Chapter 1 Functions and Linear Models Sections 1.3 and 1.4

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Chapter 1Functions andLinear ModelsSections
1.3 and 1.4
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Linear Function
A linear function can be expressed in the form
Function notation
Equation notation
where m and b are fixed numbers.
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Graph of a Linear Function
The graph of a linear function is a straight line.
This means that we need only two points to
completely determine its graph.
m is called the slope of the line and b is the
y-intercept of the line.
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Example Sketch the graph of f (x) 3x 1
y-axis
Arbitrary point
(1,2)
x-axis
(0,-1)
y-intercept
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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
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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
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Role of m and b in f (x) mx b
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Role of m and b in f (x) mx b
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Role of m and b in f (x) mx b
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The graph of a Linear Function Slope and
y-Intercept
Example Sketch the graph of f (x) 3x 1
y-axis
(1,2)
x-axis
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Graphing a Line Using Intercepts
Example Sketch 3x 2y 6
y-axis
x-intercept (y 0)
x-axis
y-intercept (x 0)
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Delta Notation
If a quantity q changes from q1 to q2 , the
change in q is denoted by ?q and it is computed
as
Example If x is changed from 2 to 5, we write
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Delta Notation
Example the slope of a non-vertical line that
passes through the points (x1 , y1) and (x2 , y2)
is given by
Example Find the slope of the line that passes
through the points (4,0) and (6, -3)
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Delta Notation
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Zero Slope and Undefined Slope
Example Find the slope of the line that passes
through the points (4,5) and (2, 5).
This is a horizontal line
Example Find the slope of the line that passes
through the points (4,1) and (4, 3).
This is a vertical line
Undefined
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Examples
Estimate the slope of all line segments in the
figure
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Point-Slope Form of the Line
An equation of a line that passes through the
point (x1 , y1) with slope m is given by
Example Find an equation of the line that
passes through (3,1) and has slope m 4
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Horizontal Lines
Can be expressed in the form y b
y 2
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Vertical Lines
Can be expressed in the form x a
x 3
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Linear Models Applications of linear Functions
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First,General Definitions
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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

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Cost Function

• 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.

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

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Linear Revenue Function
Dollars
Revenue
Units
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Nonlinear Revenue Functions
Dollars
Dollars
Revenue
Revenue
Units
Units
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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
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Linear Profit Function
Dollars
Profit
Units
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Nonlinear Profit Functions
Dollars
Dollars
Profit
Profit
Units
Units
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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)
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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.
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Break-Even Analysis
The break-even point is the level of production
that results in no profit and no loss.
Profit 0 means Revenue Cost
Dollars
Revenue
profit
loss
Cost
Break-even Revenue
Units
Break-even point
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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
C (x) 3x 3600 where x is the number of shirts
produced.
b. The revenue function
R (x) 12x where x is the number of shirts
sold.
c. The profit from 900 shirts
P (x) R(x) C(x) P (x) 12x (3x 3600)
9x 3600 P(900) 9(900) 3600 4500
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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
C (x) R (x)
Therefore, at 400 units the break-even revenue is
4800
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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.

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Linear Demand Function
q items demanded
Price p
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Nonlinear Demand Functions
q items demanded
q items demanded
Price p
Price p
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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.

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Linear Supply Function
q items supplied
Price p
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Nonlinear Supply Functions
q items supplied
q items supplied
Price p
Price p
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Market Equilibrium
Market Equilibrium occurs when the quantity
produced is equal to the quantity demanded.
supply curve
q
surplus
shortage
demand curve
p
Equilibrium Point
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Market Equilibrium
Market Equilibrium occurs when the quantity
produced is equal to the quantity demanded.
q
supply curve
surplus
shortage
demand curve
Equilibrium demand
p
Equilibrium price
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Market Equilibrium

• 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.

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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).
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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
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a. The demand function
b. The supply function
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c. The equilibrium price and equilibrium demand
The equilibrium demand is 960 units at a price of
2.44 per unit.
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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.
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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.
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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
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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
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Example Find the equation of least-squares for
the data (1 , 2), (2 , 3), (3 , 7). The scatter
plot of the points is
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Solution We complete the following table
x y xy x2
1 2 2 1
2 3 6 4
3 7 21 9
Sum 6 12 29 14
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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
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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.
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Example Find the correlation coefficient for the
least-squares line from the last example.
Points (1 , 2), (2 , 3), (3 , 7)
0.9449