1. Chapter 1 Functions and Linear Models Sections 1.3 and 1.4

2. Linear Function A linear function can be expressed in the form where m and b are fixed numbers. Equation notation Function notation

3. Graph of a Linear Function The graph of a linear function is a straight line. m is called the slope of the line and b is the y-intercept of the line. This means that we need only two points to completely determine its graph.

4. y-axis x-axis (1,2) Example: Sketch the graph of f (x) = 3x – 1 y-intercept Arbitrary point (0,-1)

5. Role of m and b in f (x) = mx + b The Role of m (slope) f changes m units for each one-unit change in x. The Role of b (y-intercept) When x = 0, f (0) = b

6. To see how f changes, consider a unit change in x. Then, the change in f is given by Role of m and b in f (x) = mx + b

10. The graph of a Linear Function: Slope and y-Intercept y-axis x-axis (1,2) Example: Sketch the graph of f (x) = 3x – 1 y-intercept Slope = 3/1

11. Graphing a Line Using Intercepts y-axis x-axis Example: Sketch 3x + 2y = 6 y-intercept (x = 0) x-intercept (y = 0)

12. Delta Notation If a quantity q changes from q 1 to q 2, the change in q is denoted by  q and it is computed as Example: If x is changed from 2 to 5, we write

13. Delta Notation Example: the slope of a non-vertical line that passes through the points (x 1, y 1 ) and (x 2, y 2 ) is given by: Example: Find the slope of the line that passes through the points (4,0) and (6, -3)

14. Delta Notation

15. Zero Slope and Undefined Slope Example: Find the slope of the line that passes through the points (4,5) and (2, 5). Example: Find the slope of the line that passes through the points (4,1) and (4, 3). Undefined This is a vertical line This is a horizontal line

16. Examples Estimate the slope of all line segments in the figure

17. Point-Slope Form of the Line An equation of a line that passes through the point (x 1, y 1 ) with slope m is given by: Example: Find an equation of the line that passes through (3,1) and has slope m = 4

18. Horizontal Lines y = 2 Can be expressed in the form y = b

19. Vertical Lines x = 3 Can be expressed in the form x = a

20. Linear Models: Applications of linear Functions

21. First, General Definitions

22. Cost Function  A cost function specifies the cost C as a function of the number of items x produced. Thus, C(x) is the cost of x items.  The cost functions is made up of two parts: C(x)= “variable costs” + “fixed costs”

23.  If the graph of a cost function is a straight line, then we have a Linear Cost Function.  If the graph is not a straight line, then we have a Nonlinear Cost Function. Cost Function

24. Linear Cost Function Dollars Units Cost Dollars Units Cost

25. Dollars Units Cost Dollars Units Cost Non-Linear Cost Function

26. Revenue Function  The revenue function specifies the total payment received R from selling x items. Thus, R(x) is the revenue from selling x items.  A revenue function may be Linear or Nonlinear depending on the expression that defines it.

27. Linear Revenue Function Dollars Units Revenue

28. Nonlinear Revenue Functions Dollars Units Revenue Dollars Units Revenue

29. Profit Function  The profit function specifies the net proceeds P. P represents what remains of the revenue when costs are subtracted. Thus, P(x) is the profit from selling x items.  A profit function may be linear or nonlinear depending on the expression that defines it. Profit = Revenue – Cost

30. Linear Profit Function Dollars Units Profit

31. Nonlinear Profit Functions Dollars Units Profit Dollars Units Profit

32. The Linear Models are Cost Function: ** m is the marginal cost (cost per item), b is fixed cost. Revenue Function: ** m is the marginal revenue. Profit Function: where x = number of items (produced and sold)

33. Break-Even Analysis The break-even point is the level of production that results in no profit and no loss. To find the break-even point we set the profit function equal to zero and solve for x.

34. Break-Even Analysis Profit = 0 means Revenue = Cost Dollars Units loss Revenue Cost profit Break-even point Break-even Revenue The break-even point is the level of production that results in no profit and no loss.

