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- Topic 1 Topic 1 : Elementary functions Reading: JacquesSection 1.1 - Graphs of linear equations Section 2.1 – Quadratic functionsSection 2.2 – Revenue, Cost and Profit 1
- 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 2
- 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) 3
- Example: Student grades• Example: • Consider the function:• y = a + bx • y = 5+ 0x• y : is the final grade, • What does this tell• x : is number of hours us? studied, • Assume different• a%: guaranteed values of x ……… 4
- Example Continued: What grade if you study 0 hours? 5 hours?• y=5+0x Linear FunctionsOutput = constant slope Input y a b X 60 50 5 5 0 0 Dependent Y Variable 40 5 5 0 1 30 5 5 0 2 20 5 5 0 3 10 5 5 0 4 0 0 1 2 3 4 5 5 0 5 Independent X Variable 5
- Example Continued…. output= constant slope input y a b X 5 5 15 0 • y=5+15x 20 5 15 1 35 5 15 2 50 5 15 3 65 5 15 4 Linear Functions 65 55 If x = 4, what gradeDependent Y Variable 45 35 will you get? 25 Y = 5 + (4 * 15) = 65 15 5 -5 0 1 2 3 4 Independent X Variable 6
- 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 12 10 If p =5, how much will be 8 demanded?Q Demand 6 D = 10 - (2 * 5) = 0 4 2 0 0 1 2 3 4 5 7 Price
- 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? 8
- 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 = 80 x = (80-5 / 15) x = 5 hours per week 9
- 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 – DThen, simplifySo P= 5 – 0.5D is the inverse function 10
- More Variables:• Student grades again: • Example: y = a + bx + cz • If y = 5+ 15x + 3z, and a• y : is the final grade, student studies 4 hours per week and completes 5• x : is number of hours questions per week, what studied, is the final grade?• z: number of • Answer: questions completed • y = 5 + 15x + 3z• a%: guaranteed • y = 5 + (15*4) + (3*5) • y = 5+60+15 = 80
- Another example: Guinness Demand.• The demand for a pint of • 6 = a + 2b Guinness in the Student => a = 6-2b bar on a Friday evening • 4 = a + 3b is a linear function of => a = 4-3b price. When the price • 6-2b = 4-3b per pint is €2, the • Solving we find that b = -2 demand ‘is €6 pints. When the price is €3, • If b = -2, then a = 6-(-4) = 10 the demand is only 4 • The function is D = 10 – 2P pints. Find the function • What does this tell us?? D = a + bP • Note, the inverse Function is • P = 5- 0.5D 12
- A Tax Example…. • Answer:• let €4000 be set as the • THP = E – 0.4 (E – 4000) target income. All income if E>4000 above the target is taxed at 40%. For every €1 • THP = E + 0.4 (4000-E) below the target, the if E<4000 worker gets a negative • In both cases, income tax (subsidy) of THP = 1600 +0.6E So 40%. i) If E = 4000 =>• Write out the linear function between take- THP = 1600+2400=4000 home pay and earnings. ii) If E = 5000 => THP = 1600+3000=4600 iii) If E = 3000 => THP = 1600+1800=3400 13
- 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 14
- Topic 1 continued:Non- linear EquationsJacques Text Book:Sections 2.1 and 2.2 15
- 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 16
- 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 17
- e.g. y = -x2+4x+5 2y a x b X C-7 -1 4 4 -2 55 -1 0 4 0 59 -1 4 4 2 55 -1 16 4 4 5-7 -1 36 4 6 5Since a<0 => ‘Hill Shaped Graph’ 18
- The graph Quadratic Functions 10 8 y=0, then x= +5 6 OR x = -1 4 Y = X2 2 0 -2 -2 0 2 4 6 -4 -6 -8 Independent X Variable 19
- Special Case: a=1, b=0 and c=0 So y = ax2+bx+c => y = x2 2 Quadratic Functionsy= a x b x c 4016 1 16 0 -4 0 354 1 4 0 -2 0 30 250 1 0 0 0 0 20 2 4 1 4 0 2 0 Y=X 1516 1 16 0 4 0 1036 1 36 0 6 0 5 0 -4 -2 -5 0 2 4 6 -10 Min. Point: (0,0) Intercept = 0 Independent X Variable 20
- 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 21
- Solving Quadratic Equations:• 2) Algebraic Approach: find the value(s), if any, of x when y=0 by applying a simple formula… x= −b ± (b 2 − 4ac ) 2a 22
- Example• e.g. y = -x2+4x+5• hence, a = -1; b=4; c=5 −4 ± (16 − 4(−1×5) ) x= −2 −4 ± (16 + 20) −4 ±6 x = = −2 −2• Hence, x = +5 or x = -1 when y=0• Function cuts x-axis at +5 and –1 23
- Example 2• y = x2-4x+4 18• hence, a = 1; b= - 4; 16 y c=4 14• If y = 0 12 10 Y 8 +4± (16 − 4(1× 4) ) 6 x= 4 2 2 4± 0 = 0 x -2 -1 0 1 2 3 4 5 2 X Function only cuts x-axis at one point, x = 2 when y = 0 where x=2 24
- Example 3• y = 3x2-5x+6 120 y• hence, a = 3; 100 b= - 5; c=6 80• If y = 0 Y 60 + 5 ± ( 25 − 4(3 × 6) ) x= 40 6 20 4 ± − 47 = 0 6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 Xwhen y = 0 there is no solution The quadratic function does not intersect the x-axis 25
- 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 26
- 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 27
- 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 – 16Q 3Q2 – 32Q+16Q +96 – 236 = 0 3Q2 – 16Q –140 = 0 − b ± ( b 2 − 4ac ) Q=• Solve the quadratic using the formula 2a where a = 3; b = -16 and c = -140• Solution: Q = +10 or Q = -4.67• Profit maximising output is +10 (negative Q inadmissable) 28
- 350 Graphically 300 MC MR 250 200MR and MC 150 100 50 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 Q 29
- 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 Q2. Determine the values of Q for which the firm breaks even.3. Sketch the graph of the profit function against Q 30
- Solution:1. Profit = TR – TC = P.Q – (FC + VC) π = (14 - 2Q)Q – (2Q + 10) π = -2Q2 + 12Q – 102. Breakeven: where Profit = 0 Apply formula to solve quadratic where π = 0 so solve -2Q2 + 12Q – 10 = 0 with Q = −b± (b − 4ac ) 2 2a• Solution: at Q = 1 or Q = 5 the firm breaks even 31
- 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 32
- 20 Profit 10 0 -2 -1 0 1 2 3 4 5 6 7 8 -10Profit -20 -30 -40 -50 Q 33
- 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 34

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