Chapter 4 - Chapter
4:

Linear
Functions
(4.1)
...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chapter
4:

Linear
Functions
(4.1)
 
 Definition:

A
linear
function
(always
degree
1)

is
defined
by
an
equation
of
the
 form
 f ( x ) = Ax + B ,

where
A
and
B
are
constants
is
called
the
Standard
form.
 
 For
a
Line
passing
through
(x1,y1)
and
(x2,y2):

Slope=
 
 
 
 
 1) 
Slope‐intercept
form
of
Line:
 
 
 
 
 2) Point‐slope
Form:
 
 
 
 
 
 3) Standard
Form:
 
 
 
 Ex:

If
f(3)
=
2
and
f(‐3)
=
‐4,
find
the
linear
function
defining
f(x)
in
the
slope‐ intercept
form.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Application
of
linear
function
concept:


 
 A
factory
owner
buys
a
new
machine
for
$20,
000.

After
eight
years,
the
machine
 has
a
salvage
value
of
$1000.

Find
a
formula
for
the
value
of
the
machine
after
t
 years
when
 0 ≤ t ≤ 8.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Application
of
linear
function
in
economics:
 
 Suppose
the
cost
($)
to
a
manufacturer
of
producing
x
units
of
a
certain
motorcycle
 is
given
by
 
 Cost
function
C(x)
=
220x
+
4000

 
(y
=
mx
+b
form)

where
m
=
220

and
b
=
4000
 
 a) Find
the
marginal
cost
=
additional
cost
to
produce
one
more
unit
 
 
 
 
 
 
 
 b) Compute
the
cost
of
500
motorcycles
 
 
 
 
 
 
 
 c) Use
the
results
from
a)
and
b)
to
find
the
cost
of
501
motorcycles
 
 
 d) Find
the
fixed
cost
(the
cost
before
starting
production)
 
 
 
 
 
 
 
 
 Ex:

Let
x
denote
a
temperature
on
the
Celsius
scale,
and
let
y
denote
the
 corresponding
temperature
on
the
Fahrenheit
scale.
 
 a) Find
the
linear
function
relating
x
and
y;
use
the
facts
that
320F
corresponds
 to
00C
and
2120F
corresponds
to
1000C.

Write
the
function
in
standard
form.
 
 
 
 
 
 
 
 
 
 
 b) What
Celsius
temperature
corresponds
to
98.60F?
 
 
 
 
 
 
 
 
 c) Find
a
number
z
for
which
z0F=z0C.
 
 
 
 
 
 
 
 
 
 
 
 
 
 Quadratic
Function
(4.2)
 
 Definition:

A
quadratic
function
(always
degree
2)

is
defined
by
an
equation
of
the
 form
 f ( x ) = ax 2 + bx + c 
where

a,
b,
c
are
constants,
with
 a ≠ 0.

The
graph
of
a
 quadratic

functions
is
a
parabola.
 
 Graphs
representing

a
quadratic
equation:
 
 
 
 
 
 
 
 
 
 
 
 
 Terms


 
 Vertex:
 
 
 
 
 
 
 
 Axis
of
symmetry
 
 
 
 Vertex
form
of
a
parabola:



 
 f ( x ) = y = a( x − h )2 + k 
 
 
 Ex:

Rewrite
the
function
 f ( x ) = x 2 + 6x − 3
in
the
vertex
form.
 
 Completing
the
square
method:
 
 
 
 
 
 
 
 
 
 
 
 
 Using
the
vertex
formula:
 
 
 
 
 
 
 
 
 
 
 
 
 
 Note:

The
maximum
or
minimum
value
of
f(x)
occurs
at
the
vertex.
 
 
 In
general
given
a
quadratic
function
of
the
form
 y = f ( x ) = ax 2 + bx + c ,
express
it
in
 the
vertex
form
by
completing
the
square.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 How
does
the
constant
a
plays
a
role?
 
 
 
 
 
 
 
 
 
 
 
 
 Now
let’s
go
back
to
the
first
example
to
find
the
x‐intercept,
y‐intercept
and
the
 general
shape
of
the
parabola.
Graph
the
parabola
and
note
how
the
value
and
sign
 of
the
constant
a
determine
the
shape.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Graph
the
parabola
 f ( x ) = −2x 2 + 20x + 15 
by
determining
its
vertex,
y‐intercept,
 axis
of
symmetry.

