Section 1.2
Linear Functions
Prepared by E. Gretchen Gascon
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Function Notation
Function is an algebraic expression where for every x (the
independent variable) there is one and only one y (the
dependent variable) corresponding to it.
A linear equation y = 4x – 3 can be written in function
notation as y = f(x) = 4x - 3.
f(x)
x
y
You enter a
value for x,
and the
function
f(x) always
outputs a
unique
value for y
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Problem # 13 page 27
Write a cost function given a set of parameters use c(x) to represent y instead of f(x)
c(x) = mx + b
OR
c(x) = (unit cost) x + (fixed cost)
Given a fixed cost of $400 (the cost regardless of how many items
are made, in other words the cost to produce 0 items c(0) = 400.
10 items cost $650 to make. (the total cost)
Fixed Cost is the yintercept (0,400) of a
linear graph
C(x) = mx + b
C(0) = m(0) + 400
C(10) = m(10) + 400
Marginal
(unit) cost is
the slope
(25/1) of the
linear graph
650
= 10m + 400
250
= 10m
25
=
m
The unknown is the marginal
cost. Substitute what you know
into the cost function and solve
for m.
therefore
C(x) = 25x + 400
Rewrite the cost
function using
the value for m
3
Graph of Problem #13
optional
Cost Function
c(x)
1000
y = 25x + 400
10, 650
500
0, 400
0
0
5
10
15
20
25
number of items
4
Problem # 19 page 29
graph the supply and demand functions on the same graph
s( q) 
3
q
2
Supply
equation
To graph
in Excel:
3
D ( q)  81  q
4
Demand
equation
Next, select the cells A1 : C13
Create a
table, let q be
arbitrary, and
solve for s(q)
and d(q)
5
How to create the graph in Excel
Choose the
Chart
wizard and
select the
xy(scatter)
Screen may
look slightly
different in
Excel 2007
Next: choose series
in “columns”, you will
see two lines that
cross. You may then
choose “Finish”
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Graph continued
Select one of the dotted lines, and right
click.
From the menu select “Add Trendline”
Repeat the process for the
other line.
Type: Linear
Check: Display
equation on
chart
The point of
intersection is the
equilibrium point.
or
Supply = Demand
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Problem #19 page 28 part b
find the equilibrium quantity and the equilibrium price
Set the supply function = the demand function and solve
for q the equilibrium quantity.
3
81 3
q  q
2
1 4
4  3  4  81 3 
 q    q
2 
1 4 
6q
 324  3q
9q  324
q  36
3
(36)
2
S ( q )  54
S (q) 
Multiply each
side of the
equation by
the common
denominator
Next: substitute q into
the supply function to
find the equilibrium
price.
36 strawberry-flavored licorice for $54
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