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Common Core Math 2 Name: ________________________

Final Exam Review Packet

Use this packet for questions from every unit that will help you prepare for the Final Exam

Honors Math 2.

Topic Lessons Packet Pages

Probability Odds, Independent/Dependent. Mutually Inclusive/Exclusive,

Permutations, Combinations, Conditional Probability 2-3

Transformations Rotations, Reflections, Translations, Dilations 4-6

Triangles Congruence, Midsegment, Isosceles Triangles 7-9

Polynomials Adding, Subtracting and Multiplying 10

Quadratics Standard Form, Factoring, Quadratic Formula, Solving, Discriminant 11-12

Exponent/Logarithms Properties of Exponents, Exponential to Radical Form, Solving

exponential equations 13-15

Advanced Functions Solving Rational Equations, Extraneous Solutions, Solving Inverse

Equations, Transformation of Functions, solving varition 16-17

Trigonometry Graphing Sine/Cosine, Right Triangle Trig, Law of Sines/Cosines,

Area of a Triangle, Pythagorean Theorem 18-19

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Probability Review:

Odds, Independent and Dependent Events, Mutually Exclusive/Inclusive, Permutations and Combinations,

Conditional Probability

Odds vs. Probability

Odds: Likelihood of an event occurring to it not occurring

Probability: Likelihood of an event occurring to total number of outcomes

Independent/Dependent (AND) vs. Mutually Inclusive/Exclusive (OR)

AND…MULTIPLY OR…ADD Independent

One event does not affect the outcome of the second event

Ex: Flipping a coin and rolling a die

P(A) x P(B)

Mutually Exclusive The events cannot happen at the same time

Ex: Being a boy vs being a girl

P(A)+P(B) Dependent

One event affects the outcome of the second event

Ex> picking a card and picking a second card without replacing the first card

P(A) x P(B) (after A happens)

Mutually Inclusive The events can happen at the same time

Ex: Being a boy and having blue eyes

P(A)+ P(B) – P(A and B) Permutations and Combinations

Permutation: Order matters nPr

Combination: Order doesn’t matter

nCr

Conditional Probability A probability where a certain prerequisite condition has already been met

P(A | B) = P(A and B)

P(B)

Practice Questions:

1. 21 students at school have an allergy to peanuts, shellfish, or both. 14 have an allergy to peanuts, 12 have an

allergy to shellfish. How many students have an allergy to both peanuts and shellfish?

A. 12 B. 7 C. 5 D. 2

2. A total of 540 customers, who frequented an ice

cream shop, responded to a survey asking if they

preferred chocolate or vanilla ice cream.

308 of the customers preferred chocolate ice

cream

263 of the customers were female

152 of the customers were males who

preferred vanilla ice cream

What is the probability that a customer chosen at

random is a male or prefers vanilla ice cream?

A. 419/540 B. 119/180

C. 197/540 D. 38/135

3. A teacher is making a multiple choice quiz. She

wants to give each student the same questions, but

have each student's questions appear in a different

order. If there are twenty-seven students in the class,

what is the least number of questions the quiz must

contain?

4. How many ways can a school pick 5 people for

student council if there are 21 people to choose from?

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5. Determine whether the following situations would

require calculating a permutation or a combination:

i. Selecting three students to attend a

conference in Washington, DC

ii. Selecting a lead and an understudy for a

school play.

iii. Assigning students to their seats on the

first day of school.

iv. Selecting a President, Vice President and

Secretary for student council.

v. Selecting 7 people to decorate for the

homecoming dance

6. A coach must choose five starters from a team of

12 players. How many different ways can the coach

choose the starters?

7. If there are 14 people applying for a job a gym,

how many different ways can the boss choose the

gymnastics instructor, desk manager and janitor?

8. What is the total number of possible 4-letter

arrangements of the letters m, a, t, h, if each letter is

used only once in each arrangement?

9. A locker combination system uses three digits

from 0 to 9. How many different three-digit

combinations with no digit repeated are possible?

10. A bag contains three chocolate, four sugar, and

five lemon cookies. Greg takes two cookies from the

bag, at random, for a snack. Find the probability that

Greg did not take two chocolate cookies from the bag.

