand is only 4 pints. Find the function

D = a + bP

• 6 = a + 2b=> a = 6-2b

• 4 = a + 3b=> a = 4-3b

• 6-2b = 4-3b

• Solving we find that b = -2

• If b = -2, then a = 6-(-4) = 10• The function is D = 10 – 2P• What does this tell us??• Note, the inverse Function is

• P = 5- 0.5D

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A Tax Example….

• let €4000 be set as the target income. All income above the target is taxed at 40%. For every €1 below the target, the worker gets a negative income tax (subsidy) of 40%.

• Write out the linear function between take-home pay and earnings.

• Answer:• THP = E – 0.4 (E – 4000)

if E>4000• THP = E + 0.4 (4000-E)

if E<4000• In both cases,

THP = 1600 +0.6E Soi) If E = 4000 => THP = 1600+2400=4000ii) If E = 5000 =>

THP = 1600+3000=4600iii) If E = 3000 =>

THP = 1600+1800=3400

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Tax example continued…. THP = 1600 +0.6E

If the hourly wage rate is equal to €3 per hour, rewrite take home pay in terms of number of hours worked?

• Total Earnings E = (no. hours worked X hourly wage)• THP = 1600 + 0.6(3H) = 1600 + 1.8H

Now add a (tax free) family allowance of €100 per child to the function THP = 1600 +0.6E

• THP = 1600 + 0.6E + 100Z (where z is number of children)

Now assume that all earners are given a €100 supplement that is not taxable,

• THP = 1600 + 0.6E + 100Z + 100 = 1700 + 0.6E + 100Z

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Topic 1 continued: Non- linear Equations

Jacques Text Book: Sections 2.1 and 2.2

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Quadratic Functions

• Represent Non-Linear Relationships

y = ax2+bx+c where a≠0, c=Intercept

• a, b and c are constants

• So the graph is U-Shaped if a>0,• And ‘Hill-Shaped’ if a<0• And a Linear Function if a=0

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Solving Quadratic Equations:

1) Graphical Approach: To find Value(s), if any, of x when y=0, plot the function and see where it cuts the x-axis

• If the curve cuts the x-axis in 2 places: there are always TWO values of x that yield the same value of y when y=0

• If it cuts x-axis only once: when y=0 there is a unique value of x

• If it never cuts the x-axis: when y=0 there is no solution for x

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e.g. y = -x2+4x+5

y a x2 b X C

-7 -1 4 4 -2 5

5 -1 0 4 0 5

9 -1 4 4 2 5 5 -1 16 4 4 5 -7 -1 36 4 6 5

Since a<0 => ‘Hill Shaped Graph’

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The graph

Quadratic Functions

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-6

-4

-2

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4

6

8

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-2 0 2 4 6

Independent X Variable

Y =

X2

y=0, then x= +5 OR x = -1

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Special Case: a=1, b=0 and c=0So y = ax2+bx+c => y = x2

y = a x2 b x c

16 1 16 0 -4 0

4 1 4 0 -2 0

0 1 0 0 0 0

4 1 4 0 2 0 16 1 16 0 4 0 36 1 36 0 6 0

Quadratic Functions

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-5

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-4 -2 0 2 4 6

Independent X Variable

Y =

X2

Min. Point: (0,0)Intercept = 0

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Practice examples

• Plot the graphs for the following functions and note (i) the intercept value (ii) the value(s), if any, where the quadratic function cuts the x-axis

• y = x2-4x+4

• y = 3x2-5x+6

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Solving Quadratic Equations:

• 2) Algebraic Approach: find the value(s), if any, of x when y=0 by applying a simple formula…

( )a

acbbx

2

42 −±−=

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Example

• e.g. y = -x2+4x+5• hence, a = -1; b=4; c=5

• Hence, x = +5 or x = -1 when y=0• Function cuts x-axis at +5 and –1

( )2

)51(4164

−×−−±−

=x

( )2

64

2

20164

−±−=

−+±−

=x

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Example 2• y = x2-4x+4• hence, a = 1; b= - 4;

c=4 • If y = 0

( )2

)41(4164 ×−±+=x

2

04 ±=x

x = 2 when y = 0

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14

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-2 -1 0 1 2 3 4 5

X

Y

y

Function only cuts x-axis at one point, where x=2

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Example 3

• y = 3x2-5x+6• hence, a = 3;

b= - 5; c=6 • If y = 0

( )6

)63(4255 ×−±+=x

6

474 −±=

when y = 0 there is no solution

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40

60

80

100

120

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5

X

Y

y

The quadratic function does not intersect the x-axis

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Understanding Quadratic Functions

intercept where x=0 is c a>0 then graph is U-shaped a<0 then graph is inverse-U a = 0 then graph is linear

• b2 – 4ac > 0 : cuts x-axis twice• b2 – 4ac = 0 : cuts x-axis once• b2 – 4ac < 0 : no solution

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Essential equations for Economic Examples:

• Total Costs = TC = FC + VC• Total Revenue = TR = P * Q∀ π = Profit = TR – TC• Break even: π = 0, or TR = TC• Marginal Revenue = MR = change in total

revenue from a unit increase in output Q• Marginal Cost = MC = change in total cost

from a unit increase in output Q• Profit Maximisation: MR = MC

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An Applied Problem

• A firm has MC = 3Q2- 32Q+96• And MR = 236 – 16Q• What is the profit Maximising Output?

Solution• Maximise profit where MR = MC

3Q2 – 32Q + 96 = 236 – 16Q3Q2 – 32Q+16Q +96 – 236 = 03Q2 – 16Q –140 = 0

• Solve the quadratic using the formula where a = 3; b = -16 and c = -140

• Solution: Q = +10 or Q = -4.67

• Profit maximising output is +10 (negative Q inadmissable)

( )a

acbbQ

2

42 −±−=

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Graphically

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-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12

Q

MR

an

d M

C

MC

MR

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Another Example….

• If fixed costs are 10 and variable costs per unit

are 2, then given the inverse demand function P = 14 – 2Q:

1. Obtain an expression for the profit function in terms of Q

2. Determine the values of Q for which the firm breaks even.

3. Sketch the graph of the profit function against Q

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Solution:

1. Profit = TR – TC = P.Q – (FC + VC)π = (14 - 2Q)Q – (2Q + 10)

π = -2Q2 + 12Q – 10

2. Breakeven: where Profit = 0Apply formula to solve quadratic where π = 0

so solve -2Q2 + 12Q – 10 = 0 with • Solution: at Q = 1 or Q = 5 the firm breaks

even

( )a

acbbQ

2

42 −±−=

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3. Graphing Profit Function

• STEP 1: coefficient on the squared term determines the shape of the curve

• STEP 2: constant term determines where the graph crosses the vertical axis

• STEP 3: Solution where π = 0 is where the graph crosses the horizontal axis

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-40

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-10

0

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-2 -1 0 1 2 3 4 5 6 7 8

Q

Prof

it

Profit

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Questions Covered on Topic 1: Elementary Functions

• Linear Functions and Tax……

• Finding linear Demand functions

• Plotting various types of functions

• Solving Quadratic Equations

• Solving Simultaneous Linear (more in next lecture)

• Solving quadratic functions