Functions and Graphs

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Chapter 3

• Functions and Graphs

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Functions and Graphs

• Graphs of functions
• Applications of linear functions
• Applications of quadratic functions

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Graphs of functions

• Linear function f(x) ax b

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f(x) 0.5x 3

• x f(x)
• 0 -3
• 2 -2
• 6 0
• 8 1

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Example 2

• f(x) x 1 for x
• -2x 7 for x 2

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f(x) x 1 for x 2

• x f(x)
• -2 -1
• 0 1
• 2 3
• 3 1
• 4 -1
• 5 -3

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Example 4
f(x) x for x0 and f(x) -x for x 8
Example

• g(x) sqrt(x 1)
• x g(x)
• -1 0
• 0 1
• 1 1.41
• 2 1.73
• 3 2

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Example 7
g(x) sqrt(x 1)
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Example 8

• f(x) 2 x3/5
• x f(x)
• -4 14.8
• -3 7.4
• -2 3.6
• -1 2.2
• 0 2
• 1 1.8
• 2 0.4
• 3 -3.4
• 4 -10.8

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g(x) 2x -1 for x0
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Problem 49
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Problem 50
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Applications of Linear Functions

• Revenue price per item number of items
• Cost fixed costs variable costs
• Profit revenue - cost

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Fixed Costs Examples

• Buildings
• Machinery
• Real estate taxes
• Product design

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Variable Costs Examples

• Labor
• Materials
• Shipping
• Variable costs depend on the number of items made.

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Graph the following
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Graph the following
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Graph the following equation

• Y 200X
• for X 0, 10, 20, 30, 40, 50

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Graph the following equation

• Y 100X 1000
• for X 0, 10, 20, 30, 40, 50

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Example 1

• An anti-clot drug can be made for 10 per unit.
  The total cost to produce 100 units is 1500.

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• a. Assume that the cost function is linear and
  find its rule.
• C mx b, where C is cost, m is the slope of
  the linear equation, x is the variable and b is a
  constant.
• C 1500, m 10, x 100
• b C mx 1500 10(100) 500
• C 10X 500

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• b. What are the fixed costs?
• C mx b, where mx is the variable portion
  and b the fixed
• Therefore, the fixed costs are 500.

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• c. What is the average cost to produce a unit?
• Caverage (mx b)/x m b/x
• Caverage 10 500/100 15

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Rates of Change

• Rate of change of a linear function is the slope
  m (y mx b)
• In economics, the rate of change in the cost
  function is called the marginal cost.
• When the cost function is linear, the marginal
  cost is equal to the slope m.
• This is the cost to produce one more item.

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Example 3

• According to the Kelley Blue Book, a Ford Focus
  ZX5 Hatchback that is worth 14,632 today, will
  be worth 10,120 in three years (if it is in good
  condition with average mileage).

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a. Assuming linear depreciation, find the
depreciation function for this car.

• m (10120 14632)/(3 0) -1504
• b 14632 (-15040) 14632
• y -1504x 14632

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b. What will the car be worth in 5 years?

• y -1504x 14632
• y -1504(5) 14632 7112

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c. At what rate is the car depreciating?

• The rate of depreciation is determined by the
  slope namely -1504 per year.

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Example 4

• An electronics company manufactures handheld PCs.
  The cost function for one of their models is
• C 160x 750000

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• a. What are the fixed costs for this product?
• C 160x 750000
• 750,000

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• b. What is the marginal cost?
• C 160x 750000
• 160

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• c. After 50,000 units have been produced, what is
  the cost of producing one more?
• C 160x 750000
• 160
• C50000 160(50000) 750000 8,750,000
• C50001 160(50001) 750000 8,750,160
• C50001 - C50000 160

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Break-Even Analysis

• A company manufactures a particular model of DVD
  player that sells to retailers for 168. The cost
  of making x of these DVD players is given by
  the function
• C 118x 800000

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• a. Find the function R that gives the revenue
  from selling x players.
• R 168x

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• b. What is the revenue from selling 40,000
  players?
• R 168x 168(40000) 6,720,000

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• c. Find the profit function, P.
• P R C 168x (118x 800000)
• 50x - 800000

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• d. What is the profit from selling 10,000
  players?
• P 50x 800000 50(10000) 800000
  -300000

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Example 7

• A company manufactures a DVD player that it sells
  to retailers for 168. The cost of making the DVD
  players is given by the function C(x) 118x
  800,000. Find the breakeven point for the DVD
  player.

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• R(x) C(x)
• 168x 118x 800,000
• 168x 118x 800,000
• 50x 800,000
• X 16,000

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Supply and Demand

• Supply and demand for an item are generally
  related to its price.
• Supply and demand can be graphed.
• Conventions
• Price p (y axis)
• Quantity q (x axis)

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Typical Demand Curve Why?
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Typical Supply Curve Why?
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Typical Supply/Demand Curves
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Example 8

• Bill Cornett, an economist, has studied the
  supply and demand for aluminum siding and has
  determined that price per unit, p, and the
  quantity demanded, q, are related by the linear
  equation
• p 60 (3/4)q

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• a. Find the demand at a price of 40 per unit.
• p 60 (3/4)q or q (60 p)(4/3)
• q (60 p)(4/3) and for p 40,
• q (60 40 )(4/3) 80/3 26 2/3 units

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• b. Find the price if the demand is 32 units.
• p 60 (3/4)q so with q 32,
• p 60 (3/4)32 36

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• c. Graph p 60 (3/4)q

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• d. What quantity is demanded at a price of 30?
• 40 units
• e. At what price will 60 units be demanded?
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• f. What quantity is demanded at a price of 60
  per unit?
• 0 units

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Example 9

• Suppose the economist of the previous example
  concludes that the supply q of siding is
  related to its price p by the equation
• p .85q

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• a. Find the supply if the price is 51 per unit.
• p .85q so that q p/.85
• q 51/.85 60

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• b. Find the price per unit if the supply is 20
  units.
• p .85q
• p .85(20) 17

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• c. Graph the supply equation
• p .85q

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• d. Use the graph to find the approximate price at
  which 35 units will be supplied.
• 29.75

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Example 10

• The supply and demand curves of examples 8 and 9
  are graphed as follows

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• Graphically determine whether there is a surplus
  or a shortage of supply at a price of 40 per
  unit.
• Surplus supply is greater than demand.

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Equilibrium Point

• Where supply and demand curves cross.
• Equilibrium price
• Equilibrium quantity

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Example 11

• In the situation described in the previous 3
  examples, what is the equilibrium price and the
  equilibrium quantity?

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• Equilibrium when
• demand supply
• 60 (3/4)q .85q
• 60 1.6q or q 37.5
• At q 37.5, equilibrium price .85q or
• .85(37.5) 31.875

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Group Work
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Applications of Quadratic Functions
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Example 1

• Anne Kelly owns and operates Aunt Emmas
  Blueberry Pies. She hired a consultant to analyze
  her business operations. The consultant tells her
  that her profits, P, from the sale of x cases
  of pies, are given by
• P 120x x2

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• How many cases of pies should she sell in order
  to maximize profit? What is the maximum profit?
• 60 cases for a profit of 3600 (see following
  graph and chart)

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Example 2

• Suppose that the price and demand for an item are
  related by
• p 150 6q2
• where p is the price and q is the number of
  items demanded (in hundreds).