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Name: ___________________________________ Grade: _____

• You are expected to know all topics from Algebra 2 and trigonometry from Geometry. • This packet is optional due on 9/6. It will be worth 4 classwork grades in September.

• The answer keys will be on the school website & IB website. • You will have several tests on these topics in the first marking period. • If you have trouble reviewing the topics over the summer, use resources such as Khan

academy, IXL, and youtube.

You will be given homework every class and the homework will be checked every class in IB Extended Math.

There is no credit for late homework and it DOES AFFECT your grade!

The topics of IB extended Math are: 1) Reviewing and extending algebra 2 topics (3 months) 2) Trigonometry (3 months) 3) Probabilities and statistics (2 weeks) 4) Sequences and series (2 weeks) 5) Exponentials and logarithms (1 month) 6) Polar coordinates (2 weeks)

Algebra 2 is crucial for you to be successful in the upcoming advanced math classes including IB Extended Math and SAT/ACT tests. If you don’t think your algebra 2 skills are strong, please spend time over the summer. You will be expected to understand everything in this packet since this is a pure algebra review.

Table of Contents:

Topic 1 – factoring & solving quadratics

Topic 2 –Rational Expressions and Equations

Topic 3 - Simplifying Radicals and Complex Numbers

Topic 4 - Solving Equations and Inequalities

Topic 5- Graphs of Functions

Topic 6 - Other topics in algebra 2

Topic 7 - Trigonometry

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Algebra 2 Review Topic 1: Factoring and Solving Quadratics 7 Factoring Rules Rule #1 - GCF 1) 2 5 3 6 2 3 63 9 12a bc a bc a b c− + Rule #2 – Difference of Perfect Squares 2)

9x 2 − 49y 2 3) 2100 100x − Rule #3 – Trinomial with 1 as a Leading Coefficient 4) 2 20a a+ − 5)

3x 2 −18x + 24 Rule #4 – Sum/Difference of Perfect Cubes

6) 3 3125x y− 7) 3 364 8a y+ Rule #5 – Perfect Square Trinomials 8)

9x 2 + 30x + 25 9)

36y 2 − 84 y + 49 Rule #6 – Factoring by Grouping 10)

a2x − b2x + a2y − b2y 11) 3 24 4 6 6x x x+ − − Rule #7 – Trinomial with Leading Coefficient >1 12) 3y2 + 5y + 2 13) 26 5 6x x− − What if it can’t be factored?

14) 2 225x y+

3

Factoring Review How do I know which factoring rule to use?

Factoring Mixed Practice 15) 22 8 3 12x x x+ + + 16) 22 7 15x x− − 17)

27x 3 + 8 18) 23 5 2x x− −

4

Solving Quadratics What are the different ways to solve quadratics? #1 – Solve by Factoring 19) 2 8 9 0w w− − = 20)

225 36 0p − = 21)

3x 2 = 16x − 5 #2 – Solve by Graphing 22) #3 – Solve by Completing the Square

23) 2 12 36p p− +

#4 – Solve by the Quadratic Formula

24) 2 6 21 0x x− + = 25) 22 1 1x x x+ + = − −

2 42

b b acxa

− ± −=

5

How do I know which way to solve? Try to solve by factoring first, if you can’t solve by factoring use the quadratic formula or solve by completing the square. Solving Quadratics Mixed Practice 26) 2 4 3x x+ = 27) 22 5 3x x+ = 28) 2 5x x= 29) 2 3 40 0x x+ − = 30) 23 5 2 0x x− + = 31) Describe the Nature of the Roots of a Quadratic

A) 2 Real Rational Roots B) 1 Real Double Root C) 2 Real Irrational Roots D) 2 Imaginary Roots

What are the nature of the roots of the following quadratics with roots? 32) �± 1

3� 33) �± 1

3𝑖𝑖� 34) {0,−5}

35) �2 ± √5� 36) 37)

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E X T R A N O T E S A N D E X A M P L E S : Factoring Examples: Rule 2 Difference of Perfect Squares

Rule 3 Trinomial w/Leading Coeff.=1

Rule 4 Sum/Difference of Perfect Cubes

Rule 5 Perfect Square Trinomials

Quadratic Example: To solve a quadratic equation, you may be asked to find the solutions, zeros, or roots. These answers will also be found on a graph (called a parabola) as x-intercepts. Note: A quadratic equation can have two solutions, one solution (a double root-touches the x-axis and turns around) or no real solutions (graph does not cross the x-axis). Solving by Factoring:

1) Get the equation equal to zero. Move everything to left side. 2) Factor the left side using an appropriate technique we have learned. 3) Set each factor = 0 and solve.

