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```Unit 5: Graphing Polynomial Functions
Review Supplement
1.
5.2: Evaluate and Graph Polynomial
Functions
Given
'(-t)
Bookwork:
None
-|rt * 10, use
direct substitution to find /(*a).
- r
+io
: t i(+)*- ?t-1))
: .t * \ \Eb) '3, { -U+)*,o
o
^-"
-t6tf
!
+ 10x3 - 27, use
/(x) =
synthetic substitution to find f (-3).
Given
6xS
"tl6 o ro o o-21
["6 *rB !4 -rq? 6lb -fng
i..--_..-a+'.*_
Given g(.r) = 2x3 + Sxz * 1!x
direct substitution
IGY)= 2fY)' * {-
14, use
-'(+
' = 2(e"5(H) :trrr i-A)__7e)1
-.rdr
=W
+_V
'
x-,rs..<.+t
-
30, use
synthetic substitution to tnA e
5.
"
Describe the degree and leading
coefficient of the polynomial function
(]).
graphedbelow. r
5z L-1O27-\$a
if-6;;{z [email protected]
-
to rino g (- i).
z
Given g(x) =Zxa -9xz +37x
4.
2.
- _rer
*€ + zrs.
4l
:. lr +128+el 81 lo
a!r*
3.
/(x) = x +*xa
t
cka6<cL
:
rr
45jA
l' I lred*.l t
l l 4e'{'l ';afr:
5 -'CI -7* ::
7 -t .-10 rZ L-lL
iI
Bd*
(-9FE4-*k*i".Ekrce#\
-/*al
Describe the dr rgree and leading
coefficient of tl re polynomial function
graphed below '
A(q.*t=
7.
8.
Describe the-en{behavior of
f (x) =13+*u:')+ 6x32r +
gd! * dt"'J*trca- {-l
neqol-tvt
*"1--*.
' n te.v&a
..
vl
,/\''i:;H,,-o
I
;
:
.
nrr*L"
-J'3:.
{
*sY q -at
A5
9.
5.5: Find Rational Zeros
++cn, f{o) -u
Describe the
g1x1
B
*-
€r* -++<
/(1) =
\$
,lb -\)-r)
t, a -1
ffi
bFz
-1s -l-8
:a
f (x) = l2x3
-
Sxz +'J.Ix
-+
-
16
i8^
fr,')
z/,
0= *'-bvz+1y ->
)'/, ,'1,,
l*+
*tx +
x?a.,ulq
U"Lt
{
VJ
I
13.
)<;J',?
=A-5X;;[z-.21
Find the exact values of all real zeros
of
g(x) = x3 +3x2'!54'=t*,'in,|,
3tt 3 o -s+
|I- 3 {g
Lo
I :o
x-r) I : 3
--i-:
Lo
r)(lx * z)
I
Ly'd"
) lq h,Yl |t?t
|Y",'l+
exagllgluelof all real
5x2 + 6 = x3 - xz +9x + 4
solutions of
":
l, )r *r t
Find the
!. \
ztx'+!3x-Z:o
IJ
12.
\/
-
?"
r
X: tla) *+
Use the Rational Zero Theorem to list
all of the possible rational zeros of
'!.
f(x) = 24x3 + 133x2 + 63x given one of the factors is r * 5
zD
( 7x
11.
-
10. Factor
Find all the zeros of
given
Txzs
*4 -r &
fts xq +el {lr)
-v +e
FS X')[email protected] 4(x)
f(x):6x3-r3x2-r3x*20
Bookwork:
P. tt05:33-34
=W,;'6:J-
,1
^
eld behavior of
-l{F
x-J-L1*{5--
ir=LlJtr;zl
E+
6 -tE-_E
X'+f y tt\$:'a
Xr+bx+t t-18*3-
ix+3)""-4
va3: { 3i
/F#.-'--{4,,1
{x'-3:3i : S
w
i
14. Find the exact values of all real zeros
ofh(x) = 3x3 - 34x2 +7Bx * 68
-]i3,
;
is taken from a polynomial function. Use the Location
Principle to determine the interval(s), one unit in length, in which there must be a
15. The table of values below
zero.
