MAC 1147 Exam #1 Retake

Name i1nswV K~

10# Sp",,);)o I 0

HONOR CODE: On my honor, I have neither given nor received any aid on this examination.

Signature: _________________

Instructions: Do all scratch work on the test itself. Make sure your final answers are clearly labelled. Be sure to simplify all answers whenever possible. SHOW ALL WORK ON THIS EXAM IN ORDER TO RECEIVE FULL CREDIT!!!

No. Score 1 /8 2 /8 3 /8 4 /8 5 /8 6 /8 7 /8 8 /8 9 /8

10 /8 11 /10 12 /10

Bonus /10 total I / 100 I

1. (a) Determine whether the relation represents a function. Justify your answer. If it is a function, state the domain and range. (3 points)

6

-----­6

9

12 --1----/ ­ - - 12

9

'(-ts bw. ~J.~ .'1\ -ttu ~i't\ i<, ~UOcA_~ witt. b,:~ ~ tA~ i~ ~~.

'Dtl~~ -::: f ~,C.j'l l'l} / R~ = ~ 'Sf (." 1, 12 J (b) Determine whether the relation represents a function.

Justify your answer. If it is a function, state the domain and range. (3 points)

{(red, Jim), (blue, Nancy), (green, Dan), (blue, Will)}

tva ~~ ~ b(vt ;k ~ ~V\ i<;. ~U W~ :J... ~f.fvw o"'~tcJ3 .,w-. '-44 ~.

(c) Determine whether the the equation defines y as a function of x. Justify your answer. (2 points)

J

,JIt-'1 3 7;-1 -til J -~

3 =1- (xl

~ ::: 6"(~bd'

2. Let f(x) be the piecewise defined function

-Ix-II + 1 if - 3 < x < 1

f (x) = 4 if x = 1{

(x - 1)2 + 1 if 1 < x < 4

(a) Find f( -2). (1 point)

- \ -~-\ \ -\- \ ::. - 1- 3\ +r = - ~ ( =~ (b) Find f(3). (1 point)

(3- \ ) 'l- +I = 2!L t ( ~ \( t-I ~~ (c) Find the domain of f(x). (2 points)

(d) Sketch the graph of f (x). (4 points)

I - 10 - - -,­r ­

I I I I I

_J__ I

~ Y' ;-+-I ,

-~ -fI

I

I

t -5­

3. The graph of f is given below. Use the graph of f to graph the function G(x) = 2f( -x) - 3.

r)srvtr- -z, - 4

w"J~wJ- j GCy,'r:- 2+

J"') ­ ~ - -3 .

1

-4 -2 0 2 4 6 8 10 x

10· . r -I! I I

1_ - y -5- - I I

I I I I

-10 .5 10 I x i

I II

f 5· ­- - t --4- --I

I I I

I Ii I

~ I Il J- -10­

4. The graph of f is given below. Use the graph of f to graph the function H(x) = - f(x + 3) + 2.

\-Ux~-_-+(Kt-3\t L! \ \"fL

tt~,t1\ C)I\ \tH"t,

o.~ x-M.\~

.. . - 0

f(x) 1

! .. ­ ~. -&.- ~ II

I ­ . - 6 ­

I t-

r I Y

I Ir- L- 4t

-3 -'2

I I 1 I I

-10

I ---- - r

I

I I 1

J

":1 0 2 3 x

I - 10- ---r l\ I If I I I I- -y- - 5- - ­I I

- ::--r---­

5 10 i x I II

I ---5­ . - I --- ­-Ir

I

1-10­

":2

I

~-5 -

I I ~ - -10

1 I ,

I

5. Determine the function whose graph is given. Be sure to write the function in t he form f (x) = ax2 + bx + c.

-1-I'lertex~ 8) -I- 10(-4-- - II I ., I I II I I I I

I I ,

---1-I)

-I-~~-I

I I

I I x I

~ ~-+- -~ -- ~~ I i I :