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M O R E P R A C T I C E A :

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M O R E P R A C T I C E B :

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Algebra 2 Review Topic 2: Rational Expressions and Equations Simplifying Rational Expressions Factor and Reduce!

1)

2

2

2 3 22 8

x xx+ −

− Multiplying and Dividing Rational Expressions

2) x2 − 11 + x

÷2x2 − 5x + 3

3 − 2x 3) Adding and Subtracting Rational Expressions YOU NEED A COMMON DENOMINATOR WHENEVER YOU ARE ADDING OR SUBTRACTING FRACTIONS! Do not cancel on top and bottom! Get a common denominator & then add or subtract the numerators.

4) 8 4

2 2x x−

− + 5) 2 22 6c d d c

cd c d+ −

+

Solving Rational Equations Multiply both sides of the equation by the LCD then cancel the fractions and simplify.

6) 1 5 1

3( 2) 6 2x xx x+

= +− − 7) 2

4 22 2 4

uu u u

− =+ − −

Mixed Practice Simplifying and Adding and Subtracting Rational Expressions

8) 2

7 4 12 6

xx x x

+− =

− + − 9)

2 2

2 2

7 12 7 106 20

x x x xx x x x

− + + +⋅

− − + −

10)

2 5 4 59 9x x

x x− −

+− − 11)

2 9 34 8

x x− −÷

6 12

2

10

5x

x

x

x+

+

10

E X T R A N O T E S A N D E X A M P L E S : Simplifying Rational Expressions 1.) Factor the numerator and denominator 2.) Divide out any common factors

Example: Multiplying and Dividing Rational Expressions 1.) Factor all numerators and denominators (GCF, Unfoil (trinomials), Diff. of Squares,etc) 2.) Divide out common factors (reduce). Example 1-Multiplying:

Ex.

2 2

2

2 24 ( 2)( 2) 2

x x x x xx x x x x x+ +

⋅ = ⋅ =− + − −

Example 2-Dividing: Change all division problems to multiplication- ‘flip and multiply’! Then follow steps above.

Ex. 6 4

4 4 4 6 6x x x x

x x x xx

x+ −

÷− − − +

= ⋅ =+

Adding/Subtracting with Unlike Denominators Multiply the numerator and denominator by what is missing from the factors of your LCD (also called the Least Common Denominator).

Ex LCD is 6. 3 1 23 2 3 2

x x+ ⋅ + ⋅ =

3 3 2 3 2 3 5 36 6 6 6

x x x x x+ + + ++ = =

Complex Fractions Find the LCD for each set of fractions. Then flip and multiply.

Ex.

1 1 3 1 33 1 3 1 35 5 4 5 4 5 4 5 44

x xx x xx x x x

x x x x xx x x x

++ + + +

= = = ⋅ =+ + ++ +

Rational Equations Step 1: Multiply each term of the equation by the LCD. Step 2: List the values that must be excluded from the solution. Step 3: Solve for the variable. Check your solution in the ORIGINAL equation!

Ex.

5 3 01 2p p

− =− + LCD is (p - 1)(p + 2)

( 1)( 2) ( 1)( 2) ( 1)( 2)5 3 0

1 25( 2) 3( 1) 05 10 3 3 0

2 13 13 / 2

p p p p p pp p

p pp p

p p

− + − + − +− =− +

+ − − =+ − + =

= − = −

2

2

4 3 ( 1)( 3) ( 1)12 ( 4)( 3) ( 4)

x x x x xx x x x x

+ + + + += =

− − − + −

12 3

x x++

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M O R E P R A C T I C E C :

M O R E P R A C T I C E D :

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Algebra 2 Review Topic 3: Simplifying Radicals and Complex Numbers Simplifying Radicals

1) 2 2 11200a b c 2)3 54 3)

4 5 1116a b 4) 3 4 9 118a b c− Operations with Radicals

5)3 372 50x y xy⋅ 6) (2 3)(2 3)− + 7) 3 3 324 81 3− +

8) 4

12 9) 18 8x x− 10)

