J

Date _____

Math Department

Review Packet - Algebra II

I Quadratics Equations

I Solve each of the following equations using factoring

a x2 -36=0 b 7x2 -I4x=O

c x3 -6x2 -7x=O d 6x2 +7x - 3 =0

e 3x2 +3x-36=0 f 32x2 -2=0

g x3 -2x2 -9x+18=0 h x 3 - 3x2 +6x -18 = 0

Quadratics Equations continued)

Factor each middot or nomial GOM~lErELY

Sum of Two Cubes la3 +b3 =(a+b(a2 -ab+b2 )j

Difference of Two Cubes ja3 b3 =(a--h)(a2 +ab+b2 )j

a x3 +27 b 8x3 -125

c x 4 +5x2 -14 d 2x5 -18x3 +40x

Solve each ofmiddotthe followina emiddotq_uations using the Quadratic FormuI~

I -b+ b -4ac IQuadratic Formulax - a2

a 4x2 +6x+l= 0 b x 2 +2x+2=0

2x2 d 3x2 -2x-7 =0c + 3x - 5 = 0

2 of 11

II Powers Roots and Radicals

Rewrite the ex ression with Evaluate where osslble

a (-3) -4 b 4 c 3x3 (2x) 2

XO +7

f 2o(a- 4 b- 2 )

s(a-2b4 t2

I

a x 5 = 2

I

d 2(x2) 4 -3=159

b 2J3x-1 +3 = 11

e J2x+4 =Jx+2

c 4x2 =64

f Vx-6=-2

3 of 11

III Simplifying Rational Expressions

I Sirrip1iflttbe RatibnBImiddotEiigtressii)niusj[ig11MilltiplicationorJ)ivisiampn bull f~ ~t~ middot I x 2 +4x-12 2a 6

x2(x2 +9x+18)deg x

3x2 -12 1b

5x-10 2x+4

x 2 -4 x+2 c x 2+4 x-2

5x2 -20 xd

25x2 x-2

f

x3 +3x2 x 2 +5x+6 g 5x32x

x 2 +x-20 llx+55h x+l x+l

x 2 +5x+6 x 2 -4i

x+3 x+l

x 2 +6x-7 x+7x2 +x-30 middot__x_ j 3x2

e x2 +6x bull 6x

4 of 11

Simplifying Rational Expressions (continued)

ISiimplify the RationaI Expression us1ngAddition or Subtraction LCD=)

a 4 2 -+shy3x2 5x

b 3 x+l --+-shy2x-2 4

c 4 x -+--shy3x3 6x3+3x2

d 5x-l 6

x 2 +2x-8 x+4

e x+l 1

x 2 +6x+9 x 2 -9

5 of 11

uatfon

IV Solving Rational Equations

ISolve each tatlorial eq

a 3 9 --=-shyx+4 x-2

I

b 4x x x-I - x2 -1

c 3 2 x --=--+-shyx2 4 x+2 x-2

d 3x-2 6 --- 2 +1x-2 x -4

e x 3x+I 4--=--+--shyi+2 x-I x 2 +x-2

6 of 11

i-l O V Function Operations

Pfrforn the indicated operation with the f11ncti9ns givenbull -

Let f(x)=x 2 -3x+4 g(x)=5x+2 and h(x)=6x

b (J-h)(x)=a (J + g)(x)= c (g middoth)(x) =

f (g middot f)O)=d (J +h)(-2)= e (h-g )(3)=

g (f og)(x) = h (f oh)(x) = i (g O f)(l) = ~GZf middotbCA f-s~

j (foh)(-7)= I f (g(h(-1))) =k g(h(J(x))) = (A stAp5 )s0p

7 of 11

Quadratics Equations continued)

