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## Linear functions

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**Linear functions**Functions in general Linear functions C. Linear (in)equalities Handbook: Haeussler E., Paul R., Wood R.(2011). Introductory Mathematical Analysis for business, economics and life and social sciences. Pearson education**A. Functions in general**Introduction In every day speech we often hear economists say things like “ interest rates are a function of oil prices”, “pension income is a function of years worked” Sometimes such usage agrees with mathematical usage, but not always. (Handbook: Section 2.1 p80, paragraph 1-2)**A. Functions in general**Example Taxidriver What does a taxi ride cost me with company A? • Base price: 5 Euro • Per kilometer: 2 Euro Price of a 7 km ride?**A. Functions in general**Example Taxidriver What does a taxi ride cost me with company A? • Base price: 5 Euro • Per kilometer: 2 Euro Price of an x km ride?**A. Functions in general**Definition • x and y : VARIABLES (length of ride in km) (price of ride in euro) • y depends on x: INPUT OUTPUT x y y: DEPENDENT VARIABLE x:INDEPENDENT VARIABLE Function: rule that assigns to each input at most 1 output (Section 2.1 p81, last 4 paragraphs)**A. Functions in general**Definition • We say: y is FUNCTION of x, or in short f of x • We denote: y(x) or y=f(x) • Outputs are also calledfunction values (Handbook: Section 2.1 p82)**A. Functions in general**Three representations First way: Most concrete form! Through a TABLE, e.g. for y = 2x + 5: But: limited number of values no overall picture**A. Functions in general**Three representations Second way: Most concentrated form! Through the EQUATION, e.g. y = 2x + 5. formula y = 2x + 5: EQUATION OF THE FUNCTION**A. Functions in general**Three representations y 7 6 5 4 3 2 1 -1 x y 0 5 1 7 -4 -3 -2 -1 0 1 2 3 4 x Third way: Most visual form! Through the GRAPH rectangular coordinate system: x-coordinate, y-coordinate (Handbook: Section 2.5 p99)**A. Functions in general**Three representations Third way: Most visual form! Through the GRAPH e.g. for y = 2x + 5: STRAIGHT LINE! Note: In this example, the graph is a only a part of a straight line**A. Functions in general**Exercises p q 10 640 12 560 14 480 The demand qof a product depends on the price p. For a local pizza parlor some weekly demands and prices are given Remark: this table is called a demand schedule (a) What is the input variable? What is the output variable ? (b) Indicate the points in the table on a graph (Handbook: Section 2.1 p85 – example 5)**A. Functions in general**Exercises Suppose a 180-pound man drinks four beers in quick succession. The graph shows the blood alcohol concentration (BAC) as a function of the time. (a) Input ? Output ? (b) How much BAC is in the blood after 5 hours ? (c) What will be the maximal BAC ? After how much time, will this maximum be attained ? (d) What’s the behavior of the BAC as a function of time ? (Section 2.1 p79)**A. Functions in general**Summary - Definition input x, output y - 3 representations : table equation y=f(x) graph in rectangular coordinate system Extra: Handbook - Problems 2.1: Ex 17, 48, 50**B. Linear functions**Example Taxidriver y = 5 + 2x FIXED PART + VARIABLE PART FIXED PART + MULTIPLE OF INDEPENDENT VARIABLE FIXED PART + PART PROPORTIONAL TO THE INDEPENDENT VARIABLE**B. Linear functions**Example Taxidriver • Examples: cost of a ride with company B, C? B base price: 4.5 euro, price per km: 2.1 euro C base price 8 euro, price per km: 0.5 euro y = 4.50 + 2.10x; y = 8 + 0.5x; • In general: y = base price + price per km x y = b + mx**B. Linear functions**Equation A function f is a linear function if and only if f(x) can be written in the form f(x)=y=mx + b where m, b are constants. Caution: m and bFIXED: parameters x and y: VARIABLES! (Section 3.1 p138)**B. Linear functions**Applications • 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 + b! • 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 + b!**B. Linear functions**Applications • The demand qof a product depends on the price p and vice versa. For a local pizza parlor the function is given by • p=26-q/40 • Note: The function p(q) is called the • demand function by economists**B. Linear functions**Exersises Rachel has saved $7250 for college expenses. She plans to spend $600 a month from this account. Write an equation to represent the situation.**B. Linear functions**Exersises For a local pizza parlor the weekly demand function Is given by p=26-q/40. (a) What will be the revenue for the pizza parlor if 400 pizza’s are ordered ? (b) Express the revenue as a function of the demand q. Note: Demand functions are not always linear ! !! Not all functions are first degree functions**B. Linear functions**Example Taxidriver y = 2x + 5 • The graph of a linear function with equation y=mx +b is • a STRAIGHT LINE**B. Linear Functions1. Equation2. Graph3. Significance**parameters b, m**B. Linear functions**Example Taxidriver A: y = 2x + 5 B: y = 4.5x + 2.1 C: y = 8x + 0.5 What’s the effect of the different values for m ? For b ?