23 2−

Rational Exponents Express the following in exponential form:

11)23 (3 )x 12) 7 13) 54 4a 14)

54 (4 )a Express the following in radical form:

15) 13r

− 17)

2 13 2a b

Powers of i 18) i = 2i = 3i = 4i = 5i = 6i = 7i = 8i = 19) 33i 20) 102i Imaginary Numbers

21) 50− 22) 3 3− ⋅ − 23)

22 − 3i

13

Operations with Complex Numbers 24) (2 )(4 3 )i i+ − 25) (1 5 ) (4 )i i i− − + + Solving Radical and Rational Exponent Equations

26) 13(5 5) 1 3a − + = 27) 1 2 1x − + = − 28)

51 7 1 25

x − − = −

Mixed Practice with Radicals and Complex Numbers SIMPLIFY OR SOLVE: 29)

42 4 12 1 3x− − + = − 30) x2 + 49 = 0 31) i93

32) −7 ⋅ −7 33)

4 944 16 p q 34)3 12 2424 p q r

35) ( )( )2 5 3 2 5+ − 36) 3 75 12 3 18 5 8− − + 37) ( )

343 2 1 25x − + =

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E X T R A N O T E S A N D E X A M P L E S : Simplifying Radicals

To simplify, use or break out into prime factors looking for the same repeated factors (2 or 3 or 4 of a kind—depending on the index).

Ex)5 4 19 4 4 19 3 4 4 34 4 432 16 2 2 2x y z x x y z z xyz x y z= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅

Simplifying Radical Expressions:

Ex) 3 43 3 3 38 8 8 16 2 2 8 4 8x x x x x x x x x x x+ + − = + − = − 3 43 8 8 16 16x x x x+ + − Multiplying: Simplify each first, then multiply.

Ex) 2 12 3 3 6 36 36⋅ = = Dividing Radicals: Answers cannot have radicals in the denominator. We need to rationalize the answer by multiplying the top and bottom by a radical that will ‘lift’ the root sign. If the denominator is a binomial you must multiply top/bottom by its conjugate.

Ex)5 5 5 5 5 5

55 5 5x x x x x x

xx x x= ⋅ = =

Radicals or i’s in the denominator: Multiply the numerator and the denominator by the conjugate.

Ex) 2

5 5 5 25 5 25 55 5 5 25 26

i i ii i i i

− − −= ⋅ = =

+ + − − Adding/Subtracting Radicals: Simplify then combine like terms.

Ex) 2 12 3 3 2 2 3 2 3 3− = ⋅ − =

Rational Exponents: Divide the exponents by the index: Examples of rewriting:

Ex) Complex Numbers Remember: If there is a negative (-) under an even root, pull it out as in i.

Ex)Cycle of powers of i:

n n nab a b= •

n n na b ab• =

ab abx x=

35 35x x=

a bi+

1 2 3 4, 1, , 1i i i i i i= = − = − =

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Algebra 2 Review Topic 4: Solving Equations and Inequalities Absolute Value Equations, Rational Equations, Radical Equations, Quadratics, Systems, Inequalities Absolute Value Equations

1) 3 2 7 2 17x − + = 2) 9 4 5 18x x+ = + Absolute Value Inequalities

3) 1 8 3 162

x − − ≤ 4) 3 2 1x − − ≤

Radical and Rational Exponent Equations

5) 12 5 1 13

x= − + 6)

132 2( 1) 4x− = − +

Rational Equations

7) 1 5

3 9 3 18x x

x− − =

8) 2

3 5 21 1

xx x

− =+ − 9) 2

1 3 12a a

− − =

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Systems Solve the system and state the number of solutions to the system.

11)

2 35

y xy x

= +

= + 10)

Quadratics Solve and describe the nature of the roots. 12) 13) 2 72x = − 14) 2 6 1x x− =

15) Find (0)r for 2( ) 1r x x= − . Mixed Practice

16) 2

1 18 7 1x

x x x= +

+ + + 17) 2 3 10 0x x+ − = 18) 3

14 1 84

x − = −

19)

3 2 34

x − ≥ 20)

2 1 13

xx

+=

21) 25 10 5 0x x− + =

2x 2 + 9x = −7

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E X T R A N O T E S A N D E X A M P L E S : Solving Absolute Value Equations:

Set up 2 cases and solve for both: Ex)

CHECK your solutions! Solving Absolute Value Inequalities LessthAND ( ) (graph is ‘in between’) versus GreatOR ( ) (graph opposite directions) Set up 2 cases—Remember to flip the sign when setting up the 2nd negative case!

