Q
u
ad
r
a
t
ic F
un
c
t
i
ons
—
F
u
ll De
pt
h S
tu
d
y
G
u
ide
(
NC Ma
t
h
2
CDM
)
1
.
GRAPHING
—
Dee
p
Di
v
e
A
qu
ad
r
a
t
ic f
un
c
t
i
on
g
r
a
p
h
s
a
s
a
p
a
r
ab
o
la
—
a U
-
s
ha
p
ed c
urv
e
(
op
e
ns
up
or
d
own
).
S
t
a
n
da
r
d F
orm
y
=
a
x²
+
b
x
+
c
a
c
ontro
l
s
di
r
ec
t
i
on
o
f
op
e
n
i
n
g a
n
d
w
id
t
h
/
st
ee
pn
e
ss
c
i
s
t
he
y
-
i
nt
e
r
ce
pt
di
r
ec
t
l
y
(
s
i
n
ce
p
l
u
ggi
n
g i
n
x
=
0
lea
v
e
s
y
=
c
)
Be
st
f
orm
f
or
qu
ickl
y
r
eadi
n
g
t
he
y
-
i
nt
e
r
ce
pt
;
not
ideal f
or
r
eadi
n
g
t
he
v
e
rt
e
x
di
r
ec
t
l
y
.
Ve
rt
e
x
F
orm
y
=
a
(
x
−
h
)
²
+
k
(
h
,
k
)
=
t
he
v
e
rt
e
x
,
r
ead di
r
ec
t
l
y
off
t
he e
qu
a
t
i
on
Wa
t
ch
t
he
s
ig
n
fl
i
p
:
if
you
s
ee
(
x
−
3
)
²
,
h
=
3
.
If
you
s
ee
(
x
+
3
)
²
,
r
e
wr
i
t
e i
t
a
s
(
x
−
(−
3
))
²
,
so
h
=
−
3
.
Be
st
f
orm
f
or
qu
ickl
y
ide
nt
if
y
i
n
g
t
he
v
e
rt
e
x
a
n
d a
x
i
s
o
f
symm
e
try
.
Fac
tor
ed F
orm
(
I
nt
e
r
ce
pt
F
orm
)
y
=
a
(
x
−
p
)(
x
−
q
)
p
a
n
d
q
=
t
he
x
-
i
nt
e
r
ce
pts
(
roots
/
z
e
ros
),
r
ead di
r
ec
t
l
y
off
t
he e
qu
a
t
i
on
Sa
m
e
s
ig
n
-
fl
i
p
ca
ut
i
on
a
s
v
e
rt
e
x
f
orm
:
(
x
+
4
)
m
ea
ns
t
he i
nt
e
r
ce
pt
i
s
x
=
−
4
.
Be
st
f
orm
f
or
qu
ickl
y
ide
nt
if
y
i
n
g
x
-
i
nt
e
r
ce
pts
.
C
onv
e
rt
i
n
g be
tw
ee
n
f
orms
:
Fac
tor
ed
/
Ve
rt
e
x
→
S
t
a
n
da
r
d
:
e
xp
a
n
d
(
FOIL
/
di
str
ib
ut
e
)
a
n
d c
om
bi
n
e like
t
e
rms
.
S
t
a
n
da
r
d
→
Ve
rt
e
x
:
c
omp
le
t
e
t
he
squ
a
r
e
(
s
ee Sec
t
i
on
2
).
S
t
a
n
da
r
d
→
Fac
tor
ed
:
fac
tor
t
he
tr
i
nom
ial
(
s
ee Sec
t
i
on
2
),
on
l
y
poss
ible if i
t
fac
tors
n
icel
y
(
r
a
t
i
on
al
roots
).
Ve
rt
e
x
The highe
st
or
l
ow
e
st
po
i
nt
on
t
he
p
a
r
ab
o
la
—
w
he
r
e i
t
cha
n
ge
s
di
r
ec
t
i
on
.
Fi
n
di
n
g
t
he
v
e
rt
e
x
f
rom
STANDARD f
orm
:
x
-
c
oor
di
n
a
t
e
:
h
=
−
b
/(
2
a
)
The
n
p
l
u
g
t
ha
t
x
-
v
al
u
e back i
nto
t
he
or
igi
n
al e
qu
a
t
i
on
to
fin
d k
(
t
he
y
-
c
oor
di
n
a
t
e
).