Explain why using the complement of the event of not

choosing two chocolate cookies might be an easier

approach to solving this problem.

11. Of 50 students going on a class trip, 35 are

student athletes and 5 are left-handed. Of the student

athletes, 3 are left-handed. Which is the probability

that one of the students on the trip is an athlete or is

left-handed?

12. There are 89 students in the freshman class at

Northview High. There are 32 students enrolled in

Spanish class and 26 enrolled in history. There are 17

students enrolled in both Spanish and history. If a

freshman is selected at random to raise the flag at the

beginning of the school day, what is the probability

that it will be a student enrolled in Spanish or

history?

13. What is the probability of rolling a 5 on the first

number cube and rolling a 6 on the second number

cube?

14. The sections on a spinner are numbered from 1

through 8. If the probability of landing on a given

section is the same for all the sections, what is the

probability of spinning a number less than 4 or

greater than 7 in a single spin?

15. A movie company surveyed 1000 people. 229

people said they went to see the new movie on

Friday, 256 said they went on Saturday. If 24 people

saw the movie both nights, what is the probability

that a person chosen at random saw the movie on

Friday or Saturday?

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Geometric Transformations Review: Rotations, reflections, translations, and dilations.

Reflections 𝑟𝑥−𝑎𝑥𝑖𝑠 (𝑥, 𝑦) → (𝑥, −𝑦) 𝑟𝑦−𝑎𝑥𝑖𝑠 (𝑥, 𝑦) → (−𝑥, −𝑦)

𝑟𝑦=𝑥 (𝑥, 𝑦) → (𝑦, 𝑥)

𝑟𝑦=−𝑥 (𝑥, 𝑦) → (−𝑦, −𝑥)

Rotations 𝑅90 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 (𝑥, 𝑦) → (−𝑦, 𝑥)

(Same as 270 clockwise) 𝑅180 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 (𝑥, 𝑦) → (−𝑥, −𝑦)

𝑅270 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 (𝑥, 𝑦) → (𝑦, −𝑥)

(Same as 90 clockwise)

Translations (𝑥, 𝑦) → (𝑥 ± #, 𝑦 ± #) 𝑥 + # = 𝑟𝑖𝑔ℎ𝑡 𝑦 + #

= 𝑢𝑝 𝑥 − # = 𝑙𝑒𝑓𝑡 𝑦 − #

= 𝑑𝑜𝑤𝑛 Dilation – a transformation that produces an image that is the same shape as the original, but is a

different size. (The image is similar to the original object) Dilation is a transformation in which each point

of an object is moved along a straight line. The straight line is drawn from a fixed point called the center

of dilation.

𝑆𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 =𝑖𝑚𝑎𝑔𝑒 𝑙𝑒𝑛𝑔𝑡ℎ

𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ

A dilation is an enlargement if the scale factor is greater than 1. A dilation is a reduction if the scale factor

is between 0 and 1.

1. Which transformation will always produce a congruent figure?

A.(𝑥, 𝑦) → (𝑥 + 2, 3𝑦) C. (𝑥, 𝑦) → (2𝑥, 2𝑦)

B.(𝑥, 𝑦) → (𝑥 − 3, 𝑦) D. (𝑥, 𝑦) → (2𝑥, 𝑦 + 1)

2. Which transformation will carry the rectangle show to

the rght onto itself?

A. reflection over line m

B. reflection over line y=1

C. rotation 90o CCW about the origin

D. rotation 270o CCW about the origin

7. ∆ABC is dilated by a scale factor of k producing ∆A’B’C’.

How does angle A compare to angle A’?

A. Angle A’ will be k time larger than Angle A

B. Angle A’ will be k times smaller than Angle A

C. Angle A’ will be the measure of Angle A + k

D. Angle A’ will be the same as Angle A

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3. Triangle EGF is graphed below.

Triangle EGF will be rotated 90 degrees CCW around the origin and will then be reflected across the y-

axis, producing an image triangle. Which additional transformation will map the image triangle back onto

the original triangle?