Ex) 1: Ex) 2:

Solving Quadratic Equations: To solve a quadratic equation you may be asked to find the solutions, zeros, or roots. These answers will also be found on a graph (called a parabola) as x-intercepts. Note: A quadratic equation can have two solutions, one solution (a double root-touches the x-axis and turns around) or no real solutions (graph does not cross the x-axis). Solving by Factoring:

1) Get the equation equal to zero. Move everything to left side. 2) Factor the left side using an appropriate technique we have learned. 3) Set each factor = 0 and solve.

Ex) Solve for x:

Solving using the Quadratic Formula: Get the equation equal to zero. Move everything to left side. Find a, b, c and plug into the

formula: Don’t forget to simplify!

Ex)

2 5 3x − =

2 5 3 2 5 32 8 2 2

4 1

x xx x

x x

− = − = −= =

= =

, <≤ , >≥

2 5 3x − < 2 5 3x − >

2 5 3 2 5 32 8 2 2

4 11 4

x xx x

x and xx

− < − > −< >

< >< <

2 5 3 2 5 32 8 2 2

4 1

x xx x

x or x

− > − < −> <

> <

x 2 − 3x −10 = 0( 5)( 2) 0

5 0 2 0

5 2

x xx x

x x

− + =− = + =

= = −

2 42

b b acxa

− ± −=

2

2

: 3 5 9 0

( 5) ( 5) 4(3)(9)2(3)

5 83 5 836 6

Solve for x x x

i

− + =

− − ± − −=

± − ±= =

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E X T R A N O T E S A N D E X A M P L E S : Rational Equations Step 1: Multiply each term of the equation by the LCD. Step 2: List the values that must be excluded from the solution. These are values of the variable that make the denominator = 0 (these values make the equation undefined). Step 3: Solve for the variable. Check your solution in the ORIGINAL equation!

Ex) 2

5 3 03 4a a

− = LCD is 212a

2

2

12 20 3 01 4

20 3 020 3

320

aa a

aa

a

⋅ − =

− ==

=

Radical Equations Steps to solving radical equations: (1) get the radical on the side by itself (“isolate the radical”) (2) square (or cube, etc) both sides of the equation (3) solve for the variable (4) check for extraneous solutions

Ex) 2 3 1 10x− + = −

( ) ( )2 2

3 1 5

3 1 5

3 1 258

x

x

xx

+ =

+ =

+ ==

Non-linear systems Ex) Solve linear quadratic system: y = x2 - 4x - 2 and y = x – 2

1. Enter the first equation into Y1. 2. Enter the second equation into Y2. 3. Hit GRAPH. 4. Use the INTERSECT option twice to find the two locations where the graphs intersect (the answers). 2nd TRACE (CALC) #5 intersect Move spider close to the intersection. Hit ENTER 3 times. 5. Answer: (5,3) and (0,-2)

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Algebra 2 Reveiw Topic 5: Graphs of Functions Logs, Exponentials, Absolute Value, Quadratics, Higher Order Polynomials, Cube, Cube Root, Square Roots, Rational Equations (Increasing, Decreasing, Domain, Range, Transformations, Asymptotes, Inequalities) Recognizing Graphs of Functions What is the name of the function show in each graph below? What is the equation of the graph? 1) 2) 3) 4) 5) 6) 7) 8)

8) Which of the above graphs have a domain or all real numbers? 9) Which of the above functions have a range of all real numbers? 10) Which of the above functions have asymptotes? What are the equations of the asymptotes?

2 4 6–2 x

2

4

6

–2

y

2 4–2–4 x

2

4

6

8

–2

y

2 4–2–4 x

2

4

–2

–4

y

2 4–2–4 x

2

4

6

y

2 4–2–4 x

2

4

6

y

2 4 6 x

2

4

6

y

2 4–2–4 x

2

4

–2

–4

y

2 4–2–4 x

2

4

–2

–4

y

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Transformation Equations For each of the following, name the function and the vertex (or pivot point). Then give the equation of the function after it has been shifted right by three and down 2.