Does
the
graph
open
up
or
down
and
why?
 
 Find
the
vertex
by
completing
the
square
method:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Check
your
answer
from
above
by
using
the
vertex
formula:
 
 
 
 
 
 
 
 
 
 
 x‐intercept:
 
 
 
 
 
 
 
 
 
 
 
 
 
 y‐intercept:
 
 
 
 
 
 
 
 Graph:


 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Setting
up
Equations
that
define
functions

(4.4)
 
 Steps:
 
 1) After
reading
the
problem
carefully
draw
a
picture
or
make
a
table
that
 conveys
the
given
information.

Figure
out
what
is
given?
 
 2) Identify
what
the
problem
is
you
to
find.


Most
of
the
time
the
problem
will
 ask
you
to
find
a
particular
quantity
or
a
formula
for
a
quantity.

Assign
a
 variable
to
denote
the
key
quantity.


Figure
out
what
are
you
looking
for?
 
 3) Label
any
other
quantities
that
appear
relevant.

Jot
down
what
you
already
 know
systematically.

Figure
out
if
there
are
expressions
that
relate
these
 quantities.

You
want
to
express
the
resultant
expression
in
terms
of
the
key
 variable
identified
in
step
2.
 
 4) Use
substitution
to
express
the
final
equation
in
terms
of
the
key
variable.
 
 Overall,
find
a
strategy
that
will
connect
what
you
know
with
what
you
are
looking
 for
to
solve.
 
 
 
 Ex:

A
rectangle
is
inscribed
in
a
circle
of
diameter
12
inches.

Express
the
perimeter
 and
the
area
of
the
rectangle
as
a
function
of
its
width
x.

Follow
the
logical
thought
 process
as
described
in
the
steps
above.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

The
product
of
two
numbers
is
16.

Express
the
sum
of
the
squares
of
the
two
 numbers
as
a
function
of
a
single
variable.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

A
right
circular
cylinder
has
a
surface
area
of
65
sq.
inches
(including
top
and
 bottom).

Find
an
expression
for
the
volume
of
the
cylinder
as
function
of
radius
r.


 


 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Maximum
and
Minimum
Problems
(4.5)
 
 
 In
this
section
we
will
learn
how
to
set
up
equations
of
degree
2
that
will
help
us
 find
the
maximum
and
minimum.
 
 Recall:

A
standard
quadratic
function
of
the
form
 y = ax 2 + bx + c 
can
be
rewritten
 into
the
vertex
form
 y = a( x − h )2 + k 
where
(h,
k)
=
vertex
of
the
parabola
and
 −b 4ac - b2 h= , k= = f(h) 
 2a 4a 
 
 
 If
a>0,
then
the
parabola
opens
upward
and
the
vertex
in
that
case
represents
the
 minimum
point.
 
 
 If
a<o,
then
the
parabola
opens
downward
and
the
vertex
in
thatcase
represents
the
 maximum
point.
 
 
 Theorem:

For
 y = ax 2 + bx + c 
representing
the
parabola,
the
x­coordinate
of
 −b the
maximum
or
the
minimum
takes
place
at
 x = h = 
 2a 
 
 Strategy
for
solving
max/min
problems
whose
model
equation
is
of
the
form
 y = ax 2 + bx + c 
(quadratic
expression
in
one
variable)


 
 Steps:
 
 1) Express
the
quantity
to
be
maximized
or
minimized
in
terms
of
a
of
a
 single
variable.

The
equation
should
be
a
quadratic.
 
 −b 2) Use
the
vertex
formula
 x = 
to
locate
the
x‐coordinate
of
the
vertex.

If
 2a −b the
parabola
opens
up,
you
got
a
minimum
at
 x = 
and
if
the
parabola
 2a −b opens
down,
you
get
a
maximum
at
 x = 
 2a 
 
 
 3) After
you
determined
the
x‐coordinate
of
the
vertex,
relate
the
 information
to
the
problem
at
hand;
often
you
will
need
to
find
the
 function
value
at
the
given
x‐value.
 
 
 
 Ex:

Suppose
you
have
600
ft.
of
fencing
with
which
to
build
three
adjacent
 rectangular
corrals
as
shown
below.