A. rotation 270 degrees CCW

B. rotation 180 degrees CCW

C. reflection across y=-x

D. reflection across y=x

4. Which line of reflection would carry the figure onto itself?

A. 𝑦 = 𝑥 C.𝑥 = −2

B. 𝑦 = −2 D. 𝑥 = 1

5. The translation (𝑥, 𝑦) → (𝑥 − 2, 𝑦 + 4) maps ∆ABC onto ∆A’B’C’. What translation maps ∆A’B’C’ onto

∆ABC?

A. (𝑥, 𝑦) → (𝑥 + 2, 𝑦 − 4) C. (𝑥, 𝑦) → (𝑥 − 2, 𝑦 + 4)

B. (𝑥, 𝑦) → (𝑥 + 2, 𝑦 + 4) D (𝑥, 𝑦) → (𝑥 − 2, 𝑦 − 4)

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6. For the figure below, what is the line of reflection that maps ∆AEY onto ∆A’E’Y’?

8. For the parallelogram below, which line of reflection would carry the parallelogram onto itself?

A. x-axis

B. y-axis

C. Line y=x

D. Line m

m

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Similarity & Congruence Review:

Triangle Congruence

Triangle Midsegment Thm Isosceles Triangles

𝑫𝑬̅̅ ̅̅ || 𝑨𝑩̅̅ ̅̅ and 𝑫𝑬 =𝟏

𝟐𝑨𝑩

1. Based on the given information in the figure at the right, how can you justify that ∆𝐽𝐻𝐺 ≅ ∆𝐻𝐽𝐼 ?

A. ASA B. AAS C. SSS D. SAS

2. Which statement cannot be justified given only that ∆𝑃𝐵𝐽 ≅ ∆𝑇𝐼𝑀 ?

A. 𝑃𝐵̅̅ ̅̅ ≅ 𝑇𝐼̅̅̅ B. < 𝐵 ≅< 𝐼 C. < 𝐵𝐽𝑃 ≅< 𝐼𝑀𝑇 D. 𝐽𝑃̅̅ ̅ ≅ 𝑀𝐼̅̅ ̅̅

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3. In the figure at the right, which theorem or

postulate can you use to prove ∆𝐴𝐷𝑀 ≅ ∆𝑍𝑀𝐷 ?

A. ASA

B. AAS

C. SSS

D. SAS

4. Which pair of triangles can be proven

congruent by the ASA postulate?

5. Which pair of triangles can be proven

congruent by the AAS postulate?

6. Which pair of triangles can be proven

congruent by SSS?

7. Which pair of triangles can be proven

congruent by SAS?

8. What additional information do you need to

prove ∆𝑁𝑂𝑃 ≅ ∆𝑄𝑆𝑅 ?

A. 𝑃𝑁̅̅ ̅̅ ≅ 𝑆𝑄̅̅̅̅ C. < 𝑃 ≅< 𝑆

B. 𝑁𝑂̅̅ ̅̅ ≅ 𝑄𝑅̅̅ ̅̅ D. < 𝑂 ≅< 𝑆

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9. Given the diagram, which of the following

must be true?

A. ∆𝑋𝑆𝐹 ≅ ∆𝑋𝑇𝐺

B. ∆𝑆𝑋𝐹 ≅ ∆𝐺𝑋𝑇

C. ∆𝐹𝑋𝑆 ≅ ∆𝑋𝐺𝑇

D. ∆𝐹𝑋𝑆 ≅ ∆𝐺𝑋𝑇

10. Solve for x

11. Use diagram at right to find XZ

a. Find XZ.

b. If XY=10, find MO.

12. Solve for x and y.

13. Solve for x and y.

14. Which statement must be true about the

triangle below?

15. Use the figure at the below.

a. What is the distance across the lake?

b. Is it shorter distance from A to B or from B to

C? Explain.

Similarity & Congruence Notes/Help:

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Polynomials: Adding & Subtracting polynomials – Add like terms, the exponents don’t change!

Ex: (3𝑥2 − 4 + 2𝑥) + (5𝑥 − 6𝑥2 + 7) =−3𝑥2 + 7𝑥 + 3

Ex: (3𝑥2 − 4 + 2𝑥) − (5𝑥 − 6𝑥2 + 7) =9𝑥2 − 3𝑥 − 11

Multiplying Polynomials – Each term in a polynomial has to be multiplied to each term in the other

polynomial. Exponents change when terms are multiplied!