11) 22( 3) 5y x= − − 12) 3( 1)y x= − 13) 6y x= − Name: _____________ Name: _____________ Name: _____________

Vertex: _____________ Vertex: _____________ Vertex: _____________

Translated Equation: Translated Equation: Translated Equation:

______________y = ______________y = ______________y =

14) 4 15y x= + + 15) log( 1) 7y x= + − 16) 3 1xy = −

Name: _____________ Name: _____________ Name: _____________

Vertex: _____________ Vertex: _____________ Vertex: _____________

Translated Equation: Translated Equation: Translated Equation:

______________y = ______________y = ______________y =

Domain, Range, Increasing Decreasing For each of the following, determine the domain, range, intervals to which the function is increasing and decreasing, is sign of the leading coefficient and the end behavior. 17) 18)

Domain: __________________ Domain: __________________

Range: ___________________ Range: ___________________

Increasing: ________________ Increasing: ________________

Decreasing: _______________ Decreasing: _______________

As , ( ) ____x f x→ ∞ → As , ( ) ____x f x→ ∞ →

As , ( ) ____x f x→ −∞ → As , ( ) ____x f x→ −∞ →

Leading Coefficient: __________ Leading Coefficient: _________

Factors: ________________

Possible Equation: _________________

2 4–2–4 x

2

4

6

–2

–4

–6

y

2 4 6 8–2 x

2

4

6

8

–2

y

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Asymptotes Find all asymptotes of the following functions.

19) log( 5)y x= − 20) 4 1xy = − 21) 4 1xy

x=

+

22) 2

13 3 18

yx x

=+ − 23)

42 6xyx+

=− 24)

4 3yx

= +

Inequalities Graph the following inequality:

25) 3 4 4y x≤ − − + Zeros: Find (0)f for the following functions. Name the # of real and imaginary solutions & degree. Remember that x-intercepts = zeros = solutions = roots. 26) 27)

(0)f = _______________ (0)f = _______________ # of Real Solutions = _____ # of Real Solutions = _____ # of Imaginary Solutions = _____ # of Imaginary Solutions = _____ Degree of Function: ______ Degree of Function: ______

2 4–2–4 x

–5

–10

–15

–20

y

2 4–2–4 x

2

4

–2

–4

y

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E X T R A N O T E S A N D E X A M P L E S : Functions Be able to recognize the graphs for the following functions: linear, quadratic, absolute value, polynomial (cube and cube root especially), exponential, and logarithm functions.

Equation examples: 2 3y x= − Linear (degree of 1), 2y x= Quadratic (degree of 2), y x=

Absolute Value, 3y x= Cube function, 3y x= Cube Root, 2xy = Exponential (a number raised

to the x power), 2logy x= Logarithm

Function Equation etc Linear

y = mx + b (SI) ax + by = c (SF)

12

12xxyym

−−

= slope

b is (0,b) y-intercept ),( 11 yx point on the

line Horizontal line HOY y = #, zero slope Vertical lines are not functions (VUX) x = # undefined slope

Quadratic “U” Parabola Find Vertex by using “Calc” key maximum or minimum

y = a(x – h)2 + k Vertex (h,k) opposite same a>0 opens up, a< 0 opens down a >1 stretch, a <1

shrink

Absolute Value “V”

khxay +−= Vertex (h,k) opposite same a>0 opens up, a< 0 opens down a >1 stretch, a <1

shrink

Square Root

khxay +−= opposite same Starting point (h, k) a>0 opens up, a< 0 reflects down a >1 stretch, a <1

shrink Cube Root

khxay +−= 3 opposite same Turning Point (h,k) a>0 as graph on left , a< 0 reflects a >1 stretch, a <1

shrink Exponential Growth

y = a(b)(x-h) + k b > 1 y = k is the horizontal asymptote e ≈ 2.72 (natural log base e)

Logarithmic

khxy b +−= )(log (inverse of exponential).

)(log ab = ba

loglog

x = h is vertical asympt. Log is log base 10 Ln is log base e

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E X T R A N O T E S A N D E X A M P L E S : Polynomials: To find the zeros of a polynomial equation, either:

1.) Graph the equation on your calculator and look at where the graph crosses/touches the x-axis Or 2.) Solve the equation by factoring and setting each factor = 0 (may need to use the quadratic formula (given to you on the ‘formula’ screen.) You must do this when you cannot tell where the graph crosses or if it doesn’t cross the x-axis.)