Fi
n
di
n
g
t
he
v
e
rt
e
x
f
rom
VERTEX f
orm
:
r
ead
(
h
,
k
)
di
r
ec
t
l
y
,
r
e
m
e
m
be
r
i
n
g
t
he
s
ig
n
fl
i
p
.
A
x
i
s
o
f S
ymm
e
try
The
v
e
rt
ical li
n
e
t
ha
t
sp
li
ts
t
he
p
a
r
ab
o
la i
nto
two
m
i
rror
-
i
m
age hal
v
e
s
.
I
t
al
w
a
ys
p
a
ss
e
s
t
h
rou
gh
t
he
v
e
rt
e
x
.
E
qu
a
t
i
on
:
x
=
h
(
s
a
m
e
x
-
v
al
u
e a
s
t
he
v
e
rt
e
x
)
F
rom
st
a
n
da
r
d f
orm
:
x
=
−
b
/(
2
a
)
Ma
x
i
mum
/
Mi
n
i
mum
The
v
e
rt
e
x
IS
t
he
m
a
x
or
m
i
n
v
al
u
e
o
f
t
he f
un
c
t
i
on
.
If
a
>
0
(
pos
i
t
i
v
e
)
→
p
a
r
ab
o
la
op
e
ns
UPWARD
→
v
e
rt
e
x
i
s
a MINIMUM
If
a
<
0
(
n
ega
t
i
v
e
)
→
p
a
r
ab
o
la
op
e
ns
DOWNWARD
→
v
e
rt
e
x
i
s
a MAXIMUM
The k
-
v
al
u
e
(
y
-
c
oor
di
n
a
t
e
o
f
v
e
rt
e
x
)
i
s
t
he ac
tu
al
m
a
x
/
m
i
n
VALUE
o
f
t
he f
un
c
t
i
on
.
x
-
i
nt
e
r
ce
pts
(
roots
/
z
e
ros
)
Whe
r
e
t
he
p
a
r
ab
o
la c
ross
e
s
t
he
x
-
a
x
i
s
(
y
=
0
).
A
p
a
r
ab
o
la ca
n
ha
v
e
2
,
1
,
or
0
r
eal
x
-
i
nt
e
r
ce
pts
.
Fi
n
di
n
g
t
he
m
:
s
e
t
y
=
0
a
n
d
so
l
v
e
us
i
n
g fac
tor
i
n
g
,
squ
a
r
e
root
m
e
t
h
o
d
,
c
omp
le
t
i
n
g
t
he
squ
a
r
e
,
or
t
he
qu
ad
r
a
t
ic f
ormu
la
(
Sec
t
i
on
2
).
F
rom
fac
tor
ed f
orm
,
s
e
t
each fac
tor
e
qu
al
to
0
.
y
-
i
nt
e
r
ce
pt
Whe
r
e
t
he
p
a
r
ab
o
la c
ross
e
s
t
he
y
-
a
x
i
s
(
x
=
0
).
Fi
n
di
n
g i
t
:
s
e
t
x
=
0
a
n
d e
v
al
u
a
t
e
.
F
rom
st
a
n
da
r
d f
orm
,
t
he
y
-
i
nt
e
r
ce
pt
i
s
s
i
mp
l
y
t
he
v
al
u
e
o
f
c
.
Di
r
ec
t
i
on
o
f O
p
e
n
i
n
g
C
ontro
lled e
nt
i
r
el
y
b
y
t
he
s
ig
n
o
f
a
(
s
a
m
e i
n
all
t
h
r
ee f
orms
):
a
>
0
→
op
e
ns
UP
(
sm
ile
s
ha
p
e
)
a
<
0
→
op
e
ns
DOWN
(
f
rown
s
ha
p
e
)
La
r
ge
r
|
a
|
→
n
a
rrow
e
r
/
st
ee
p
e
r
p
a
r
ab
o
la
;
sm
alle
r
|
a
|
→
w
ide
r
/
fl
a
tt
e
r
p
a
r
ab
o
la
2
.
SOLVING QUADRATICS
—
Dee
p
Di
v
e
"
S
o
l
v
i
n
g
"
a
qu
ad
r
a
t
ic
m
ea
ns
fin
di
n
g i
ts
x
-
i
nt
e
r
ce
pts
/
roots
/
z
e
ros
—
t
he
x
-
v
al
u
e
s
t
ha
t
m
ake
y
=
0
.
