1 Functions and Linear Models Chapter 1 Functions: Numerical, Algebraic and Graphical Linear Functions Linear Models Linear Regression Lecture 1

2 Functions A real-valued function f assigns to each real number x in a specified set of numbers (the domain of f ), a unique real number f (x), read “f of x.” * The natural domain of f is the largest set of numbers for which f makes sense. NOTE: It is not f times x

3 Numerically Specified Function Ex. t01234 V(t)V(t)V(t)V(t) The data represents the velocity of an object V, in feet/sec, after t seconds have elapsed. * Note: at 2 seconds the object is going 4.9 ft/sec. Given numerical values for the function evaluated at certain values of the independent variable.

4 Algebraically Defined Function is a function.Ex. A function represented by a formula.

5 Mathematical Modeling Representing a situation in mathematical terms. Ex. The monthly payment, M, necessary to repay a home loan of P dollars, at a rate of r % per year (compounded monthly), for t years, can be found using

6 Common Types of Algebraic Functions Linear Quadratic Polynomial (a, b, m, and each a i constant) (a not 0) (a n not 0)

7 Common Types of Algebraic Functions Exponential (A, b constant, b >0) Rational (P, Q polynomials)

8 Piecewise Function Several formulas to define a single function. Ex. = 26.5 = 53.8 Use when x values are less than or equal to 2 Use when x values are greater than 2 Notice

9 Graphically Specified Function The graph of a function is the set of all points (x, f (x)) such that x is in the domain of f. Given the graph of y = f (x), find f (1). f (1) = 2 (1, 2)

10 Graph of a Function Vertical Line Test: The graph of a function can be crossed at most once by any vertical line. FunctionNot a Function It is crossed more than once.

11 Sketching a Piecewise Function Sketch the portion of the formula on its domain

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

13 Role of m and b in the Linear Function 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

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

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

16 Slope – the slope of a non-vertical line that passes through the points is given by: and Ex. Find the slope of the line that passes through the points (4,0) and (6, -3)

17 Zero Slope; Undefined Slope Ex. Find the slope of the line that passes through the points (4,5) and (2, 5). Ex. 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

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

19 **Two lines are parallel if and only if their slopes are equal or both undefined Ex. Find an equation of the line that passes through (3,5) and is parallel to the line So m = 2/3 and we have the point (3, 5):

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

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

22 Linear Models Cost Function: x = number of items ** m is the marginal cost (cost per item), b is fixed cost. Revenue Function: ** m is the marginal revenue. Profit Function:

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

24 Cost, Revenue, and Profit Functions Ex. 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 Cost: C(x) = 3x where x is the number of shirts produced. Revenue: R(x) = 12x where x is the number of shirts sold. Profit: P(x) = Revenue – Cost = 12x – (3x ) = 9x – 3600 P(900) = 9(900) – 3600 = $4500

25 Cost: C(x) = 3x Ex. 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. If x is the number of shirts produced and sold Revenue: R(x) = 12x At 400 units the break-even revenue is $4800

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

27 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

28 Ex. 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. Both 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 quantity

29 a. The demand function b. The supply function

30 c. The equilibrium price and quantity Solveand simultaneously. The equilibrium quantity is 960 units at a price of $2.44 per unit.

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

32 Linear Regression The method of least squares is 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

33 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

34 Ex. Find the equation of least-squares for the data = 2.5xyxy x2x2x2x Sum: = –1

35 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.

36 Ex. Find the correlation coefficient for the least-squares line from the last example. Points: =