Factor each middot or nomial GOM~lErELY

Sum of Two Cubes la3 +b3 =(a+b(a2 -ab+b2 )j

Difference of Two Cubes ja3 b3 =(a--h)(a2 +ab+b2 )j

a x3 +27 b 8x3 -125

c x 4 +5x2 -14 d 2x5 -18x3 +40x

Solve each ofmiddotthe followina emiddotq_uations using the Quadratic FormuI~

I -b+ b -4ac IQuadratic Formulax - a2

a 4x2 +6x+l= 0 b x 2 +2x+2=0

2x2 d 3x2 -2x-7 =0c + 3x - 5 = 0

2 of 11

II Powers Roots and Radicals

Rewrite the ex ression with Evaluate where osslble

a (-3) -4 b 4 c 3x3 (2x) 2

XO +7

f 2o(a- 4 b- 2 )

s(a-2b4 t2

I

a x 5 = 2

I

d 2(x2) 4 -3=159

b 2J3x-1 +3 = 11

e J2x+4 =Jx+2

c 4x2 =64

f Vx-6=-2

3 of 11

III Simplifying Rational Expressions

I Sirrip1iflttbe RatibnBImiddotEiigtressii)niusj[ig11MilltiplicationorJ)ivisiampn bull f~ ~t~ middot I x 2 +4x-12 2a 6

x2(x2 +9x+18)deg x

3x2 -12 1b

5x-10 2x+4

x 2 -4 x+2 c x 2+4 x-2

5x2 -20 xd

25x2 x-2

f

x3 +3x2 x 2 +5x+6 g 5x32x

x 2 +x-20 llx+55h x+l x+l

x 2 +5x+6 x 2 -4i

x+3 x+l

x 2 +6x-7 x+7x2 +x-30 middot__x_ j 3x2

e x2 +6x bull 6x

4 of 11

Simplifying Rational Expressions (continued)

ISiimplify the RationaI Expression us1ngAddition or Subtraction LCD=)

a 4 2 -+shy3x2 5x

b 3 x+l --+-shy2x-2 4

c 4 x -+--shy3x3 6x3+3x2

d 5x-l 6

x 2 +2x-8 x+4

e x+l 1

x 2 +6x+9 x 2 -9

5 of 11

uatfon

IV Solving Rational Equations

ISolve each tatlorial eq

a 3 9 --=-shyx+4 x-2

I

b 4x x x-I - x2 -1

c 3 2 x --=--+-shyx2 4 x+2 x-2

d 3x-2 6 --- 2 +1x-2 x -4

e x 3x+I 4--=--+--shyi+2 x-I x 2 +x-2

6 of 11

i-l O V Function Operations

Pfrforn the indicated operation with the f11ncti9ns givenbull -

Let f(x)=x 2 -3x+4 g(x)=5x+2 and h(x)=6x

b (J-h)(x)=a (J + g)(x)= c (g middoth)(x) =

f (g middot f)O)=d (J +h)(-2)= e (h-g )(3)=

g (f og)(x) = h (f oh)(x) = i (g O f)(l) = ~GZf middotbCA f-s~

j (foh)(-7)= I f (g(h(-1))) =k g(h(J(x))) = (A stAp5 )s0p

7 of 11

II Powers Roots and Radicals

Rewrite the ex ression with Evaluate where osslble

a (-3) -4 b 4 c 3x3 (2x) 2

XO +7

f 2o(a- 4 b- 2 )

s(a-2b4 t2

I

a x 5 = 2

I

d 2(x2) 4 -3=159

b 2J3x-1 +3 = 11

e J2x+4 =Jx+2

c 4x2 =64

f Vx-6=-2

3 of 11

III Simplifying Rational Expressions

I Sirrip1iflttbe RatibnBImiddotEiigtressii)niusj[ig11MilltiplicationorJ)ivisiampn bull f~ ~t~ middot I x 2 +4x-12 2a 6

x2(x2 +9x+18)deg x

3x2 -12 1b

5x-10 2x+4

x 2 -4 x+2 c x 2+4 x-2

5x2 -20 xd

25x2 x-2

f

x3 +3x2 x 2 +5x+6 g 5x32x

x 2 +x-20 llx+55h x+l x+l

x 2 +5x+6 x 2 -4i

x+3 x+l

x 2 +6x-7 x+7x2 +x-30 middot__x_ j 3x2

e x2 +6x bull 6x

4 of 11

Simplifying Rational Expressions (continued)

