1. science
2. mathematics
3. algebra

Common Core Math 2
Final Exam Review Packet
Name: ________________________
Use this packet for questions from every unit that will help you prepare for the NC Final
Exam for Common Core Math 2.
Topic
Lessons
Odds, Independent/Dependent. Mutually Inclusive/Exclusive,
Permutations, Combinations, Conditional Probability
Packet Pages
Transformations
Rotations, Reflections, Translations, Dilations
5-7
Triangles
Congruence, Midsegment, Isosceles Triangles
8-10
Polynomials
Adding, Subtracting and Multiplying
11
Quadratics
Standard Form, Factoring, Quadratic Formula, Solving, Discriminant
12-14
Probability
Exponent/Logarithms
Advanced Functions
Trigonometry
Properties of Exponents, Exponential to Radical Form, Solving
exponential equations
Solving Rational Equations, Extraneous Solutions, Solving Inverse
Equations, Transformation of Functions, solving varition
Graphing Sine/Cosine, Right Triangle Trig, Law of Sines/Cosines,
Area of a Triangle, Pythagorean Theorem
1-4
15-17
18-19
20-22
WHS Jan. 2015 Exam Calendar
Monday
Tuesday
Wednesday
Thursday
Friday
1/12
1st Period Exams
All Non-EOC
Exams
1/13
Eng II EOCs (All
Periods)
Non-EOC Makeup
Exams
1/14
3rd Period Exams
All Non-EOC
Exams
1/15
Biology EOCs (All
Periods)
& CC Math 1 &
1B—
Non-EOC Makeup
Exams
1/16
Make-Up Exams
(Non-EOCs/NC Final
Exams/VOCATs/EOCs)
for ALL Periods
Main Campus
7:25 – 9:45
North Campus
7:36 – 9:53
2nd Period Exams
All Non-EOC
Exams
Main Campus
10:00 – 12:20
North Campus
9:58 – 12:15
Main Campus
7:25 – 11:35
North Campus
7:36 – 11:25
Lunch Available
in Main Campus
Commons Area
Only
Main Campus
7:25 – 9:45
North Campus
7:36 – 9:53
4th Period Exams
All Non-EOC
Exams
Main Campus
10:10 – 12:30
North Campus
9:58 – 12:15
Main Campus
7:25 – 11:35
North Campus
7:36 – 11:25
**All NC Final Exam,
EOC and VOCAT
make-ups will be on
main campus only**
Lunch Available
in Main Campus
Commons Area
Only
1|Page
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
Mutually Exclusive
One event does not affect the outcome of the
The events cannot happen at the same time
second event
Ex: Being a boy vs being a girl
Ex: Flipping a coin and rolling a die
P(A) x 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)+P(B)
Mutually Inclusive
The events can happen at the same time
Ex: Being a boy and having blue eyes
P(A) x P(B) (after A happens)
P(A)+ P(B) – P(A and B)
Permutations and Combinations
Permutation: Order matters
Combination: Order doesn’t matter
nPr
nCr
Conditional Probability
A probability where a certain prerequisite condition has already been met
• For example:
• What is the probability of selecting a queen given an ace has been drawn and not replaced.
• What is the probability that a student in the 10th grade is enrolled in biology given that the
student is enrolled in CCM2?
P(A | B) = P(A and B)
P(B)
1. Suppose that Jamal can choose to get home from work by taxi or bus.
When he chooses to get home by taxi, he arrive home after 7 p.m. 8 percent of the time
When he chooses to get home by bus, he arrives home after 7 p.m. 15 percent of the time
Because the bus is cheaper, he uses the bus 60 percent of the time
What is the approximate probability that Jamal chose to get home from work by bus, given that he arrived home
after 7 p.m.?
A. 0.09
B. 0.14
C. 0.60
D. 0.74
2. 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|Page
3. 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
6. 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.
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
4. 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?
5. For a carnival game, a jar contains 20 blue marbles
and 80 red marbles. Children take turns randomly
selecting marbles from a jar. If a blue marble is
chosen, a child wins a prize. After each turn the
marble is replaced. Sally has drawn six red marbles in
a row. Which statement is true?
A. If Sally selects another red marble, then 2
of her next 3 picks will be blue marbles
because 2 blue marbles are selected for every
8 red marbles.
B. The probability that Sally selects a blue
marble on the next turn is higher than it was
on her last turn because she has chosen so
many red marbles in a row.
C. The probability that Sally selects a blue
marble on her next turn is the same as it was
on the last turn because selections are
independent of each other.
D. If Sally draws 4 more times, she will select
2 blue marbles because the probability that a
blue marble will be selected is 2 out of every
10 turns.
7. A coach must choose five starters from a team of
12 players. How many different ways can the coach
choose the starters?