_.1 2+ _LV
5 ,iL f(z Lg
x
-4
-3
-,
-1
0
1
,
3
4
f(x)
121
-3
-13
5
-3
-L6
12
78
2r3
:.:, -t i.1i
-, ..t *^ s
.i
,,. }
f"
-a t -b
y, - t 2 .t -, .*?.:: ..
'Y - L-i"r '\ - r: ; 1,.:.
i
17. Where does the given zero from Q16
5.7: Fundamental Theorem of Algebra
appear on the graph of f(r)? Where
do the other zeros appear?
ti
Bookwork:
P.405: 35-38
:?( :
t(
tla' 'Wi
_
a .- :.
tl
18. Let ftx): x'(x
-
a)(x
-
19.
b)3.
a)
How many zeros does
b)
How many distinct zeros does
/(x) have?
'1't I'ii
A.''\ \t'
c)
Describethebehaviorof thegraphatx =
0. t!"
d)
Describethebehaviorofthegraph alx =
e. I .rr-:-r''
Let
/
: b.
'
o
:i"
'
,
.,i
''
Min times
tt
crosses
t-axis
crosses
t-axis
tl
L7
22
22. Given f (x) = x(6x3
,f,
1
-
2x),
determine whethe_r /(x) is even,
I
L
odd, or neither. :;';;" ' i ,
t.J1.f,,
Maxtimes
t.
." "'-'
-.
J'.''
l,\.+*'
I'
:- in''
-
,t
/
tangent to.r-
Min times
tangent to.r-
Max number
of turning
axis
axis
ooints
r"'
tb
L
,{-
.'ir
f(x) = rzxa + ut * *' - {
i-
be a polynomial of degree zr. For each value of n, find each of the following:
Max times
dir
Use Descartes Rule of Signs to create
r
ii
l"\
i'l
;)u" o'; -
21.
t
q,.
'
)
4.i4
I
inl
- . .\_ _
\ -i
Use Descartes Rule of Signs to create.-t
table of the possible numbers of
positive real zeros, negative real
zeros, and imaginary zeros for
a
?t
il
23. Given f (x) = 3x4 +'L1xz
determine whether
odd, or neither.
:--"-
iFvcn
'
f(x)
-31,
is even,
2a.
",
f
I
20.
_")
table of the possible numbers of
positive real zeros, negative real
zeros, and imaginary zeros for
f(x) have? 3
Describe the behavior of the graph alx
ii
a
f,e
I
e)
*,-,f
ff
f (-x) = f (x),what conclusions
can you draw
-il,'. r.-,'i'ni:. Tt'r !,.up't' ",
,)' ty,n,.,l . r (' ii^r ;" {. t,,.},
,,f;
.;
25. lf f
(-x) : -f
can you draw about
/(x)?
.\
F(*. ,. :i;, {.1 , ''
'
l-''
i
i
{"i
,iir:
Without
27
y
'
cjlct lator, gr aph
x:?L)
'
n:3.
il4r,
Bookwork:
P.406: 39-40
_.t
inl
-{lbl
/i
I
ri
VpLr l0 rt irl
; r," :1:
i ..-r/
1i
a grapling
y:*(;Y'ff("+t
Functions
t.
r
26. Without
5.8: Analyze the Graphs of Polynomial
(x), what conclusions
a grap-hing
- -!tr:'zlt
:,
i
-
28. Let's cut a square from the corner of a piece of 11" by 14" construction paper and
then fold the resulting net into a box without a lid. Find a function 7(r) for the
calculgtor, gr aph
'i
si;l
i-
,;
\i:'.
ta.
volume of the box. State the domain. Find the value of x that maximizes the
-r.f5;:volume. Finally, find the max volume of the box.
i
,i8
Fx
IIti 1:
Ili {r
,i-!
.tr
i,
Y!
|
..
29.
Use
x
fk\
,:/
.-
i
r.te,l1.i.r.-.,:
i
-2
-605
-276
r.
;,"
find the zeros of the function. Then sraph it.
the table below to
-3
r- *'ft ":1a,t
-1
, -3s
x
=f
0
1
?
3
10
-9
-20
49
-32x2*x*10
-',
[?