ISiimplify the RationaI Expression us1ngAddition or Subtraction LCD=)

a 4 2 -+shy3x2 5x

b 3 x+l --+-shy2x-2 4

c 4 x -+--shy3x3 6x3+3x2

d 5x-l 6

x 2 +2x-8 x+4

e x+l 1

x 2 +6x+9 x 2 -9

5 of 11

uatfon

IV Solving Rational Equations

ISolve each tatlorial eq

a 3 9 --=-shyx+4 x-2

I

b 4x x x-I - x2 -1

c 3 2 x --=--+-shyx2 4 x+2 x-2

d 3x-2 6 --- 2 +1x-2 x -4

e x 3x+I 4--=--+--shyi+2 x-I x 2 +x-2

6 of 11

i-l O V Function Operations

Pfrforn the indicated operation with the f11ncti9ns givenbull -

Let f(x)=x 2 -3x+4 g(x)=5x+2 and h(x)=6x

b (J-h)(x)=a (J + g)(x)= c (g middoth)(x) =

f (g middot f)O)=d (J +h)(-2)= e (h-g )(3)=

g (f og)(x) = h (f oh)(x) = i (g O f)(l) = ~GZf middotbCA f-s~

j (foh)(-7)= I f (g(h(-1))) =k g(h(J(x))) = (A stAp5 )s0p

7 of 11

III Simplifying Rational Expressions

I Sirrip1iflttbe RatibnBImiddotEiigtressii)niusj[ig11MilltiplicationorJ)ivisiampn bull f~ ~t~ middot I x 2 +4x-12 2a 6

x2(x2 +9x+18)deg x

3x2 -12 1b

5x-10 2x+4

x 2 -4 x+2 c x 2+4 x-2

5x2 -20 xd

25x2 x-2

f

x3 +3x2 x 2 +5x+6 g 5x32x

x 2 +x-20 llx+55h x+l x+l

x 2 +5x+6 x 2 -4i

x+3 x+l

x 2 +6x-7 x+7x2 +x-30 middot__x_ j 3x2

e x2 +6x bull 6x

4 of 11

Simplifying Rational Expressions (continued)

ISiimplify the RationaI Expression us1ngAddition or Subtraction LCD=)

a 4 2 -+shy3x2 5x

b 3 x+l --+-shy2x-2 4

c 4 x -+--shy3x3 6x3+3x2

d 5x-l 6

x 2 +2x-8 x+4

e x+l 1

x 2 +6x+9 x 2 -9

5 of 11

uatfon

IV Solving Rational Equations

ISolve each tatlorial eq

a 3 9 --=-shyx+4 x-2

I

b 4x x x-I - x2 -1

c 3 2 x --=--+-shyx2 4 x+2 x-2

d 3x-2 6 --- 2 +1x-2 x -4

e x 3x+I 4--=--+--shyi+2 x-I x 2 +x-2

6 of 11

i-l O V Function Operations

Pfrforn the indicated operation with the f11ncti9ns givenbull -

Let f(x)=x 2 -3x+4 g(x)=5x+2 and h(x)=6x

b (J-h)(x)=a (J + g)(x)= c (g middoth)(x) =

f (g middot f)O)=d (J +h)(-2)= e (h-g )(3)=

g (f og)(x) = h (f oh)(x) = i (g O f)(l) = ~GZf middotbCA f-s~

j (foh)(-7)= I f (g(h(-1))) =k g(h(J(x))) = (A stAp5 )s0p

7 of 11

Simplifying Rational Expressions (continued)

ISiimplify the RationaI Expression us1ngAddition or Subtraction LCD=)