Fac
tor
i
n
g
O
n
l
y
wor
k
s
clea
n
l
y
w
he
n
t
he
qu
ad
r
a
t
ic fac
tors
w
i
t
h
r
a
t
i
on
al
/
i
nt
ege
r
num
be
rs
.
P
ro
ce
ss
(
f
or
x²
+
b
x
+
c
,
a
=
1
):
1
.
Fi
n
d
two
num
be
rs
t
ha
t
mu
l
t
i
p
l
y
to
c a
n
d add
to
b
.
2
.
W
r
i
t
e a
s
(
x
+
first
num
be
r
)(
x
+
s
ec
on
d
num
be
r
).
3
.
Se
t
each fac
tor
to
0
a
n
d
so
l
v
e
(
Ze
ro
P
ro
d
u
c
t
P
rop
e
rty
).
E
x
a
mp
le
:
x²
+
7x
+
10
=
0
→
(
x
+
2
)(
x
+
5
)
=
0
→
x
=
−
2
or
x
=
−
5
P
ro
ce
ss
(
w
he
n
a
≠
1
):
us
e g
roup
i
n
g
—
mu
l
t
i
p
l
y
a
×
c
,
fin
d
two
num
be
rs
t
ha
t
mu
l
t
i
p
l
y
to
t
ha
t
pro
d
u
c
t
a
n
d add
to
b
,
sp
li
t
t
he
m
iddle
t
e
rm
,
t
he
n
fac
tor
b
y
g
roup
i
n
g
.
S
qu
a
r
e R
oot
Me
t
h
o
d
Be
st
w
he
n
t
he
r
e
'
s
NO li
n
ea
r
(
b
x
)
t
e
rm
—
i
.
e
.,
t
he e
qu
a
t
i
on
i
s
al
r
ead
y
i
n
t
he f
orm
a
(
x
−
h
)
²
+
k
=
0
or
a
x²
+
c
=
0
.
P
ro
ce
ss
:
1
.
I
so
la
t
e
t
he
squ
a
r
ed
t
e
rm
.
2
.
Take
t
he
squ
a
r
e
root
o
f b
ot
h
s
ide
s
.
3
.
Re
m
e
m
be
r
:
±
—
b
ot
h a
pos
i
t
i
v
e a
n
d
n
ega
t
i
v
e
root
.
4
.
S
o
l
v
e f
or
x
.
E
x
a
mp
le
:
(
x
−
3
)
²
=
25
→
x
−
3
=
±
5
→
x
=
8
or
x
=
−
2
C
omp
le
t
i
n
g
t
he S
qu
a
r
e
T
urns
st
a
n
da
r
d f
orm
i
nto
v
e
rt
e
x
f
orm
,
a
n
d ca
n
so
l
v
e f
or
roots
di
r
ec
t
l
y
.
W
or
k
s
on
ANY
qu
ad
r
a
t
ic
(
e
v
e
n
w
he
n
i
t
d
o
e
sn
'
t
fac
tor
n
icel
y
).
P
ro
ce
ss
(
f
or
x²
+
b
x
+
c
=
0
,
a
=
1
):
1
.
M
ov
e c
to
t
he
ot
he
r
s
ide
:
x²
+
b
x
=
−
c
2
.
Take half
o
f b
,
squ
a
r
e i
t
:
(
b
/
2
)
²
3
.
Add
t
ha
t
v
al
u
e
to
BOTH
s
ide
s
.
4
.
The lef
t
s
ide i
s
now
a
p
e
r
fec
t
squ
a
r
e
tr
i
nom
ial
—
r
e
wr
i
t
e a
s
(
x
+
b
/
2
)
²
.
5
.
S
o
l
v
e
us
i
n
g
t
he
squ
a
r
e
root
m
e
t
h
o
d
.
E
x
a
mp
le
:
x²
+
6x
+
5
=
0
→
x²
+
6x
=
−
5
→
half
o
f
6
i
s
3
,
squ
a
r
ed i
s
9
→
add
9
to
b
ot
h
s
ide
s
→
x²
+
6x
+
9
=
4
→
(
x
+
3
)
²
=
4
→
x
+
3
=
±
2
→
x
=
−
1
or
x
=
−
5
End of Document
329