Polynomials Example: Cubic Degree 3

Zeros 1. Real Zeros are the x values of the x intercepts. 2. Zeros are also called roots, or solutions 3. If the zero is x = h, then its factor is (x-h) 4. The number of zeros = the degree (this includes real, imaginary and double roots)

Types 1. If there are no x intercepts there are no real zeros, (all zeros will be imaginary) 2. A tangent implies a double root (repeated solution) 3. Irrational zeros come in pairs as do imaginary zeros

Turns 1. The maximum number of turns is equal to the degree – 1.

End Behavior

1. If the leading coefficient (LC) is ‘+’ the right behavior rises, if the LC is ‘-‘ the right behavior falls 2. If the degree is even, right and left behavior will be the same, if the degree is odd right and left behavior is opposite.

Finding Domain/Range, A ‘Function’ means that x-values do not repeat---it must pass the vertical line test. Domain – set of all x-values vs. Range – set of all y-values

Ex 1: Find the Domain/Range of 2 3y x= − . From the graph shown: (Note: ℜ symbol means “all reals”)

Domain = All Real Numbers Range = All Numbers Greater than -3

Increasing/Decreasing Intervals As x increases from - infinity to + infinity (read from left right), do y values increase or decrease? The intervals will be the x values in these areas. Ex 3: What are the decreasing intervals? The function decreases from (0, 2)

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E X T R A N O T E S A N D E X A M P L E S : Leading Coefficients If the function ends up, the leading coefficient is positive. If the function ends down, the leading coefficient is negative. Transformations What is the new equation shown in bold in the graph to the right?

The parent graph is the cube root function 3y x= . The function is shifted

down by 2 therefore the new equation is 3 2y x= − Rational Functions: See the chart for information on rational graphs: Rational Function

y = )()(

xqxp

where p(x) and q(x) are polynomial functions q(x) ≠ 0 discontinuous

Domain all real numbers except the values that make q(x)= 0 Zeros of function set p(x)= 0 and solve

Vertical Asymptotes: Set q(x) = 0 and solve. Look at domain restrictions. Horizontal Asymptotes: 1. Degree of p(x) < Degree of q(x) y = 0 2. Degree of p(x) > Degree of q(x) None 3. Degree of p(x) = Degree of q(x) y = LC of p(x)/LC of Q(x)

Factors, Zeros and Equations What is the sign of leading coefficient of the graph to the right? The leading coefficient is positive because the function ends up. Determine the end behavior. As , ( )x f x→ +∞ → +∞ and As , ( )x f x→ −∞ → +∞ What are the factors?

, ( 3), ( 1)x x x+ + What is a possible equation?

2( ) ( 3)( 1)f x x x x= + + What are the zeros of the function? Remember - (0)f ’s = x-intercepts = zeros = solutions = roots.

{ }3, 1,0− −

Parent Graph

1 2 3 4–1–2–3–4 x

1

2

3

4

–1

–2

–3

–4

y

2–2 x

1

2

–1

–2

y

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Algebra 2 Review Topic 6: Other! Sequences and Series, Statistics, Composition of Functions, Variation, Inverses, Properties Sequences and Series How do I know when to use each formula?

Mixed Sequences and Series Practice

1) Find 1st 3 terms: 1 14, 2 1 1n na a a for n+= = + ≥ 2) Find 20a for 1 12,1, , ....2 4

3) Find the 3 arithmetic means: 5, ___, ___, ___, -3 4) Find the 17th term if 1 20 & 4a d= − =

36

Mixed Sequences and Series Practice - Continued 5) 97 is the ___?__ th term of –3, 1, 5, 9, … 6) Find sum of 1 2 4 8 16...− + − + to 15 terms

7) Find the sum of geometric series 1 10, 270, 4na a n= = = 8) Find the sum of 1 12,1, , ....2 4