a 4 2 -+shy3x2 5x

b 3 x+l --+-shy2x-2 4

c 4 x -+--shy3x3 6x3+3x2

d 5x-l 6

x 2 +2x-8 x+4

e x+l 1

x 2 +6x+9 x 2 -9

5 of 11

uatfon

IV Solving Rational Equations

ISolve each tatlorial eq

a 3 9 --=-shyx+4 x-2

I

b 4x x x-I - x2 -1

c 3 2 x --=--+-shyx2 4 x+2 x-2

d 3x-2 6 --- 2 +1x-2 x -4

e x 3x+I 4--=--+--shyi+2 x-I x 2 +x-2

6 of 11

i-l O V Function Operations

Pfrforn the indicated operation with the f11ncti9ns givenbull -

Let f(x)=x 2 -3x+4 g(x)=5x+2 and h(x)=6x

b (J-h)(x)=a (J + g)(x)= c (g middoth)(x) =

f (g middot f)O)=d (J +h)(-2)= e (h-g )(3)=

g (f og)(x) = h (f oh)(x) = i (g O f)(l) = ~GZf middotbCA f-s~

j (foh)(-7)= I f (g(h(-1))) =k g(h(J(x))) = (A stAp5 )s0p

7 of 11

uatfon

IV Solving Rational Equations

ISolve each tatlorial eq

a 3 9 --=-shyx+4 x-2

I

b 4x x x-I - x2 -1

c 3 2 x --=--+-shyx2 4 x+2 x-2

d 3x-2 6 --- 2 +1x-2 x -4

e x 3x+I 4--=--+--shyi+2 x-I x 2 +x-2

6 of 11

i-l O V Function Operations

Pfrforn the indicated operation with the f11ncti9ns givenbull -

Let f(x)=x 2 -3x+4 g(x)=5x+2 and h(x)=6x

b (J-h)(x)=a (J + g)(x)= c (g middoth)(x) =

f (g middot f)O)=d (J +h)(-2)= e (h-g )(3)=

g (f og)(x) = h (f oh)(x) = i (g O f)(l) = ~GZf middotbCA f-s~

j (foh)(-7)= I f (g(h(-1))) =k g(h(J(x))) = (A stAp5 )s0p

7 of 11

i-l O V Function Operations

Pfrforn the indicated operation with the f11ncti9ns givenbull -

Let f(x)=x 2 -3x+4 g(x)=5x+2 and h(x)=6x

b (J-h)(x)=a (J + g)(x)= c (g middoth)(x) =

f (g middot f)O)=d (J +h)(-2)= e (h-g )(3)=

g (f og)(x) = h (f oh)(x) = i (g O f)(l) = ~GZf middotbCA f-s~

j (foh)(-7)= I f (g(h(-1))) =k g(h(J(x))) = (A stAp5 )s0p

7 of 11

--- -

VI Inverses

--ltmiddot middot- 1 v iampelti ea middotfgftefi 11Fifld the tl lUL J -middot- Q

a J(x)= 2x+5

c J(x)=5-~x2

bull bullmiddot I~ bull

b J(x)= V2x+4

x-2d ( )=shyf x

4

a J(x)=x+7 g(x)=x-1

y

1-i--t-+middotmiddot--shy - - - middot--r---middot shy

shy ~ shy - - - 1shy - - -shy

______ - -shy - -+--+-t-+---l

__--+--+--+--1---- -- -shy -

x

c J(x)=__x2 g(x) = J3x

3

1b J(x)==-x+1 g(x)=2x-2

2

y

d xs+2 ~-shyJ(x)= g(x)=V7x-2

7

1-shy

gt--shy

-

middotmiddotmiddotshy

+-shy

t

+--shy

--arshy

+shy+-shy

middotshy

+--l

+-shy+-r

- 1--middot 1

---+-middotshy

-+-middotmiddotmiddot

~

~-shy

-

-

shy

-shy

---shy -- +-gt--- x

The graph of the inverse function is the reflection of the original function over what line

8 of 11

bull

middot VII Exponential amp Logarithmic Functions - - - -

Selve each eauation - - - shy- middot- - middotshy~

y =logb x if and only if x = bY

Think of y =logb x as the answer to To what power must b be raised to obtain x

ca b 53x = 25x-1 4 x = 025 (1r =27

d IOX =15

g llog 64 =shy

x 2

j log7 (2x +5) =log7 (x-3)

rn 256e 2x = 1400

3e e x = 24

h log3 x =5

k log2 (2x 2 )= 5

n 75 =21 ( 105)

f ln3x = -05003

i log4 256 = x

log x =2096910013

o 10x2+ 3x - 7 =1000 =f ()3

9 of 11

Exponential amp Logarithmic Functions (continued)

a 21ogx-xlogy

c logx+21ogy

e 1Iogx+-logy-2logz

2

g logx+logy+logz-2logw

d log(x2 + t) z

f x2 logshy

z6

10 of 11

Exponential amp Logarithmic Functions (continued)

Use the eQuation laquoi iven and the pmiddotroperties of logs to solve the problems below where

(1) A= p (1 + n) nt bull P = original amount deposited or the initial investment bull r = the interest rate expressed as a decimal (5 7 005) bull n = the number of times a year the interest is paid

(2) A= Pe rr (quarterly 7 means n = 4) bull t = the number of years the investment spans

a Find the value of a $1000 investment b If you invest $30000 at 4 76 interest at 6 interest after 10 years paid quarterly how long would it take compounded you to double your money Round your

answer to the nearest hundredth (a) annually

(b) quarterly

(c) monthly

(d) continuously

c Suppose $2000 is invested in a 3shy d You invest $200 at 1225 earning year certificate of deposit (CD) that continuous interest How many years earns 6 interest compounded does it take for your money to increase 5 continuously How much will the times its original value Round your investment be worth after 3 years answer to the nearest tenth