Ex: 4𝑏(𝑐𝑏 − 𝑧𝑑) =4𝑏2𝑐 − 4𝑏𝑧𝑑

Ex: (4𝑥 − 5)(𝑥 + 2) = 4𝑥2 + 8𝑥 − 5𝑥 − 10 = 4𝑥2 + 3𝑥 − 10

Ex: (2𝑥2 − 6𝑥 + 1)(𝑥 + 3) =2𝑥3 + 6𝑥2 − 6𝑥2 − 18𝑥 + 𝑥 + 3 =2𝑥3 − 17𝑥 + 3

Ex: (𝑥 + 5)(𝑥 − 2)(3𝑥 + 4) =(𝑥2 + 3𝑥 − 10)(3𝑥 + 4) = 3𝑥3 + 4𝑥2 + 9𝑥2 + 12𝑥 − 30𝑥 − 40 = 3𝑥3 + 13𝑥2 − 18𝑥 − 40

1. (3𝑥5 + 17𝑥3 − 1) + (−2𝑥5 − 6)

2. (6𝑥2 − 3𝑥 + 2) − (−6𝑥2 + 3𝑥 − 5)

3. (𝑥 + 2)(𝑥2 + 2𝑥 + 3)

4. (3𝑥 − 4)(6𝑥 + 7)

5. (4𝑥2 − 3𝑦2 + 5𝑥𝑦) − (8𝑥𝑦 + 3𝑦2)

6. Which expression is equivalent to 𝑡2 − 36?

A.(𝑡 − 6)(𝑡 − 6) C. (𝑡 + 6)(𝑡 − 6)

B.(𝑡 − 12)(𝑡 − 3) D. (𝑡 − 12)(𝑡 + 3)

7. Which of the following is equivalent to

(5𝑡 + 3)2 ?

A. 10𝑡 + 9

B. 25𝑡2 + 9

C. 25𝑡2 + 30𝑡 + 9

D. 10𝑡2 + 30𝑡 + 9

8. Which expression is equivalent to (𝑥 +

1)(3𝑥 − 2)(𝑥 + 4) ?

A. 5𝑥 + 3

B. 3𝑥3 − 8

C. 3𝑥3 + 13𝑥2 + 2𝑥 − 8

D. 16𝑥2 + 2𝑥 − 8

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Quadratic Review Standard Form 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐

c is the y-intercept of a quadratic, positive a(faces up like a U), negative a(faces down)

Solutions (known as x-intercepts, zeros, or roots) of a quadratic can be found three ways:

Method 1) Graphing – Graph the function in y=, 2nd, trace, zero (left bound, enter, right bound, enter, guess,

enter)

Method 2) Factoring – transform a quadratic from standard form into factored form then use zero-product

property

Ex: Solve 𝑓(𝑥) = 3𝑥2 + 7𝑥 − 6

Factored form: (3𝑥 − 2)(𝑥 + 3) = 0

Set each factor equal to zero and solve for variable.

3𝑥 − 2 = 0 𝑥 + 3 = 0

3𝑥 = 2 𝑥 = −3

𝑥 =2

3

Method 3) Quadratic Formula – works for every quadratic!! 𝑥 =−𝑏±√𝑏2−4𝑎𝑐

2𝑎 use the a, b, c, from standard form.

Ex: Solve 𝑓(𝑥) = 3𝑥2 + 7𝑥 − 6 𝑎 = 3, 𝑏 = 7 𝑎𝑛𝑑 𝑐 = −6

𝑥 =−(7)±√(7)2−4(3)(−6)

2(3)=

−7±11

6 𝑠𝑜 𝑥 = 2/3 𝑎𝑛𝑑 𝑥 = −3

Discriminant 𝒃𝟐 − 𝟒𝒂𝒄

If 𝑏2 − 4𝑎𝑐 > 0 the quadratic has TWO real solutions.

If 𝑏2 − 4𝑎𝑐 = 0 the quadratic has ONE real solutions.