Find
the
dimensions
so
that
the
enclosed
area
 is
as
large
as
possible.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

A
baseball
is
thrown
straight
upward
and
its
height
as
a
function
of
time
t
is
 given
by
 h( t ) = −16t 2 + 32t .

Find
the
maximum
height
of
the
ball
and
the
time
at
 which
the
height
is
attained.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Find
the
point
on
the
curve
 y = x 
that
is
nearest
to
the
point
(4,
0)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Polynomial
Functions

(4.6)
 
 Recall:

A
function
f(x)
is
called
linear
if
 
 
 
 
 
 
 
 A
function
f(x)
is
called
quadratic
if

 
 
 
 
 
 
 
 
 
 Definition:

A
polynomial
function
is
a
function
of
the
form

 
 f ( x ) = an x n + an −1x n −1 + an −2 x n −2 + ........... + a2 x 2 + a1x + a0 ,



 an ≠ 0 
 
 n
is
the
degree
of
the
polynomial.

n
is
a
positive
integer
and
 a1 ,a2 ,a3 ,........an 
are
 constants.

The
largest
exponent
n
is
the
degree
of
the
polynomial.
 
 
 Ex:

What
are
degrees
of
the
following
polynomial
functions:
 
 f ( x ) = x 3 − 5x 2 + 2x − 1 g( u ) = 3u7 − 5u 2 + π 
 Note:

By
definition,
n
is
a
positive
integer.
 
 Definition:

A
power
function
is
of
the
form
 y = x n 
where
n
is
any
real
number
 constants
(do
not
have
to
be
integers).
 
 Ex:
 
 
 
 
 Graphs:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Note:

There
is
no
similarity
in
the
way
power
functions
look
as
the
degree
increases
 unlike
polynomial.
 
 
 Ex:

Sketch
the
graph
of
 f ( x ) = −( x − 2)2 + 1
by
specifying
x
and
y
intercepts.

 Remember
translations
(horizontal
and
vertical)
and
reflections
while
sketching
the
 graph.
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Do
the
same
for
 y = −( x − 1)3 − 1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Property
summary
for
Polynomial
Functions:
 
 1) The
graph
of
a
polynomial
function
of
degree
2
or
above
is
an
unbroken
 smooth
curve
(deg
1
or
0
will
yieldthe
graph
of
a
straight
line).
 2) The
graph
of
a
polynomial
of
deg
n
has
at
most
(n‐1)
turning
points.
 
 ex:


Quadratic
function
(deg
2)
has
one
turn
 
 Cubic
function
(deg3)
 
 
 
 
 
 
 
 
 can
have
two
or
no
turn.
 
 
 
 
 
 
 
 
 3) Consider
 f ( x ) = an x n + an −1x n −1 + an −2 x n −2 + ........... + a2 x 2 + a1x + a0 ,
 
 For
n=
even
degree:

As
x
gets
larger
(positively
or
negatively)

y
only
gets
positively
 larger.
 
 Graphically
the
end
behavior
will
be
as
follows:
 
 
 
 For
n
=
odd
degree:

As
x
gets
positively
larger,
y
also
gets
positively
larger.

As
x
 gets
negatively
larger,
y
also
gets
negatively
larger.


 
 Graphically
the
end
behavior
be
as
follows:
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Let
 f ( x ) = − x 3 + x 2 + 9x + 9 
 
 Determine
which
of
the
following
graph
could
be
a
possible
representation
of
f(x).

If
 the
graph
is
not
a
representation
of
f(x),
which
property
of
polynomial
function
is
 violated.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:


Let
 f ( x ) = x 3 − 5x 2 − x + 5.

First
find
the
x
and
y
intercepts.

Describe
the
 behavior
of
f(x)
at
each
x‐intercept
immediate
vicinity
by
figuring
out
intervals
 where
f(x)
>0,

or
f(x)<0.
 
 Note:

Factors
of
f(x)

correspond
to
the
roots
(zeroes)
of
the
graph
of
the
function.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Rational
Functions
(4.7)
 
 Definition:

The
next
family
of
functions
that
we
are
going
to
learn
about
are
the
 rational
functions.

They
are
of
the
form:

 f ( x) y= , where f(x) and g(x) are both polynomials and g(x) ≠ 0
 g( x ) 
 Ex:
 
 
 
 
 The
major
difference
between
rational
and
polynomial
functions
is
Rational
 Functions
have
breaks
(discontinuous)
unlike
the
polynomials
which
are
always
 smooth
and
continuous.
 