8. There are fourteen juniors and twenty-three
seniors in the Service Club. The club is to send four
representatives to the State Conference.
i. How many different ways are there to select
a group of four students to attend the
conference?
ii. If the members of the club decide to send
two juniors and two seniors, how many
different groupings are possible?
9. 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?
10. There are 12 boys and 14 girls in Mrs.
Schultzkie's math class. Find the number of ways
Mrs. Schultzkie can select a team of 3 students from
the class to work on a group project. The team is to
consist of 1 girl and 2 boys.
11. A locker combination system uses three digits
from 0 to 9. How many different three-digit
combinations with no digit repeated are possible?
3|Page
12. 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.
13. A four person committee is to be chosen from
Department A and Department B. Department A has
15 employees and Department B has 20 employees.
i. What is the probability of choosing 3 from
Department A and 1 from Department B.
ii. Determine the odds for selecting 3 from A and 1
from B.
14. 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?
15. 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?
16. What is the probability of rolling a 5 on the first
number cube and rolling a 6 on the second number
cube?
17. 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?
18. At a school carnival one of the booths has 12
plastic ducks floating in a tub of water. Each duck has
a zero, one, or two printed on the bottom, indicating
the number of prize tickets you receive if you select
that duck. Six of the ducks have a zero on the bottom,
three of them have a one printed on the bottom, and
three of them have a two printed on the bottom. If
you randomly select a duck, and then randomly select
another duck without returning the first to the tub,
what is the probability that you will receive four
prize tickets?
19. The probability that a city bus is ready for service
when needed is 84%. The probability that a city bus
is ready for service and has a working radio is 67%.
Find the probability that a bus chosen at random has
a working radio given that it is ready for service.
Round to the nearest tenth of a percent.
20. Events A and B are independent. Find the missing
probability.
P(A) = _?_
P(B) = 0.3
P(A and B) = 0.06
21. 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|Page
Geometric Transformations Review
Rotations, reflections, translations, and dilations.
Reflections
Rotations
Translations
(Same as 270 clockwise)
(Same as 90 clockwise)
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.
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.
Reflectional Symmetry
Rotational Symmetry
If a line can be drawn through a figure for which it
divides the figure into congruent halves.
A rotation which the figure is its own image. To find
the rotational degrees where a regular polygon will
rotate onto its own image find
360/3=120 degrees
5|Page
1. Which transformation will always produce a
congruent figure?
A.
C.
B.
D.
4. Triangle EGF is graphed below.
2. Which transformation will carry the rectangle
show below onto itself?
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
A.
B.
C.
D.
reflection over line m
reflection over line y=1
rotation 90o CCW about the origin
rotation 270o CCW about the origin
3.
has points F(2, 4) and G(6, 1). If
is
dilated with respect to the origin by a factor of k,
to produce F’G’, which statement must be true?
A. The line that passes through F’ and G’
intersects the y-axis at (0, 5.5+k)
B.The line that passes through F’ and G’
intersects the y-axis at (0, 5.5)
5. Which rotation will carry a regular hexagon
onto itself?
A. 30 degrees CCW
B. 90 degrees CCW
C. 120 degrees CCW
D. 270 degrees CCW
6. Which rotation will carry a regular octagon
onto itself?
A. 80 degrees CCW
B. 90 degrees CCW
C. 120 degrees CCW
D. 200 degrees CCW
7. Which line of reflection would carry the figure
onto itself?
C.The line that passes through F’ and G’ has a
slope of
k
D.The line that passes through F’ and G’ has a
slope of -
A.
B.
C.
D.
6|Page
8. The translation
maps
∆ABC onto ∆A’B’C’. What translation maps
∆A’B’C’ onto ∆ABC?
13. The vertices of triangle PQR are located at
P(2, 5), Q(12, 20), and R(12, 5). The vertices of
the triangle will undergo the transformation
A.
B.
described by the rule
C.
D
Which
statement about their image triangle is true?
9. What is the image of E(2, -3) after a reflection
first across
and then across
?
A. (-6, 5)
B. (0, -1)
C. (2, 5)
D. (-6, -1)
10. For the figure below, what is the line of
reflection that maps ∆AEY onto ∆A’E’Y’?
A. The perimeter of the image will be 5 times the
perimeter of the preimage.
B. The area of the image will be 5 times the area
of the preimage.
C. The perimeter of the image will be 1/5 the
perimeter of the preimage.
D. The area of the image will be 1/5 the area of
the preimage.
14. ∆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
11. Rectangle A(1, 1), B(4, 1), C(4, 2), D(1, 2) is
similar to rectangle A’B’C’D’ under a dilation
centered at the origin. If A’B’=6 units. What is
the scale factor?
A. -2
B. 2
C.
15. For the parallelogram below, which line of
reflection would carry the parallelogram onto
itself?
m
D.
12. Which rotation will carry a parallelogram
onto itself?