If 𝑏2 − 4𝑎𝑐 < 0 the quadratic has NO real solutions. (2 imaginary)

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1. Which function has exactly one solution? A. 4𝑥2 − 12𝑥 − 9 = 0

B. 4𝑥2 + 12𝑥 + 9 = 0

C. 4𝑥2 − 6𝑥 − 9 = 0

D. 4𝑥2 + 6𝑥 + 9 = 0

2. The heights of two different projectiles after they

are launched are modeled by f(x) and g(x). The

function f(x) is defined as 𝑓(𝑥) = −16𝑥2 + 42𝑥 + 12.

The table contains the values for the quadratic g(x).

What is the approximate difference in the maximum

heights achieved by the two projectiles?

A. 0.2 feet C. 5.4 feet

B. 3.0 feet D. 5.6 feet

3. A company found that its monthly profit, P, is given

by 𝑃 = −10𝑥2 + 120𝑥 − 150 where x is the selling

price for each unit of the product. Which of the

following is the best estimate of the maximum price

per unit that the company can charge without losing

money?

A. $300 C. $11

B. $210 D. $6

4. A ball is thrown from the top of a building. The

table shows the height, h, (in feet) of the ball above

the ground t seconds after being tossed.

t 1 2 3 4 5 6 h 299 311 291 239 155 39

How long after the ball was tossed was it 80 feet

above the ground?

A. about 5.1 seconds C. about 5.7 seconds

B. about 5.4 seconds D. about 5.9 seconds

5. Which of the following is a factor of

4𝑎𝑏 + 2𝑎 + 6𝑏 + 3 ?

A. (2a-3) C. (2b-1)

B. (2a+3) D. (2b+3)

6. If t is an unknown constant, which binomial must

be a factor of 7𝑚2 + 14𝑚 − 𝑡𝑚 − 2𝑡?

A. (7m+t) C. (m+2)

B. (m-t) D. (m-2)

7. What is the equation of a parabola with the vertex

(3, -20) and passes through the point (7, 12)?

A.𝑦 = 2𝑥2 + 12𝑥 − 2 C. 𝑦 = −2𝑥2 + 12𝑥 − 38

B.𝑦 = 2𝑥2 − 12𝑥 − 2 D. 𝑦 = 2𝑥2 − 12𝑥 + 38

8. The function 𝐶 = 75𝑥 + 2600 gives the cost, in dollars,

for a small company to manufacture x items. The function

𝑅 = 225𝑥 − 𝑥2 gives the revenue, also in dollars, for

selling x items. How many items should the company

produce so that the cost and revenue are equal?

9. What is the discriminant of 4𝑥2 + 28𝑥 = −49?

10. The graph of the function x2 will be shifted down 2

units and to the right 3 units. Write an equation in vertex

form that corresponds to the resulting graph.

11. Brian used the quadratic formula to solve a quadratic

equation and his result is below. Write the original

quadratic equation he started with in standard form.

𝑥 =8 ± √(−8)2 − 4(1)(−2)

2(1)

12. A rocket is launched. The function that models this

situation is ℎ(𝑡) = −16𝑡2 + 96𝑡 + 180

i. What is the height of the rocket 2 seconds after launch?

ii. What is the max value?

iii. When is the rocket 100 feet above ground?

x g(x) 0 9 1 33 2 25

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Exponent Rules:

Product of powers: 𝑥𝑚 ∗ 𝑥𝑛 = 𝑥𝑚+𝑛

Quotient of powers: 𝑥𝑚

𝑥𝑛 = 𝑥𝑚−𝑛

Negative exponents: 𝑥−𝑛 =1

𝑥𝑛 or 1

𝑥−𝑛 = 𝑥𝑛

Power of power: (𝑥𝑚)𝑛 = 𝑥𝑚∗𝑛

Power of a quotient: (𝑥𝑚

𝑥𝑛 )𝑝

=𝑥𝑚𝑝

𝑥𝑛𝑝

Power of a product: (𝑥𝑚𝑦)𝑛 = 𝑥𝑚𝑛𝑦𝑛

Zero exponents: 𝑥0 = 1, 𝑥 ≠ 0

Exponent Form: 𝑥2

3 Radical Form: √𝑥23𝑜𝑟 (√𝑥

3)2

1. Simplify (16𝑥5𝑦−3𝑧2)−1/4

2. Simplify (4𝑥−3𝑦4𝑧−2)−3/2

3. Simplify (8𝑤7𝑥−5𝑦3𝑧−9)−2/3

4. Which expression is equivalent to (16𝑥

16𝑦−2

𝑥−

16𝑦6

)

3

2

?

