1.1 - MATH
1081
 Wednesday,
January
12
 
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Unformatted text preview: MATH
1081
 Wednesday,
January
12
 
 Section
1.1
 
 Slopes
and
Equations
of
Lines
 
 Homework
#1
(due
1/24):

 Section
1.1
#36,
64,
70,
72
 
 
 
 
 
 
 
 
 Section
2.1
#36,
44,
46,
64 What
is
the
rate
of
change
of
the
graph
at
each
of
the
labeled
 points?
 
 In
order
to
formalize
the
notion
of
the
rate
of
change
of
a
curve
 at
a
given
point,
we
are
going
to
first
make
the
question
a
 simpler
one
and
go
back
to
something
with
which
we
are
 already
familiar.
 
 What
is
the
rate
of
change
of
the
line
shown
here?
 
 Slope
of
a
Line
 
 The
rate
of
change
of
a
line,
also
called
the
slope
of
the
line,
is
 defined
as
the
vertical
change
(the
“rise”)
over
the
horizontal
 change
(the
“run”)
as
one
travels
along
the
line.
 
 That
is,
if
( x1 , y1 ) 
and
( x2 , y2 )
are
different
points
on
the
line,
 then
the
slope
is
 
 Change in y Δy y2 − y1 ,
 m= = = Change in x Δx x2 − x1 
 where
 x1 ≠ x2 .
 
 What
kind
of
line
can
have
two
different
points
with
the
same
 x ‐coordinate?

What
is
the
slope
of
such
a
line?
 
 
 What
kind
of
line
can
have
two
different
points
with
the
same
 y ‐coordinate?

What
is
the
slope
of
such
a
line?
 
 
 The
phrase
“no
slope”
should
not
be
used,
because
it
is
 ambiguous.
 Example:

Determine
the
slope
of
the
line
through
the
given
 points.
 
 a.

(1, 4 ) 
and
( 4, −5 ) 
 
 b.

( −2, 3) 
and
( 5, 5 ) 
 
 c.

( 8, 9 ) 
and
( 8,1) 

 
 d.

(10, 3) 
and
( −15, 3) 
 
 Equations
of
Lines
 
 In
order
to
determine
the
equation
of
a
line,
we
need
to
know
 the
slope
and
the
coordinates
of
one
point.

Of
course,
as
we
 just
reviewed,
we
can
find
the
slope
of
a
line
given
any
two
 different
points.
 
 One
point
of
particular
interest
is
the
point
where
the
line
 crosses
the
 y ‐axis,
as
every
non‐vertical
line
must.

The
 coordinates
of
this
point
will
have
the
form
( 0, b ) 
for
some
 number
 b 
(which
may
also
be
0).
 
 Equation
 
 
 
 
 y = mx + b 
 
 
 
 





 
 y − y1 = m ( x − x1 )
 
 
 x = k
 

 
 y = k
 

 
 
 
 
 
 
 
 
 
 
 
 
 Description
 Slope‐intercept
form:
 
 slope
 m 
and
 y ‐intercept
 b 
 Point‐slope
form:
 slope
 m 
and
passing
 through
the
point
( x1 , y1 ) 
 Vertical
line:
 
 slope
undefined
 Horizontal
line:
 
 slope
0
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Example:
 Determine
the
equation
of
the
line
with
the

 indicated
properties.

Also,
determine
if
the
line
 passes
through
the
origin
(the
point
( 0, 0 ) ).
 
 a. 


Slope
 −2 
and
 y ‐intercept
3
 
 
 b. Slope
 5 
passing
through
the
point
(1, 2 ) 
 
 
 c. 


Horizontal
line
through
the
point
( −1, −2 ) 
 
 4⎞ ⎛ d. Through
the
points
( −3,1) 
and
 ⎜ 4, − ⎟ 
 ⎝ 3⎠ Parallel
and
Perpendicular
Lines
 
 Two
different
lines
are
parallel,
if
either
they
are
both
vertical
 or
if
they
have
the
same
slope.
 
 Two
lines
are
perpendicular,
if
they
intersect
at
a
90‐degree
 angle.

When
the
neither
line
is
vertical,
the
product
of
their
 slopes
isequal
to
 −1,
or
equivalently,
the
slope
of
one
line
is
the
 negative
reciprocal
of
the
slope
of
the
other.
 

 Example:
 Determine
the
equation
of
the
line
with
the

 indicated
properties.
 
 a. 

Passing
through
the
point
( −2, 6 ) 
and
parallel
to



 the
line
 2 x − 3y = 5 .
 
 
 b. Passing
through
the
origin
and
perpendicular
to
 the
line
through
the
points
( −3, 0 )
and
(1, 5 ) .
 
 Example:
 The
sales
of
a
small
company
were
$27,000
in
its
 second
year
of
operation
and
$63,000
in
its
fifth
 year
of
operation.

Let
 y 
represent
sales
in
the
 x th
 year
of
operation,
and
assume
that
the
data
can
be
 approximated
with
a
linear
equation.
 
 a. 


Find
an
equation
for
the
sales
 y 
in
year
 x .
 
 b. According
to
this
model,
what
will
the
sales
be
in

 


the
eighth
year
of
operation?
 
 c. 


According
to
this
model,
in
which
year
will
sales

 


reach
$250,000?
 
 
 Tomorrow
in
recitation:
Workshop
#1
 
 Next
time:
 Section
2.1

 We
will
begin
using
clickers.
 ...
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This note was uploaded on 05/01/2011 for the course MATH 1081 taught by Professor Johanson during the Spring '08 term at Colorado.

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