11 of 11

bull

middot VII Exponential amp Logarithmic Functions - - - -

Selve each eauation - - - shy- middot- - middotshy~

y =logb x if and only if x = bY

Think of y =logb x as the answer to To what power must b be raised to obtain x

ca b 53x = 25x-1 4 x = 025 (1r =27

d IOX =15

g llog 64 =shy

x 2

j log7 (2x +5) =log7 (x-3)

rn 256e 2x = 1400

3e e x = 24

h log3 x =5

k log2 (2x 2 )= 5

n 75 =21 ( 105)

f ln3x = -05003

i log4 256 = x

log x =2096910013

o 10x2+ 3x - 7 =1000 =f ()3

9 of 11

Exponential amp Logarithmic Functions (continued)

a 21ogx-xlogy

c logx+21ogy

e 1Iogx+-logy-2logz

2

g logx+logy+logz-2logw

d log(x2 + t) z

f x2 logshy

z6

10 of 11

Exponential amp Logarithmic Functions (continued)

Use the eQuation laquoi iven and the pmiddotroperties of logs to solve the problems below where

(1) A= p (1 + n) nt bull P = original amount deposited or the initial investment bull r = the interest rate expressed as a decimal (5 7 005) bull n = the number of times a year the interest is paid

(2) A= Pe rr (quarterly 7 means n = 4) bull t = the number of years the investment spans

a Find the value of a $1000 investment b If you invest $30000 at 4 76 interest at 6 interest after 10 years paid quarterly how long would it take compounded you to double your money Round your

answer to the nearest hundredth (a) annually

(b) quarterly

(c) monthly

(d) continuously

c Suppose $2000 is invested in a 3shy d You invest $200 at 1225 earning year certificate of deposit (CD) that continuous interest How many years earns 6 interest compounded does it take for your money to increase 5 continuously How much will the times its original value Round your investment be worth after 3 years answer to the nearest tenth

11 of 11

Exponential amp Logarithmic Functions (continued)

a 21ogx-xlogy

c logx+21ogy

e 1Iogx+-logy-2logz

2

g logx+logy+logz-2logw

d log(x2 + t) z

f x2 logshy

z6

10 of 11

Exponential amp Logarithmic Functions (continued)

Use the eQuation laquoi iven and the pmiddotroperties of logs to solve the problems below where

(1) A= p (1 + n) nt bull P = original amount deposited or the initial investment bull r = the interest rate expressed as a decimal (5 7 005) bull n = the number of times a year the interest is paid

(2) A= Pe rr (quarterly 7 means n = 4) bull t = the number of years the investment spans

a Find the value of a $1000 investment b If you invest $30000 at 4 76 interest at 6 interest after 10 years paid quarterly how long would it take compounded you to double your money Round your

answer to the nearest hundredth (a) annually

(b) quarterly

(c) monthly

(d) continuously

c Suppose $2000 is invested in a 3shy d You invest $200 at 1225 earning year certificate of deposit (CD) that continuous interest How many years earns 6 interest compounded does it take for your money to increase 5 continuously How much will the times its original value Round your investment be worth after 3 years answer to the nearest tenth

11 of 11

Exponential amp Logarithmic Functions (continued)

Use the eQuation laquoi iven and the pmiddotroperties of logs to solve the problems below where

(1) A= p (1 + n) nt bull P = original amount deposited or the initial investment bull r = the interest rate expressed as a decimal (5 7 005) bull n = the number of times a year the interest is paid

(2) A= Pe rr (quarterly 7 means n = 4) bull t = the number of years the investment spans

a Find the value of a $1000 investment b If you invest $30000 at 4 76 interest at 6 interest after 10 years paid quarterly how long would it take compounded you to double your money Round your

answer to the nearest hundredth (a) annually

(b) quarterly

(c) monthly

(d) continuously

c Suppose $2000 is invested in a 3shy d You invest $200 at 1225 earning year certificate of deposit (CD) that continuous interest How many years earns 6 interest compounded does it take for your money to increase 5 continuously How much will the times its original value Round your investment be worth after 3 years answer to the nearest tenth

11 of 11