A. 24𝑥9

2𝑦9

2 C 64

𝑥12𝑦8

B. 24𝑥

34

𝑦9 D. 64𝑥

12

𝑦12

5. Which expression is equivalent to (4𝑥)1/2 ∗ 361/2

A. 12x B. 36x C. 12√𝑥 D. 24√𝑥

6. Which expression is equivalent to (16𝑥4𝑦−2

25𝑥12𝑦−4

)

−1

2

?

A. 5

4𝑥74𝑦

C 12.5

8𝑥74𝑦

B. 5𝑥

74

4𝑦2 D. 16𝑦

25𝑥2

7. Simplify √𝑏35

𝑏43

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Exponential Functions:

Exponential Growth: Exponential Decay:

𝑦 = 𝑎𝑏𝑥 𝑤ℎ𝑒𝑟𝑒 𝑎 > 0 𝑎𝑛𝑑 0 < 𝑏 < 1 𝑦 = 𝑎𝑏𝑥 𝑤ℎ𝑒𝑟𝑒 𝑎 > 0 𝑎𝑛𝑑 𝑏 > 1

b=1+r b=1-r

Compound Interest Interest Compounded Continuously Half Life

𝐴 = 𝑃(1 +𝑟

𝑛)𝑛𝑡 𝐴 = 𝑃𝑒𝑟𝑡 𝑦 = 𝑎 (

1

2)

𝑥

Solving Exponential Equations

𝑏𝑥 = 𝑏𝑦 𝑡ℎ𝑒𝑛 𝑥 = 𝑦 because bases are same

Ex: Solve for x. 103𝑥−1 = 100,000

103𝑥−1 = 105

3x-1=5

x=2

When bases aren’t the same: Isolate the exponential expression, take the log of both sides and solve. Check solutions!!

Ex: 5(10)2𝑥 = 60

Step 1: Isolate the exponential expression.

(10)2𝑥 = 12

Step 2: Take logarithm of both sides. Remember the exponent gets moved to multiply by the log(base).

2𝑥 ∗ log(10) = log (12)

Step 3: Simplify & Solve.

2𝑥∗log(10)

log (10)=

log (12)

log (10)

2𝑥 = 1.0792

x=0.5396

1. In 1950, a U.S. population model was

𝑦 = 151(1.013)𝑡−1950 million people, where t is the

year. What did the model predict the U.S. population

would be in the year 2000?

2. Copper production increased at a rate of about

4.9% per year between 1988 and 1993. In 1993,

copper production was approximately 1.801 billion

kilograms. If this trend continued, which equation

best models the copper production (P) in billions of

kilograms, since 1993? (Let t=0 for 1993)

A. 𝑃 = 1.801(4.900)𝑡 C. 𝑃 = 1.801(1.049)𝑡

B. 𝑃 = 1.801(1.490)𝑡 D. 𝑃 = 1.801(0.049)𝑡

3. The population of a small town in North Carolina is

4,000, and it has a growth rate of 3% per year. Write

an expression which can be used to calculate the

town’s population x years from now?

4. Alan has just started a job that pays a salary of

$21,500. At the end of each year of work, he will get a

5% salary increase. What will his salary be after

getting his fifth increase?

5. The value, V of a car can be modeled by the

function 𝑉(𝑡) = 13000(0.82)𝑡 where t is the number

of years since the car was purchased. To the nearest

tenth of a percent, what is the monthly rate of

depreciation?

15 | P a g e

6. The function 𝑉(𝑡) = 1000(1.06)2𝑡 models the

value of an investment after t years.

i. What is the initial value of the investment?

ii. As a percent, what interest rate is the

investment earning each year?

7. If the equation 𝑦 = 2𝑥 is graphed, which of the

following values of x would produce a point closest to

the x-axis?

A. ¼ B. ¾ C. 5/3 D. 8/3

8. Suppose a hospital patient receives medication

that is used up in the body according to the equation

𝑀 = 200(0. 8𝑡) with M in milligrams and t in hours.

What does the 0.8 represent in the equation?