I I I 1 -'- ~ ~

f()( ):: tt CX .p.(\1. -+ <B .N~.l =:--i {xL+8kf{l,)fg

-+{O)~O ~ ~(D)-= alO~\'(+~ =-0 - - J. ~t -tt>c -;(+.Y...... 2­

(.{ ({l91 ~ =0

n- _-L ""t- Z

~K\::- ~(K+4)2.~

6. Graph the polynomial function which has the following proper­ties: x-intercepts:

y-intercept: end behavior: positive: negative:

x = -2, multiplicity = 4 x = 3, multiplicity = 2 x = 6, multiplicity = 3 y= -3

9y = x(6 ,00) (- 00, -2) U (-2, 3) U (3,6)

-10- - - ­-1 - -lr r I

,I I ,I

-1 I

I, -~ y 5· -, I

1 I I !

-10 10

I I I

-5- - ­~ ~ I

- -10­ - - ­

I I I I I I l_~ J

7. Solve each inequality.

(a) IF < 81x4 (4 points)

~-~ + Eo

+ o ~ ~h(t.i - ( ~

'2..-o ""::. ~h/L . ( (.:,

3

:: (,e·-'i)(, xLf~)

::: (3}_i))x-rz)(q x1.--P1 )

t 2. \,

K=- !:. t:: - 3 t\Al ~\vn<M ~ 3 ~

"-'"'* ~I I\\u \t ~ t (b) ~~=: < 3 (4 points)

~ + I r + '3

5 Kt'-1 3("2.)(-' (,) (.. _ _ -0 &)c. - (a 2-~-~ X-::.o : ~t\' ~j

~X:~ -({ox - l~) '­p ­ _0 leo~ -fJ,. ~At1~

z,X-- b

(31

;;td-Jt;x.+'4 -Co(t-l<i __- - .. ~o ~~

vx- to-=-o 1 ~~3 MsJ \+-'t1

8. Let f(x) be given below. Find the following.

f(x) = 6x4 - 5x3 + 5x2

- 7x - 18

(a) The maximum number of real roots of f. (1 point)

(b) The number of possible positive and negative zeros of f. (3 points)

ptn>'r\W(. : 3 or I

-Pi-x) = ,,(->c)'i - es f:>.)'\ +S" (_>c)l - =t (->c)-If

-= ~ '( "f +5'('l -r- S'Ie ,. ,.... ~- (~

(c) All the possible rational zeros of f. (4 points)

It 2, 3, c.... '" '8

I", ~~ C.

- I I \ , ~ ~, ~ 2. 2. .i .J., 1 ,I"'if~'b' ,'" )',(;,,' 2..' ~ c,.'

9. Let f(x) be the polynomial function below. Find all the real zeros of f (x) and use them to factor f over the reals.

j(x) = X4 - x 3 + 2X2 - 4x - 8

~ -=-- '-I

PO'); ~~ -=- ?> or ,

.f(-x) ~ '(4. ... ,? +2>:2- -+ 'f)r. ~ g

Ttl.,. -=-1

I, <-I '1 (~ - -- -:::.l, 2, '1,1 I ~ .tl

----- ­

..r(l,}~V

~ 1-( 2­

J, 2.. L

'1 LEI

3(-'):::.{)

I I '1 ~~ i -- \ \:)

0 ~ @=]' 1

+Cx):: ex -"L)C x.} ~ (xl-t-tt)

~s: x::: '2.,-'

10. Let f(x) be the polynomial function given below. Use the given zero to find the remaining zeros. Then use that to write f in factored form.

f(x) = x4 - 7x3 + 14x2 + 2x - 20 zero: 3 + i

~~: ~ - ~

+(~)= (x - [3 -,-)1 [ x-{}tt)l ( ?) :: (X-3-t"l')('( - 3- t)(?)