,/
':1',
j
i:
.-...1..."....
l
t,.i
i :l;i
\i:
.:'!..'......,......:.
For e30-31, you can use the related quadratic equation (called the first derivative) to determine where the local minimums and
maximums occur for the given cubic function. Solve the first derivative equation. The solutions are the r-coordinates of the
minimums or maximums. To determine the y-coordinates, evaluate the cubic functions at the solutions to the first der-iV;ltive.
t
31. Cubic Function:
30. Cubic Function:
/*i
)
l-,i
:'a'' :'l
f(x)=x3+7xz-5x*f
First Derivative:
3x2+14x-5=0
Io"'
f (x) = -2x3 +
15x2
First Derivative:
+ 36x -'J-
-6x2+30x+36=0 - .
vt -'5y:*L
^1..
I
,-''- '''
'1
!
323
;'. i,-/-o',
J
t
i
\-f;;t r.^,,..i, I
\_
_.
32. From Q30-31, how
is the coefficient of the x3-term in the cubic function related to
the coefficient of the xz-term in the first derivative? How is the coefficienl of x2term of the cubic function related to the r-term of the first derivative? How is the
coefficient of the r-term of the cubic function related to the constant term of the
first derivative?
( oe&r t"''"So{ *"
p3 -t€'o"''
"{**^
',)';';";;efi. ,*&r[
*
:
o{ y-{-e'**
) z . ct ep€t'Jof \"*{'c'*.*- cse'f,;dit,.*-(
* *'{ c''* *' Ce"*'\$r1':4 +"'G'rv1k
A coe-lS''ci* J *t-
33. Find the first derivative of
f(x) = x3 +9x2 *24x * 3, and
then determine the location of the
local minimums and maximums.
F,vsf
i3rr + l9v +2T'o il
Y- +bx +8:?
Ne-v+ltuc :
( *.::){*:
f\'
x-? .j
o
I
-?
t bxtr C.Y *d
+ Ct s* dre",*afu"C 62-- 3a-*2 * 2d'
(ybrc-> y:
{r<t
rl t" 2,.-)
5.9: Write Polynomial Functions and
Models
34. Find the cubic function through the
points {-5,0), (3,0), (4, 0), {11, -1)
11
Bookwork:
P.405:41
1,
Y'1/v'-(rt.
- 7x rbo
--+vL
'4..?'
yi ,
*? 3s r lt)
.^ 1
-1i., _ i)
4ts
+{ i
yz+.tf
-t-,t
&Y
-t c"? -tx?
35_
t3
cxx}
-t /b)
) - 1]oxl -b\X
io rL -x?S:K -!Ao
2.r? +x3
-4txL -L81x-Qa
-t
t a(x +s)(x-B)fx -t)
brflr{- *r-l);i-l-i;
-,_,_q\
flf
-\ =a(d+)(-t)-s)
=
";€F
:Ja-"i-
r,,
-22>
The table shows the average speed y
(in feet per second) of a space shuttle
for different times t (in seconds) after
launch. Use a graphing calculator to
!
=, roA5v
1-
?5,?
"13X5*\1+69bfcy- Z
37. When the
space shuttle reaches a
speed of approximately 4400 feet per
second, its booster rockets fall off.
Use your polynomial modelto
determine how long after launch this
happens.
lFo
N
*
atz : 4 { rx-t i{ -:X
-q2=-8T*
O
[email protected]
!:
o
lob"l Se*r--)s
-:,)
h u*t)(x"'n)
+l}-9a'9
-* a(-++sf,-+*3)(2'- x
a<-(r+ 5h+ 3)
LCA)
-- ?ea
t
_,-,r=|orF,o)>--- .\._
/'
Y-a(x+sXx+3)(x + )-';(
u?1
=
-t
35. Find the quartic ftmctiqn through the
points (-5, 0),t(-4, -21))(-3,0),
1" *
l+ 6\ *
r*
z)
,'?rr -
i
fr"
?-l
t)
2^l= tr(2*1 ,*3-7xz-Zffix -tz.)
g;Lt zx't-r- Y-3 *Wk' - 2\t \- lr')
Ly
+Li
v
-r
= Lr(v'rii
*: )(zx
+
r)(x-
```
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