A. The medication is used up in 0.8 hours.

B. The medication is used up in 0.8 milligrams per

hour.

C. The patient started out with 0.8 milligrams of

medication.

D. There is 80% of the medication remaining after

each hour.

9. Solve 100𝑥+6 = 10002𝑥+3

10. A city’s population, P (in thousands), can be

modeled by the equation 𝑃 = 130(1.03)𝑥 where x is

the number of years after January 1, 2000. For what

value of x does the model predict that the population

of the city will be approximately 170,000 people?

11. A new automobile is purchased for $20,000. If

𝑉 = 20,000(0.8)𝑥 gives the car’s value after x years,

about how long will it take for the car to be worth

half its purchase price?

12. Solve for x: 35𝑥 = 92𝑥−1

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Solving Advanced Equations:Direct Variation 𝒚 = 𝒌𝒙

“y varies directly with x” Solve: 𝑦

𝑥=

𝑦

𝑥

Ex: y varies directly with x. Find y If y is 2 when x is 3

find y when x is 6. 2

3=

𝑦

6 y=4

Inverse Variation 𝒚 =𝒌

𝒙

“y varies inversely with x” Solve: 𝑥𝑦 = 𝑥𝑦

Ex: Suppose y varies inversely with x. Find x when y

is 7, if y is 14 when x is 2.

𝑥(7) = 2(14) x=4

Direct/Inverse Variation (combined) 𝒚 =𝒌𝒙

𝒛

“y varies directly with x and inversely with z”

Ex: If y varies directly as x and inversely as z, and

y=24 when x=48 and z=4, find x when y=44 and

z=6.

𝟐𝟒 =𝒌(𝟒𝟖)

(𝟒)→ 𝒌 = 𝟐 → 𝟒𝟒 =

𝟐𝒙

𝟔

𝒙 = 𝟏𝟑𝟐

Solving Rational and Radical Equations.

Ex: Solve 2𝑥 = √5𝑥 − 1 + 1

Step 1: Subtract 1 from each side to isolate the radical

term.

2𝑥 − 1 = √5𝑥 − 1

Step 2: Square both sides to eliminate the radical.

4𝑥2 − 4𝑥 + 1 = √5𝑥 − 1

Step 3: Set the right side equal to 0.

4𝑥2 − 9𝑥 + 2 = 0

Step 4: Solve for x (quadratic so use factoring,

graphing or quadratic formula)

𝑥 =1

4 𝑎𝑛𝑑 𝑥 = 2

Step 5: Check solutions in the original equation and

check for extraneous solutions.

2 (1

4) = √5 (

1

4) − 1 + 1 2(2) = √5(2) − 1 + 1

1

2≠ 1

1

2 4=4

so x=1/4 is not a solution. So x=2 is a solution.

The solution ¼ is an extraneous solution because it is

a solution to the transformed equation, not to the

original equation.

Ex. Solve 𝑥

𝑥−1− 1 =

𝑥

2

Step 1: Get a common denominator, in this case

2(x-1) It will eliminate the denominators altogether.

2𝑥 − 2(𝑥 − 1) = 𝑥(𝑥 − 1)

Step 2: Simplify.

2𝑥 − 2𝑥 + 2 = 𝑥2 − 𝑥

0 = 𝑥2 − 𝑥 − 2

Step 3: Solve for x.

0 = (𝑥 − 2)(𝑥 + 1)

𝑥 = 2 𝑎𝑛𝑑 𝑥 = −1

Step 4: Check solutions in the original equation and

check for extraneous solutions (or excluded values).

2

2−1− 1 =

2

2

−1

(−1)−1− 1 =

−1

2

1=1 𝑠𝑜 𝑥 = 2 𝑖𝑠 𝑎 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 −1

2≠ −1

𝑠𝑜 𝑥 = −1 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛

The solution -1 is an extraneous solution because -1

is an excluded value.

1. Solve for x: 𝑥+1

5− 2 =

−4

𝑥

2. Solve for x: 2 + √3𝑥 + 7 = 6

3. For the function 𝑦 = √𝑥 − 2 + 3

i. Sketch a graph

ii. State the transformations

17 | P a g e

4. For the function 𝑦 = √𝑥 − 33

i. Sketch a graph

ii. State the transformations

5. Solve for x: 4

𝑥−2=

−1

𝑥−3

6. Solve for x: 2

𝑥+2−

1

𝑥=

−4

𝑥(𝑥+2)

7. Suppose that y varies inversely with the square of

x, and y=50 when x=4. Find y when x=5.

