Transcript 1.2 Linear functions

1.2 Linear functions &
Applications
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Linear Functions: y=mx+b, y-y1=m(x-x1)
Supply and Demand Functions
Equilibrium Point
Cost, Revenue, and Profit Functions
Break-even Point (quantity, price)
Linear functions - good for
supply and demand curves.
•If price of an item increases, then
consumers less likely to buy so the
demand for the item decreases
•If price of an item increases,
producers see profit and supply of
item increases.
Linear Function f defined by
y  f ( x)  m x  b
(for real numbers m and b)
x=independent variable
y=dependent variable
Cranberry example and explanation of
quantity (x-axis), price (y-axis)
See Page 18 of e-text
 Cranberry example of late 1980’s early
1990’s.
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Explanation of why price is on the yaxis.
Demand Function
•defined by p = D(q)
•The function that gives the relationship
between the number of units (q) that
customers are willing to purchase at a given
price (p).
•The graph of a demand function is typically
decreasing.
EXAMPLE
If
p  D(q)  0.04q  72
is the relationship between p, the price
per unit in dollars and q, the quantity
demanded, what is the demand when
the price is $50 per unit?
ANSWER
p = -0.04q + 72
50 = -0.04q +72
-22 = -0.04q
550 = q
EXAMPLE:
Find the price when the level of
demand is 500.
Answer:
p
p
p
p
=
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=
-0.04q + 72
-0.04 (500) +72
-20 + 72
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Supply Function
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defined by p = S(q)
gives the relationship between the
number of units (q) that suppliers are
willing to produce at a given price (p).
The graph of a supply function is
typically increasing.
EXAMPLE
If p = 5 + 0.04q is the
relationship between the price (p)
per unit and the quantity (q)
supplied, When the price is set at
$73 per unit, what quantity will be
supplied?
Answer
p = 5 + 0.04q
73 = 5 + 0.04q
68 = 0.04q
1700 = q
Example 2 page 22
Part c shows (6, $4.50) as the
intersection of the supply and the
demand curve.
 If the price is > $4.50, supply will
exceed demand and a surplus will
occur.
 If the price is < $4.50, demand will
exceed supply and a shortage will
occur.
Graph of example 2
Equilibrium Point
The price at the point where the supply
and demand graphs intersect is called
the equilibrium price.
The quantity at the point where the
supply and demand graphs intersect is
called the equilibrium quantity.
To find the equilibrium
quantity algebraically, set the
supply and the demand
functions equal and solve for
quantity.
Example
Using
demand function p = 74 - .08q
supply function p = .02q + 3
to find…
(a) the equilibrium quantity
(b) the equilibrium price
(c) the equilibrium point
Answer
a) 74 – 0.08q = 0.02q + 3
71 = 0.10q
710 = q
c) (710, $17.20)
b) p = 0.02q + 3
p = 0.02(710) + 3
p = 17.2
Fixed costs (or overhead)
costs that remain constant regardless of
the business’s level of activity.
Examples
 rental fees
 salaries
 insurance
 rent
Variable Costs
costs that vary based on the
number of units produced or sold.
Examples
 wages
 cost of raw materials
 taxes
Cost Function
Total cost
C(x)= variable cost + fixed cost
Example
A company determines that
the cost to make each unit is
$5 and the fixed cost is
$1200. Find the total cost
function
Answer
C(x) = 5x + 1200
Marginal Cost is the rate of change
of cost C(x) at a production level of x
units and is equal to the slope of the
cost function at x (in linear functions)
It approximates the cost of producing
one additional item.
Example
The marginal cost to make x
capsules of a certain drug is $15 per
batch, while it cost $2000 to make
40 batches. Find the cost function,
given that it is linear.
Answer
y  y  m( x  x ) or y  mx  b
Use
and slope = 15, point (40, $2000)
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y – 2000 = 15 (x - 40)
y = 15x + 1400
Revenue, R(x)
money from the sale of x units
R(x) = p x
p is price per unit
x is number of units
Profit, P(x)
the difference between the
total revenue realized and the
total cost incurred:
P(x)= R(x) – C(x)
Example
If the revenue from the sale of x units of
a product is R(x) = 90x and the cost of
obtaining x units is C ( x)  50x  800
(a)determine the profit function.
(b)find the profit from selling 300 units.
Answer
a) P(x) = R(x) – C(x)
P(x) = 90x – (50x + 800)
P(x) = 40x – 800
b) P(300) = 40 (300) – 800
P(300) = $11,200
Review of Profit, Revenue, and Cost
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P(x)= R(x) – C(x)
when R(x) > C(x) then P(x)> 0 or a
gain.
If R(x) < C(x) then P(x) < 0 or a loss.
Finding breakeven quantity
If R(x) = C(x), then P(x) = 0. Where this happens
is the breakeven point
To find the breakeven quantity (x-value of the
break even point) either use a or b below.
a)
Set R(x)=C(x) and solve for x.
b) Set P(x)=0 and solve for x.
Always round the breakeven quantity up to the
next whole number.
A manufacturer can produce x units for
(240 + 0.18x) dollars. They can sell the
product for $3.59 per unit.
(a) find the cost function
(b) find the revenue function
(c) find the profit function
(d) the break-even quantity
(e) the profit from producing 250 units.
(f) number of units for profit of $1000.