Linear functions A. Functions in general A. Functions in general - PowerPoint PPT Presentation

1 2 Linear functions A. Functions in general A. Functions in general 1. definition B. Linear functions C. Linear (in)equalities Handbook: E. Haeussler, R. Paul, R. Wood (2011). Introductory Mathematical Analysis for business, economics and life and social sciences. Pearson education 3 4 A. Functions in general A. Functions in general Example Taxidriver Introduction What does a taxi ride cost me with company A? In every day speech we often hear economists • Base price: 5 Euro say things like “ interest rates are a function of oil prices”, • Per kilometer: 2 Euro “pension income is a function of years worked” Price of a 7 km ride? Sometimes such usage agrees with mathematical usage, but not always. (Handbook: Section 2.1 p80, paragraph 1-2)

5 6 A. Functions in general A. Functions in general Example Taxidriver Definition x and y : VARIABLES • What does a taxi ride cost me with company A? (length of ride in km) (price of ride in euro) • Base price: 5 Euro • y depends on x : INPUT � OUTPUT • Per kilometer: 2 Euro x y y : DEPENDENT VARIABLE Price of an x km ride? x : INDEPENDENT VARIABLE Function: rule that assigns to each input at most 1 output (Section 2.1 p81, last 4 paragraphs) 7 8 A. Functions in general Definition • We say: y is FUNCTION of x, A. Functions in general or in short f of x • We denote: y ( x ) or y=f(x) 1. definition • Outputs are also called function values 2. Three representations (Handbook: Section 2.1 p82)

9 10 A. Functions in general A. Functions in general Three representations Three representations First way: Most concrete form! Second way: Most concentrated form! Through a TABLE, e.g. for y = 2 x + 5: Through the EQUATION, e.g. y = 2 x + 5. x y 0 5 1 7 formula y = 2 x + 5: 2 9 EQUATION OF THE FUNCTION … … But: limited number of values � no overall picture 11 12 A. Functions in general A. Functions in general Three representations Three representations x y y Third way: Third way: 0 5 Most visual form! Most visual form! 7 1 7 6 Through the GRAPH Through the GRAPH 5 e.g. for y = 2 x + 5: rectangular coordinate system: 4 x-coordinate, y-coordinate STRAIGHT LINE! 3 2 1 Note: In this example, the graph is a only a part of a straight line -4 -3 -2 -1 0 1 2 3 4 x -1 (Handbook: Section 2.5 p99)

13 14 A. Functions in general A. Functions in general Exercises Exercises The demand q of a product depends on the price p . Suppose a 180-pound man drinks four beers in quick succession. For a local pizza parlor some weekly demands and prices are given The graph shows the blood alcohol p q Remark: this table is called concentration (BAC) as a function 10 640 of the time. a demand schedule 12 560 (a) Input ? Output ? 14 480 (b) How much BAC is in the blood after 5 hours ? (c) What will be the maximal BAC ? (a) What is the input variable? What is the output variable ? After how much time, will this maximum be attained ? (b) Indicate the points in the table on a graph (d) What’s the behavior of the BAC as a function of time ? (Handbook: Section 2.1 p85 – example 5) (Section 2.1 p79) 15 16 A. Functions in general Summary - Definition B. Linear Functions input x , output y - 3 representations : 1. equation table equation y=f(x) graph in rectangular coordinate system Extra: Handbook - Problems 2.1: Ex 17, 48, 50

17 18 B. Linear functions B. Linear functions Example Taxidriver Example Taxidriver • Examples: cost of a ride with company B, C? y = 5 + 2x B � base price: 4.5 euro, price per km: 2.1 euro FIXED PART + VARIABLE PART C � base price 8 euro, price per km: 0.5 euro y = 4.50 + 2.10 x ; y = 8 + 0.5 x ; FIXED PART + MULTIPLE OF INDEPENDENT VARIABLE • In general: y = base price + price per km × x FIXED PART + PART PROPORTIONAL y = b + m x TO THE INDEPENDENT VARIABLE 19 20 B. Linear functions B. Linear functions Equation Applications A function f is a linear function if and only if • Cost y to purchase a car of 20 000 Euro and f(x) can be written in the form drive it for x km, if the costs amount to 0.8 Euro per km? f(x)=y=mx + b y = 20 000 + 0.8 x hence … y = mx + b ! where m, b are constants. • Production cost c to produce q units, if the fixed cost is 3 and the production cost is 0.2 Caution: m and b FIXED: parameters per unit? x and y : VARIABLES! c = 3 + 0.2 q hence y = mx + b ! (Section 3.1 p138)