8. Suppose that y varies directly with x and inversely

with z2, and x=48 when y=8 and z=3. Find x when

y=12 and z=2.

9. A salesperson’s commission varies directly with

sales. For $1000 in sales, the commission is $85.

i. What is the constant of variation (k)?

ii. What is the variation equation?

iii. What is the commission for a $2300 sale?

10. If y varies directly with x and y is 18 when x is 6,

which of the following represents this situation?

A. y=24x B. y=3x

C. y=12x D. y=1/3x

11. The number of bags of grass seed n needed to

reseed a yard varies directly with the area a to be

seeded and inversely with the weight w of a bag of

seed. If it takes two 3-lb bags to seed an area of 3600

square feet, how many 3-lb bags will seed 9000

square feet?

A. 3 bags B. 4 bags

C. 5 bags D. 6 bags

12. The volume, V, of a certain gas varies inversely

with the amount of pressure, P, placed on it. The

volume of this gas is 175 cm3 when 3.2 kg/cm2 of

pressure is placed on it. What amount of pressure

must be placed on 400 cm3 of this gas?

A. 1.31 B. 1.40

C. 2.86 D. 7.31

18 | P a g e

Trigonometry: Graphing Sine and Cosine

Amplitude: Distance the max or min is from the midline. Always

positive.

Midline: The line that cuts through the middle of the curve, the

vertical shift in the curve

Pythagorean Theorem

Right Triangle Trig

adjacent

oppositeand

hypotenuse

adjacent

hypotenuse

opposite tan,cos,sin

1. Label the sides of the triangle based on the given angle

2. Set up the trig ratio based on the information given.

3. Solve for the missing side or angle. If solving for a missing side use cross multiplication. If solving for a missing

angle, use inverse trig functions.

Area of Oblique Triangles Area=(1/2)a* b*sin(C)

1. Find the length of both of the missing sides on the

following right triangle:

2. Find the value of k, correct to 1 decimal place.

Show all work.

3. An escalator at an airport slopes at an angle of 30°

and is 20 m long. Through what height would a

person be lifted by travelling on the escalator?

4. The top of a flagpole is connected to the ground by a cable 12 meters long. The angle that the cable makes with the ground is 40. Find the height of the flagpole. 5. A ship’s navigator observes a lighthouse on a cliff. She knows from a chart that the top of the lighthouse

9.5

k

72

19 | P a g e

is 35.7 meters above sea level. She measures the angle of elevation of the top of the lighthouse to be 0.7.The coast is very dangerous in this area and ships have been advised to keep at least 4 km from this cliff to be safe. Is the ship safe?

6. A school soccer field measures 45 m by 65 m. To

get home more quickly, Urooj decides to walk along

the diagonal of the field. What is the angle of Urooj’s

path, with respect to the 45-m side, to the nearest

degree?

7. A roof is shaped like an isosceles triangle. The slope

of the roof makes an angle of 24 with the horizontal,

and has an altitude of 3.5 m. Determine the width of

the roof, to the nearest tenth of a meter.

8. Which of the following functions is graphed below?

A. 3sin(x) B. 3cos(x) C. sin(3x) D. cos(3x)

9. In the right triangle LMN, LN=728 cm and LM=700

cm. What is the approximate measure of <NLM?

10. What is the amplitude of y=3sin(4x)?

11. Graph 𝑦 = −3𝑐𝑜𝑠𝜃 + 1. Identify they amplitude

and midline.

12. Electronic instruments on a treasure-hunting ship

detect a large object on the sea floor. The angle of

depression is 29, and the instruments indicate that

the direct-line distance between the ship and the

object is about 1400 ft. About how far below the

surface of the water is the object, and how far must

the ship travel to be directly over it?

13. From the top of a 120 foot tower, an air traffic

controller observes an airplane on the runway at an

angle of depression of 19o. How far from the base of

the tower is the airplane?

14. Find the area of the oblique triangle.