Statistics 1 – Finding Regression Equation

9) Jean invested $380 in stocks. Over the next 5 years, the value of her investment grew, as shown in the accompanying table. Write the regression equation for this set of data, rounding all values to two decimal places. Using this equation, find the value of her stock, to the nearest dollar, 10 years after her initial purchase. Statistics 2 – Fundamental Counting Rule, Permutations, Combinations 10) In the next Olympics, the United States can enter four athletes in the diving competition. How many different teams of four divers can be selected from a group of nine divers? 11) Find the total number of different twelve-letter arrangements that can be formed using the letters in the word PENNSYLVANIA. 12) A four-digit serial number is to be created from the digits 0 through 9. How many of these serial numbers can be created if 0 can not be the first digit, no digit may be repeated, and the last digit must be 5? 13) A multiple choice test has 10 questions where each question has 4answers. If you select one of the four answers for each question, how many different ways can you answer the questions?

37

Statistics 3 – Normal Distribution and Z-Scores 14) The width of shark jaws are normally distributed with a mean of 15.7 and a standard deviation of 2.8 inches. What is the probability that a shark that you examine at random has a jaw width less than 18.5 inches? 15) What is the probability that a shark that you examine at random has a jaw greater than 20 inches? Composition of Functions

16) If ( ) 1f x x= + and ( ) 3g x x= + , then find f g .

17) If 1( )f xx

=and 2( )g x x x= − , find ( ( 1))f g − .

Inverses

18) Find the inverse of 1 22

y x= −. 19) Is 2 2y x= − a one-to-one function?

20) Graph the inverse of the line segment. 21) What is the range of the graphed line segment? 22) What is the domain of the inverse?

2 4–2–4 x

2

4

–2

–4

y

38

20 40 60 80 100–20–40–60–80–100 x

20

40

60

80

100

–20

–40

–60

–80

–100

y

23) Graph 10xy = and the inverse of 10xy = . Polynomial / Synthetic division

24) Divide: 25) Divide: Log/exponential equations Convert each log expression into an exponential expression.

26) 27) 28) Convert each exponential expression into a log expression.

29) 30) 31)

39

E X T R A N O T E S A N D E X A M P L E S : Arithmetic/Geometric Sequences/Series

Ex)

2 31

1 3 3 33 ....2 2 2 2

3 / 2 31 (1/ 2)

n

n

∞

=

= + + +

= =−

∑

Inverses Switch x and y and re-solve for y! Ex) y= 5x + 8 Inverse: x = 5y + 8 x – 8 = 5y

Polynomial division Synthetic division Divisor has to be first degree & coefficient is 1

85

xy −=

40

Functional Inverse graphs: (1) Reflected over the line y = x, (2) x and y coordinates are switched, (3) Domain/Range are switched Popular inverse graphs created from other functions: LinesLines, QuadraticsSquare Roots, , CubesCube Roots, ExponentialLogarithms Special notes about Exponentials/Logs: Exponential graphs have horizontal asymptotes---Log graphs have vertical asymptotes The equation of the inverse of an exponential is written using ‘log’:

Ex) The inverse of 5xy = is written as 5logy x= Composition of Functions: ‘Compose’ one equation inside the other:

Ex) : Given the following functions, what is the composite function ( ( ))g f x ?

( ) 2 1( ) 3

f x xg x x

= −= Answer: ( )( ) 3(2 1) 6 3g f x x x= − = −

Remember, there is an ‘inside’ function and an ‘outside’ function. In this case, the f (x) is the inside function. Take the f(x) function and plug into the x in the g(x) function

Statistics:

• Given data • enter the set of data into a list (or lists) on your graphing calculator. • Look at the scatterplot graph, decide which model is most reasonable (linear, quadratic, cubic,

logarithmic (LN), or exponential) • calculate the appropriate regression formula. STATCalc Write the equation of this particular

equation and use it to predict appropriate values not already included in the data. Ex) {(1, 2.1), (3, 3.1), (5, 4.0), (7, 5.2), (9, 5.9)}

• Plug x into List 1 and y into List 2 • Graph the scatterplot. • It should represent the line .485 1.635y x= +

Using the equation for the line of best fit, predict the y value when x = 6: Plug 6 in for x.

.485(6) 1.635 4.545y = + =

41

Normal Distribution: A normal distribution shows data in a symmetrical, bell-shaped curve. Data is centered around the mean ( µ ). The standard deviation (σ ) tells how each data value in the set differs (deviates) from the mean. Know from memory that the Empirical Rule tells us the probability distribution of the standard normal curve. 68% of the data fall within one standard deviation of the mean. 95% of the data fall within two standard deviations of the mean. 99.7 % of the data fall within three standard deviations of the mean.