21 22 B. Linear functions B. Linear functions Applications Exersises Rachel has saved $7250 for college expenses. • The demand q of a product depends on the She plans to spend $600 a month from this account. price p and vice versa. For a local pizza Write an equation to represent the situation. parlor the function is given by p=26-q/40 Note: The function p(q) is called the demand function by economists 23 24 B. Linear functions Exersises For a local pizza parlor the weekly demand function B. Linear Functions Is given by p =26- q /40. (a) What will be the revenue for the pizza parlor 1. Equation if 400 pizza’s are ordered ? 2. Graph (b) Express the revenue as a function of the demand q . !! Not all functions are first degree functions Note: Demand functions are not always linear !

25 26 B. Linear functions Example Taxidriver B. Linear Functions y = 2 x + 5 1. Equation 2. Graph The graph of a linear 3. Significance parameters b , m function with equation y=mx +b is - a STRAIGHT LINE 27 28 B. Linear functions B. Linear functions Example Taxidriver Significance of the parameter b • Taxi company A: y = 2 x + 5. A: y = 2 x + 5 Here b = 5: the base price. B: y = 4.5 x + 2.1 • Numerically: C: y = 0.5 x + 8 b can be considered as the VALUE OF y WHEN x = 0. What’s the effect of • graphically: the different values b shows where the graph cuts for m ? For b ? the Y -axis: Y -INTERCEPT

29 30 B. Linear functions B. Linear functions Significance of the parameter m Significance of the parameter m • Taxi company A: y = 2 x + 5, m = 2: the price per • Graphically: km. • Numerically: m is CHANGE OF y WHEN x IS if x is increased by 1 unit, INCREASED BY 1 y is increased by m units INPUT OUTPUT x y 3 11 ∆ x = 1 ∆ y = 2 4 13 m is the RATE OF CHANGE of m is the SLOPE of the straight line the linear function 31 32 B. Linear functions B. Linear functions Significance of the parameter m Significance of the parameter m • Taxi company A: y = 2 x + 5, m = 2: the price per if x is increased by ∆ x units, y is increased by m ∆ x km. units • 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). INPUT OUTPUT x y ∆ x = 3 ∆ y = 3x2=6 3 11 6 17 • Always: ∆ y = m ∆ x Increase formula: (INCREASE FORMULA)

33 34 B. Linear functions B. Linear functions Significance of the parameters b and m Exercises The graph of a linear function with equation 1. The cost c in terms of the quantity q y=mx +b is produced of a good is given by c = 200 + 15 q . - a STRAIGHT LINE • Give a formula for the change of cost ∆ c . - with y-intercept b • Use this formula to determine how the - and slope m cost changes when the production of the The equation y=mx +b is called the slope- good is increased by 12 units. intercept form of the line with slope m and • Use this formula to determine how the cost changes when the production of the intercept b. It is also called an explicit good is decreased by 2 units. equation of the line . 35 36 B. Linear functions B. Linear functions Supplementary exercises Exercises • Exercise 1 • Exercise 2 A, B, D (only the indicated points are to be used!) Exercise 2

37 38 B. Linear functions B. Linear functions Slope of the line m Slope of the line m (Section 3.1 p128-129) Sign of m determines whether the linear function is Consider again supplementary Exercise 2 - increasing / constant(!!) / decreasing - Compare the slopes of lines A and D - What is the slope of line C ? - Compare the slopes of line A and B - Compare the slopes of lines D and E - Note: what about a vertical line ? (Section 3.1 p131- Example 6) (Section 3.1) 39 40 B. Linear functions B. Linear functions Slope of the line m Parallel lines (Section 3.1 p128-129) Size of m determines how steep the line is Parallel lines have the same slope Perpendicular lines (Section 3.1 p133-134) Two lines with slopes m1 and m2 are perpendicular Note: the slope and thus the steepness of the line to each other if and only if depends on the scale of the axes Note: any horizontal line and any vertical line are perpendicular to each other (Section 3.1 p128-129)

41 42 B. Linear functions Slope of the line m B. Linear Functions Remember: ∆ y = m ∆ x (INCREASE FORMULA). 1. Equation 2. Graph 3. Significance parameters b , m Therefore: 4. Determining a line based on the slope and a point / two points (Section 3.1 p128) 43 44 B. Linear functions B. Linear functions Slope of the line m Exercises John purchased a new car in 2001 for $32000. Slope of a straight In 2004, he sold it to a friend for $26000. line given by two You may assume that the price is a linear function points: of time. Find and interpret the slope. (Section 3.1 p128)