 1 Ex:

 f ( x ) = , x ≠ 0
 x 
 
 
 
 
 
 
 
 
 
 
 The
vertical
dotted
line
where
the
function
is
broken
(discontinuous)
as
shown
 above
is
called
the
vertical
asymptote.
 
 Rational
Functions
have
another
kind
of
asymptote
called
the
horizontal
asymptote.


 
 f ( x) To
figure
out
the
vertical
asymptote
for
rational
function
of
the
form
 y = 
all
we
 g( x ) have
to
do
is
to
find
the
x‐values
that
will
make
g(x)=0
.

These
x‐values
will
make
y
 go
to
infinity

(positively
or
negatively
larger
and
larger)
which
in
turn
make
the
 vertical
asymptotes.
 
 x + 2 f ( x) = Ex:

Find
the
vertical
asymptotes
of
the
rational
function
 y = 
 x − 3 g( x ) 
 
 
 Similarly
Horizontal
Asymptotes
are
defined
as
x
gets
positively
or
negatively
larger
 and
larger
(approaches
infinity)
what
y‐value
does
the
rational
function
 x + 2 f ( x) y= = 
approach.
 x − 3 g( x ) 
 x + 2 f ( x) = Ex:
Find
the
horizontal
asymptotes
of
the
rational
function
 y = 
 x − 3 g( x ) 
 1 Observe:

Any
term
of
the
form
 n → 0
as
x
gets
positively
or
negatively
larger
and
 x larger.


 
 
 
 
 
 
 
 
 
 
 
 Note:

Dividing
a
constant
by
a
large
number
makes
the
ratio
go
to
zero.
 
 
 
 
 
 
 
 a x m + .................... + a1x + a0 In
General
:

Consider
the
rational
function
 f ( x ) = m n 
 bn x + .........................b1x + b0 
 1) If
m<n,
 2) If
m=n,
 3) If
m>n,

 
 
 
 
 
 
 
 
 The
last
important
thing
to
remember
while
graphing
rational
functions
is
can
the
 function
ever
cross
the
Horizontal
Asymptote?
 
 
 
 
 
 
 
 
 
 
 
 Use
the
example
above
to
find
at
what
values
of
x
if
any,
the
rational
function
is
 crossing
the
Horizontal
Asymptote.
 
 
 
 
 
 
 
 
 
 x + 2 f ( x) = Graph
the
function
 y = 
by
labeling
all
the
asymptotes.
 x − 3 g( x ) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Important
Note:

A

function
can
never
cross
the
Vertical
Asymptote
 x −1 
.

Specify
the
intercepts,
asymptotes
and
whether
 x2 − 4 the
graph
ever
crosses
the
Horizontal
Asymptote.

Include
all
the
above
information
 in
your
sketch.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Note:

A
rational
function
usually
consists
of
several
branches
separated
by
 asymptotes.

An
asymptote
is
a
line
such
that
the
distance
between
the
line
and
 curve
approaches
zero
as
it
gets
larger
and
larger.
 Ex:

Sketch
the
graph
of
 y = x 2 − 5x + 6 Ex:

Sketch
the
graph
of
 y = 2 
.

Specify
the
intercepts,
asymptotes
and
 x − 2x − 3 whether
the
graph
ever
crosses
the
Horizontal
Asymptote.

Include
all
the
above
 information
in
your
sketch.
 
 Note:

Pay
special
attention
to
the
domain.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Slant
Asymptote
will
show
up
in
the
homework
but
not
in
your
exams:
 1 − x2 Ex:

Show
that
the
line
 y = −x 
is
a
slant
asymptote
for
the
graph
 y = .

Then
 x sketch
the
graph
with
the
asymptotes
and
intercepts.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 x2 + x − 6 Ex:

Graph
the
function
 y = 
by
finding
out
the
x‐int(s),
y‐int
(s),
All
 x−3 Asymptotes
(Vertical,
Horizontal,
or
Slant)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ...
View Full Document

This note was uploaded on 02/08/2011 for the course MATH 3 taught by Professor Staff during the Winter '08 term at University of California, Santa Cruz.

Ask a homework question - tutors are online