**B. Linear functions**Significance of the parameter b • Taxi company A: y = 2x + 5. Here b = 5: the base price. • Numerically: b can be considered as the VALUE OF y WHEN x = 0. • graphically: b shows where the graph cuts the Y-axis: Y-INTERCEPT**B. Linear functions**Significance of the parameter m • Taxi company A: y = 2x + 5, m = 2: the price per km. • Numerically: m is CHANGE OF y WHEN x IS INCREASED BY 1 INPUT OUTPUT x y 3 11 4 13 x= 1 y= 2 m is the RATE OF CHANGE of the linear function**B. Linear functions**Significance of the parameter m • Graphically: if x is increased by 1 unit, y is increased by m units m is the SLOPE of the straight line**B. Linear functions**Significance of the parameter m • Taxi company A: y = 2x + 5, m = 2: the price per km. • 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 3 11 6 17 • Always: x= 3 y= 3x2=6 y= mx (INCREASE FORMULA)**B. Linear functions**Significance of the parameter m if x is increased by x units, y is increased by m x units Increase formula:**B. Linear functions**Significance of the parameters b and m • The graph of a linear function with equation y=mx +b is • a STRAIGHT LINE • with y-intercept b • and slope m The equation y=mx +b is called the slope-intercept form of the line with slope m and intercept b. It is also called an explicit equation of the line.**B. Linear functions**Exercises • The cost c in terms of the quantity q produced of a good is given by c = 200 + 15 q. • Give a formula for the change of cost Δc. • Use this formula to determine how the cost changes when the production of the good is increased by 12 units. • Use this formula to determine how the cost changes when the production of the good is decreased by 2 units.**B. Linear functions**Supplementary exercises • Exercise 1 • Exercise 2 A, B, D (only the indicated points are to be used!)**B. Linear functions**Exercises Exercise 2**B. Linear functions**Slope of the line m Consider again supplementary Exercise 2 - 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 (Section 3.1)**B. Linear functions**Slope of the line m (Section 3.1 p128-129) Sign of m determines whether the linear function is - increasing / constant(!!) / decreasing • Note: what about a vertical line ? (Section 3.1 p131- Example 6)**B. Linear functions**Slope of the line m Size of m determines how steep the line is Note: the slope and thus the steepness of the line depends on the scale of the axes (Section 3.1 p128-129)**B. Linear functions**Parallel lines Perpendicular lines (Section 3.1 p128-129) Parallel lines have the same slope (Section 3.1 p133-134) Two lines with slopes m1 and m2 are perpendicular to each other if and only if Note: any horizontal line and any vertical line are perpendicular to each other**B. Linear functions**Slope of the line m Remember: y= mx (INCREASE FORMULA). Therefore: (Section 3.1 p128)**B. Linear Functions1. Equation2. Graph3. Significance**parameters b, m4. Determining a line based on the slope and a point / two points**B. Linear functions**Slope of the line m Slope of a straight line given by two points: (Section 3.1 p128)**B. Linear functions**Exercises John purchased a new car in 2001 for $32000. In 2004, he sold it to a friend for $26000. Find and interpret the slope.**B. Linear functions**Equation of lines A straight line through a given point (x0, y0)and with a given slope m satisfies the equation: This equation is called the point- slope form of the line Remember: The equation y=mx+b is called the slope-intercept form of the line (Section 3.1 p129-131)**B. Linear functions**Exercises John purchased a new car in 2001 for $32000. In 2004, he sold it to a friend for $26000. Find the equation that expresses the price as a function of time. You may assume that the price is a linear function of time Supplementary exercises • Exercise 3 • Exercise 4**B. Linear Functions1. Equation2. Graph3. Significance**parameters b, m4. Determining a line based on the slope and a point / two points5. Implicit equation**B. Linear functions**Equation of lines Note that e.g. the vertical line with equation x=2 can not be written in the slope-intercept form nor in the slope-point form The equation of a straight line can always be written using the general linear form Ax+By+C=0 (A and B not both 0). This is also called an implicit equation. (Section 3.1 p129-131)**B. Linear functions**Equation of lines Remember : Point-slope form Slope intercept form y=m x + b General linear from Ax + By + C = 0 note: vertical line: x=a horizontal line: y=b (Section 3.1 p129-131)**B. Linear functions**Exercises Find an equation of the line that has slope 2 and passes through (1, -3) using the - Point-slope form - Slope-intercept form - General linear form Supplementary exercises: • Exercise 5