35. Example: A shirt producer has a fixed monthly cost of $3600. If each shirt has a cost of $3 and sells for $12 find: a.The cost function b.The revenue function c.The profit from 900 shirts C (x) = 3x + 3600 where x is the number of shirts produced. R (x) = 12x where x is the number of shirts sold. P (x) = R(x) – C(x) P (x) = 12x – (3x + 3600) = 9x – 3600 P(900) = 9(900) – 3600 = $4500

36. C (x) = R (x) Example: A shirt producer has a fixed monthly cost of $3600. If each shirt has a cost of $3 and sells for $12 find the break-even point. The break even point is the solution of the equation Therefore, at 400 units the break-even revenue is $4800

40. Demand Function  A demand function or demand equation expresses the number q of items demanded as a function of the unit price p (the price per item).  Thus, q(p) is the number of items demanded when the price of each item is p.  As in the previous cases we have linear and nonlinear demand functions.

41. Linear Demand Function q = items demanded Price p

42. Nonlinear Demand Functions q = items demanded Price p

43. Supply Function  A supply function or supply equation expresses the number q of items, a supplier is willing to make available, as a function of the unit price p (the price per item).  Thus, q(p) is the number of items supplied when the price of each item is p.  As in the previous cases we have linear and nonlinear supply functions.

44. Linear Supply Function q = items supplied Price p

45. Nonlinear Supply Functions q = items supplied Price p

46. Market Equilibrium Market Equilibrium occurs when the quantity produced is equal to the quantity demanded. q p supply curve demand curve Equilibrium Point shortage surplus

47. q p shortage supply curve demand curve surplus Equilibrium price Equilibrium demand Market Equilibrium occurs when the quantity produced is equal to the quantity demanded. Market Equilibrium

48.  To find the Equilibrium price set the demand equation equal to the supply equation and solve for the price p.  To find the Equilibrium demand evaluate the demand (or supply) function at the equilibrium price found in the previous step. Market Equilibrium

49. Example of Linear Demand The quantity demanded of a particular computer game is 5000 games when the unit price is $6. At $10 per unit the quantity demanded drops to 3400 games. Find a linear demand equation relating the price p, and the quantity demanded, q (in units of 100).

50. Example: The maker of a plastic container has determined that the demand for its product is 400 units if the unit price is $3 and 900 units if the unit price is $2.50. The manufacturer will not supply any containers for less than $1 but for each $0.30 increase in unit price above the $1, the manufacturer will market an additional 200 units. Assume that the supply and demand functions are linear. Let p be the price in dollars, q be in units of 100 and find: a. The demand function b. The supply function c. The equilibrium price and equilibrium demand

51. a. The demand function b. The supply function

52. c. The equilibrium price and equilibrium demand The equilibrium demand is 960 units at a price of $2.44 per unit.

53. Linear Change over Time A quantity q, as a linear function of time t: If q represents the position of a moving object, then the rate of change is velocity. Rate of change of q Quantity at time t = 0

54. Linear Regression We have seen how to find a linear model given two data points. We find the equation of the line passing through them. However, we usually have more than two data points, and they will rarely all lie on a single straight line, but may often come close to doing so. The problem is to find the line coming closest to passing through all of the points.

55. Linear Regression We use the method of least squares to determine a straight line that best fits a set of data points when the points are scattered about a straight line. least squares line

56. The Method of Least Squares Given the following n data points: The least-squares (regression) line for the data is given by y = mx + b, where m and b satisfy: and

57. Example: Find the equation of least-squares for the data (1, 2), (2, 3), (3, 7). The scatter plot of the points is

58. Solution: We complete the following table xyxyx2x2 1221 2364 37219 Sum: 6 12 29 14

59. Example: Find the equation of least-squares for the data (1, 2), (2, 3), (3, 7). The scatter plot of the points and the least squares line is

60. Coefficient of Correlation A measurement of the closeness of fit of the least squares line. Denoted r, it is between –1 and 1, the better the fit, the closer it is to 1 or –1.

61. Example: Find the correlation coefficient for the least-squares line from the last example. Points: (1, 2), (2, 3), (3, 7) = 0.9449