Post on 07-Mar-2018
1 | P a g e
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
2 | P a g e
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?
3 | P a g e
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?
4 | P a g e
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
5 | P a g e
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)
6 | P a g e
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
7 | P a g e
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 ̅̅
8 | P a g e
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
9 | P a g e
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:
10 | P a g e
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
11 | P a g e
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)
12 | P a g e
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
13 | P a g e
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
14 | P a g e
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
16 | P a g e
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.