Linear Function : A function f defined by a linear
equation of the form:
y = f (x) = mx + b where m and b are constants
Observations:
(1) linear functions have graphs that are straight lines.
(2) Any non-vertical straight line is the graph of a linear function.
Slope of a line : The slope of a line passing through (
and 
is:
Equations of a line with slope m which passes through
point 
Point-Slope equation: 
Slope- Intercept equation : y = mx + b where
Special cases:
Horizontal Line: y = c where c is a constant
Vertical Line: x = c where c is a constant
Example: Find the equation of a straight line passing
through points (3,3) and (− 4,1).
a) In point-slope form
b) In slope-intercept form
Example: A manufacturing plant buys a new molding machine
for $225,000. Its useful
life is 8 years, at which time it will be worth $25,000.
a) Find a linear function that gives the value of the
machine as a function of time.
b) How much is the machine worth after 4 years?
Perpendicular and Parallel Lines: For non-vertical lines
and 
with
slopes 
and
respectively:
a) 
and
are parallel if and only if
b) 
and
are perpendicular if and only if
Cost functions specify the cost C as a function of the
number of items produced (C(x)).
If the relationship is linear, we can write the cost function as a linear
function
y = C( x) = mx + b where mx is the variable cost and b, the y-intercept, is the
fixed cost.
Example: It cost a company $25,000 to build 30 boats on
Monday and $30,000 to build
40 boats on Tuesday.
a) Find a linear cost function based on this information.
b) What is the daily fixed cost?
c) Give the inter pretation of the slope of the cost
function.
d) The revenue that results from sales is the total
payment received. The revenue
function is the linear function resulting from the number of items built sold
multiplied
by the selling price per item, R(x) = xgp . If this company sells its boats for
$1500 per
unit, give the revenue function:
R( x) = _____________
e) The profit is what remains of the revenue after costs
are subtracted .
Find the profit function:
P( x) = R( x) −C( x) = ___________________