=(y,>-- 3>< -r - >,x +h/t -t;j_I_~7.)(?) - (

::. G2_c,X HD)[? ')

')<'1 - 1x'3+I"tx~-t--2X- 2o ::: (~2..-fo)ctl . (7) -

-2fr.~ t 1~ --7f (! ~2-~(~7\ ~io

o -f (~\:: (I( -- 3f\J) ex -3 -"(, )(~~ ... )(-1-) ~_ C)o..-1-\i,)('~-~-")(l(-"2)()(-H) j Ttfos : 3-t1.) hj - 'J~l

11. Let f (x) be the quadratic function given below. Answer the following questions about f(x). Justify all your answers.

f(x) = 2X2 - 4x

(a) Does the graph of f open up or down? How do you know? (1 point)

(b) What is the vertex (h, k) of f? (1 point)

l., -\..( ­X~ - ~:: - ;l.("2.) - ,

(c) What are the intercepts of f? (2 points)

"l( - ', t\.t-', y==0 '! -11\+ '. 'X= 0

r= 2(0)2 - l( (0)0':: 2x2 -l{)c = 2)c (,( - 2-)

EJ

(d) What is the domain of f? (1 point)

(- oc ,co)

(e) What is the range of f? (1 point)

[-JI OO)

(f) What are the int ervals of increase and decrease of f? (2 points)

.,V\tc"~~ : ((, "., )

tkc~~ ~ (- 00, I)

(g) What does the graph of f look like? (2 points)

10 -- ­r- -r I I I -y----II

I I

-10 -'5 I

I I

-5­~ I I I I I I I I I I I

_IL -10­ -- - 1

T I I I I !

5- ­

10 x I I I I

12. Graph the rational funct ion which has the following properties:

Domain: lR\{-1,3, 5} Vertical asymptotes: . x = -1, multiplicity = 2

x = 3, mult iplicity = 1 Hole: x = 5 x-intercept: x = -4, multiplicity = 1 y-intercept: y = 2 Horizontal Asymptote: y =-2 positive: (-4, -1) U (-1,3) negative: (-00, -4) U (3,00)

1 - - f-

T I I I I I

t I

I -10 o .5 Ib

1 __ ­

---- - -1-! -- - -l-- -1 -------.

I I I

--L~-+-I I

I I I

I I I II

I - , ___1___­L 1 ---'

r I

Bonus. Find t he domain of t he function below.

f (x) = X4 - 2x3 - 7x 2 + 20x - 12

~() 'f l 'lJ '( ':. '( - 2~ ~~ +20x-I'2

POl)ffi-rl: 3 or \

T'\It : \ . fOS~i~~ hh{~ '. :il, t 2., ±~I ~YJ !:(./ :!: 11

.u I - 1 -1 'U> -no i I - \ - i 1'2..

-~ IL lQI*

~ (I<.~= ("'-I) V:'1." -;.'6 ~ .j-~ f()C)

p(~) -:: 0

;J I -( -<b 11

~ 1- 1.. -{2_

-~ \QJ

~(,)-:: (Y;_;)(l< _i)(~1 n.-fo)

:: (~. \)('K;1.)CXT~)(~-'i)

3(x)= (l<-M~-7.)"(x.-il

~') '. x-=-- t, Mil \t ::.\ '( ~~, fV\v\\- =-1

x-= -~ ,Mu \-t:;.'

x 3 + 5x2 + 8x + 4

hGc)~ ~; +Sx lt~~tY

fo\rn-.d '. 0

t'l~1\~ "' 3 or \

1'\'.1- . 'T\ "'"2 +"po~<) 'I"'1 I-n-t"~ . - 1- (_"'1

=!l I '5" 8 If

~ -\ -'1 ~ l4 \iL2l

h(~ \= (}.+\)(./-ttfk f4)

-=- (X4- i)( X. ~~y"~4 z,)

h(X.)-=- (}r.~') (~-t'2-)7.

Loo~1 -hr -P05·'''~ .

f-"'~LJV(-2, -I) V (\/ ~)

1