Sections 4.1 and 4.2 Linear Functions and Their Properties Linear Models

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

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.

y-axis x-axis (1,2) Example: Sketch the graph of f (x) = 3x – 1 y-intercept Arbitrary point (0,-1) Graph of a Linear Function

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 Role of m and b in f (x) = mx + b

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

y-axis x-axis (1,2) Example: Sketch the graph of f (x) = 3x – 1 y-intercept Slope = 3/1 Using the Slope and y-Intercept

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

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

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)

Delta Notation

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 Zero Slope and Undefined Slope

Estimate the slope of all line segments in the figure Examples

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

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

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

Linear Models: Applications of linear Functions

First, General Definitions

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

 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

Linear Cost Function Dollars Units Cost Dollars Units Cost

Dollars Units Cost Dollars Units Cost Non-Linear Cost Function

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.

Linear Revenue Function Dollars Units Revenue

Nonlinear Revenue Functions Dollars Units Revenue Dollars Units Revenue

Profit Function  The profit function specifies the net proceeds P. Thus 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

Linear Profit Function Dollars Units Profit

Nonlinear Profit Functions Dollars Units Profit Dollars Units Profit

The Linear Models are Cost Function: * m 1 is the marginal cost (cost per item), b is fixed cost. Revenue Function: * m 2 is the marginal revenue. Profit Function: where x = number of items (produced and sold)

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.

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.

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 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 ) = 9x – 3600 P(900) = 9(900) – 3600 = $4500

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

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.

Linear Demand Function q = items demanded Price p

Nonlinear Demand Functions q = items demanded Price p

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.

Linear Supply Function q = items supplied Price p

Nonlinear Supply Functions q = items supplied Price p

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

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

 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

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).

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

a. The demand function b. The supply function

Graphical Interpretation

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

More Linear Models from Verbal Descriptions

Straight-line Depreciation

(d) The slope is the average rate of change and is  4000 so this means that for each additional year that passes, the book value of the car decreases by $4000. The car will have a book value of $8000 when it is 5 years old. Straight-line Depreciation