Transcript Chapter 1 Linear Functions

Chapter 1
Linear Functions
Section 1.2
Linear Functions and Applications
Linear Functions
 Many situation involve two variables related




by a linear equation.
When we express the variable y in terms of
x, we say that y is a linear function of x.
Independent variable: x
Dependent variable: y
f(x) is used to sometimes denote y.
Linear Function
Example 1
 Given a linear function f(x) = 2x – 5, find
the following.
a.) f(-2)
f(-2) = 2 (-2) – 5 = -4 – 5 = -9
b.) f(0)
c.) f(4)
Supply and Demand
 Linear functions are often good choices for
supply and demand curves.
 Typically, there is an inverse relationship
between supply and demand in that as
one increases, the other usually
decreases.
Supply and Demand Graphs
 While economists consider price to be the
independent variable, they will plot price, p, on
the vertical axis. (Usually the independent
variable is graphed on the horizontal axis.)
 We will write p, the price, as a function of q, the
quantity produced, and plot p on the vertical
axis.
 Remember, though: price determines how
much consumers demand and producers
supply.
Example 2
 Suppose that the demand and price for a certain
model of electric can opener are related by
p = D(q) = 16 – 5/4 q Demand
where p is the price (in dollars) and q is the demand
(in hundreds).
a.) Find the price when there is a demand for
500 can openers.
b.) Graph the function.
Example 2 continued
 Suppose the price and supply of the electric can
opener are related by
p = S(q) = 3/4q
Supply
where p is the price (in dollars) and q is the demand
(in hundreds).
c.) Find the demand for electric can openers with a
price of $9 each.
d.) Graph this function on the same axes used
for the demand function.
NOTE: Most supply/demand problems will have the same
scale on both axes. Determine the x-and y-intercepts to
decide what scale to use.
Supply and Demand Graph
D(q)
Equilibrium point
(8,6)
S(q)
Equilibrium Point
The equilibrium price of a commodity is the price found at the
point where the supply and demand graphs for that commodity
intersect.
The equilibrium quantity is the demand and supply at
that same point.
Example 2 continued
p = D(q) = 16 – 5/4 q
p = S(q) = 3/4q
Demand
Supply
Use the functions above to find the equilibrium
quantity and the equilibrium price for the can
openers.
Cost Analysis
 The cost of manufacturing an item
commonly consists of two parts: the fixed
cost and the cost per item.
 The fixed cost is constant (for the most
part) and doesn’t change as more items
are made.
 The total value of the second cost does
depend on the number of items made.
Marginal Cost
 In economics, marginal cost is the rate of
change of cost C(x) at a level of
production x and is equal to the slope of
the cost function at x.
 The marginal cost is considered to be
constant with linear functions.
Cost Function
Example 3
 Write a linear cost function for each
situation below. Identify all variables used.
a.) A car rental agency charges $35 a day
plus 25 cents a mile.
b.) A copy center charges $4.75 to create
a flier and 10 cents for every copy
made of the flier.
Example 4
 Assume that each situation can be expressed
as a linear cost function. Find the cost
function in each case.
a.) Fixed cost is $2000; 36 units cost $8480
b.) Marginal cost is $75; 25 units cost $3770
Break-Even Analysis
 The revenue R(x) from selling x units of an
item is the product of the price per unit p and
the number of units sold (demand) x, so that
R(x) = p(x).
 The corresponding profit P(x) is the difference
between revenue R(x) and cost C(x).
P(x) = R(x) - C(x)
Break-Even Analysis
 A profit can be made only if the revenue
received from its customers exceeds the
cost of producing and selling its goods and
services.
 The number of units x at which revenue
just equals cost is the break-even
quantity; the corresponding ordered pair
gives the break-even point.
Break-Even Point
 As long as revenue just equals cost, the
company, etc. will break even (no profit
and no loss).
R(x) = C(x)
Example 5
 The cost function for flavored coffee at an
upscale coffeehouse is given in dollars by
C(x) = 3x + 160, where x is in pounds. The
coffee sells for $7 per pound.
a.) Find the break-even quantity.
b.) What will the revenue be at that point?
c.) What is the profit from 100 pounds?
d.) How many pounds of coffee will produce a
profit of $500?
Example 6
 In deciding whether or not to set up a new
manufacturing plant, analysts for a popcorn
company have decided that a linear function
is a reasonable estimation for the total cost
C(x) in dollars to produce x bags of
microwave popcorn.
They estimate the cost to produce 10,000
bags as $5480 and the cost to produce
15,000 bags as $7780.
Find the marginal cost and fixed cost of the
bags of microwave popcorn to be produced
in this plant, then write the cost function.