Z-Score A “z-score” represents the number of standard deviations away from the mean

• A z-score with a negative value lies below the mean. • A z-score of 0 lies at the mean • A z-score with a positive value lies above the mean.

Z-scores are a way to compare different normal distributions,

To calculate the value of a z-score, ,xz µ

σ−

= where µ is the mean and σ is the standard deviation. “x”

is the number you are seeking.

Reading/Applying the Z-Score Table: Suppose I want to find out ( 1.28)P z < .

• Find the z-score of the data point (use the formula) if it is not given to you. • Sketch and shade a graph of what data you are looking • Go to the positive z-table since 1.28 is positive. • Find 1.2 in the left row and • Read across until you are under the .08 column • What value did you find? 0.8897 • This is the probability that z < 1.28.

42

P R A C T I C E O :

43

P R A C T I C E P :

44

P R A C T I C E Q :

45

Topic 7 Trigonometry Review: Pythagorean Theorem, SOH CAH TOA and word problems

Pythagorean Theorem example: set hypotenuse (longest side or across from right angle) to “c”

46

1. A side of an equilateral triangle is 20 cm long. What is the height/altitude of this triangle in the simplest radical form? (Do you see a right triangle?)

2. A side of a square is 4 cm long. What is the diagonal of the square in the simplest radical form? (Do you see a right triangle?)

3. Solve for x. Round to the tenth.

1) 2) 3)

4. Solve for x. Round to the tenth.

1) 2)

5. Which window with the following dimensions is too small to allow a 35-inch piece of glass to fit through it?

A. 28 ×45 inches B. 16 ×33 inches C. 20 ×28 inches D. 40 ×42 inches

6. Stephen is planning a right triangular garden. He marked two sides that measure 24 feet and 25 feet. He wants to know the perimeter of the garden so that he can find out how many bricks he should buy at the store. Find the perimeter of the garden.

7. The trainer adjusted a 6-foot long bench press so that the angle of elevation was 8°. How many inches did the trainer raise the bench press?

7 x

15°

4 8

x

6 45°

x

6 feet x

8°

28

17

x°

28

x

25°

47

8. From the top of a 145-foot high tower, an air traffic controller observes an airplane on the runway at an angle of depression of 22°. How far from the base of the tower is the airplane?

9. Chelsea whose eyes are 5 feet above the ground is standing on the runway of an airport 100 feet from the control tower. She observes an air traffic controller at the window of the control tower. The angle of elevation from the person to the air traffic controller is 35°. How tall is the control tower?

10. Liz is building a rectangular gate. The dimensions of the gate are 6 feet high and 4 feet wide. She wants to fasten a thin brace diagonally at the corners to keep the gate sturdy. Approximately, how long is the brace?

11. Rosemary is cutting 3 wooden sticks to build part of a kite frame. The part she is building must be a right triangle. Which choice below could be the lengths, in inches, of the sticks Rosemary cut? Choose all that apply.

A. 4, 5, 6 B. 4, 3, 5 C. 10, 15, 12 D. 12, 13, 5

E. √7,√5, 4 F. √3,√6, 3 G. 9, 40, 41 H. √5, 2√2, 13

12. The angle of depression of an object on the ground is 14° from the top of the tallest building in the world, one of Petronas towers in Malaysia, which is 1,483 feet high. What is the distance from the object to the base of the tower to the nearest foot? 13. You are at the air show in Virginia Beach. You are looking up at a British Harrier Jet at an angle of elevation of 59°. If the plane is hovering 1100 ft above the water, how far are you from the jet (direct distance)? Round to the nearest foot.

Tower

x

145 ft

22°

Airplane on runway

100 ft

Control

Tower

5 ft 35°

48

1. Given circle T with WP = 36 cm. Calculate the exact area of the shaded sector.

2. Find the length of the balcony.

3. Calculate the area of the shaded sector, to the nearest tenth.

4. The minute hand on a clock is 10 centimeters long and travels through an arc of 108° every 18 minutes. Find the measure of the length of the arc the minute hand travels through during this 18-minute period.

130°

W

T

P

45˚

15 cm