A. 45 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
A. x-axis
B. y-axis
C. Line y=x
D. Line m
7|Page
Similarity & Congruence Review
Triangle Congruence
Triangle Midsegment Thm
Isosceles Triangles
and
1. Based on the given information in the figure at the right, how can you justify that
A. ASA
B. AAS
C. SSS
?
D. SAS
2. Which statement cannot be justified given only that
A.
B.
C.
?
D.
8|Page
3. In the figure at the right, which theorem or
postulate can you use to prove
?
6. Which pair of triangles can be proven
congruent by SSS?
A. ASA
B. AAS
C. SSS
D. SAS
4. Which pair of triangles can be proven
congruent by the ASA postulate?
7. Which pair of triangles can be proven
congruent by SAS?
5. Which pair of triangles can be proven
congruent by the AAS postulate?
8. What additional information do you need to
prove
?
A.
B.
C.
D.
9|Page
9. Given the diagram, which of the following
must be true?
A.
B.
C.
D.
10. Solve for x
13. Solve for x and y.
14. Which statement must be true about the
triangle below?
11. Use diagram at right to find XZ
a. Find XZ.
15. Use the figure at the below.
b. If XY=10, find MO.
12. Solve for x and y.
a. What is the distance across the lake?
b. Is it shorter distance from A to B or from B to
C? Explain.
10 | P a g e
Polynomials
Adding & Subtracting polynomials – Add like terms, the exponents don’t change!
Ex:
Ex:
=
=
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:
=
Ex:
=
=
Ex:
=
=
Ex:
=
=
=
8. The floor of a rectangular cage has a length of
4 feet greater than the width, w. James will
increase both dimensions of the floor by 2 feet.
Which equation represents the new area, N, of
the floor of the cage?
A.
B.
C.
D.
1.
2.
3.
4.
9. Find the volume of a rectangular prism with a
length (4x-2), width (x+1), and height (x-5).
5.
6. Which expression is equivalent to
A.
C.
B.
D.
?
10. Which expression is equivalent to
7. Which of the following is equivalent to
?
A.
B.
C.
D.
A.
B.
C.
D.
11 | P a g e
Quadratic Review
Standard Form
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
Factored form:
Set each factor equal to zero and solve for variable.
Method 3) Quadratic Formula – works for every quadratic!!
use the a, b, c, from standard form.
Ex: Solve
=
Discriminant
If
If
If
the quadratic has TWO real solutions.
the quadratic has ONE real solutions.
the quadratic has NO real solutions. (2 imaginary)
12 | P a g e
1. Which function has exactly one solution?
A.
B.
C.
D.
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
The table contains the values for the quadratic g(x).
x
0
1
2
g(x)
9
33
25
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. The number of bacteria in a culture can be
modeled by the function
,
where t is the temperature, in degrees Celsius, the
culture is being kept. A scientist wants to have fewer
than 200 bacteria in a culture in order to test a
medicine effectively. What is the approximate
domain of temperatures that will keep the number of
bacteria under 200?
A.
B.
C.
D.
4. A flying squirrel jumps from one tree to the next.
The height in feet x seconds into the jump is given by
. When is the squirrel more than 12
feet above the ground?
A. x=0.29
C.
B. x=1.71
D.
5. A company found that its monthly profit, P, is given
by
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
6. 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
7. Which of the following is a factor of
?
A. (2a-3)
C. (2b-1)
B. (2a+3)
D. (2b+3)
8. If t is an unknown constant, which binomial must
be a factor of
?
A. (7m+t)
B. (m-t)
C. (m+2)
D. (m-2)
9. If a is an unknown constant, which binomial must
be a factor of
?
A. (3b+2)
B. (3a-2)
C. (a+1)
D. (a-1)
13 | P a g e
10. A rectangular rug is placed on a rectangular floor.
The width of the floor is 4 feet greater than the
length, x, of the floor. The width of the rug is 2 feet
less than the width of the floor. The length of the rug
is 4 feet less than the width of the rug. Which
function, R(x), represents the area of the floor not
covered by the rug.
A.
B.
C.
D.
11. What is the equation of a parabola with the vertex
(3, -20) and passes through the point (7, 12)?
A.
B.
17. Write
in standard form.
18. 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.
19. The towers of a suspended bridge are 800 feet apart
and rise 162 feet higher than the road. Suppose that the
cable between the towers has the shape of a parabola and
is 2 feet higher than the road at the point halfway between
the towers.
C.
D.
12. For the function
(-1, 8). What is c?
A. -13
B. -3
, the vertex is
C. 3
D. 13
13. A town is planning a playground. It wants to fence in a
rectangular space using an existing wall. What is the
greatest area it can fence in using 100 ft of donated
fencing?
14. The function
gives the cost, in dollars,
for a small company to manufacture x items. The function
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?
15. What is the discriminant of
What is the approximate height of the cable 120 feet from
either tower?
A. 80 ft
B. 74 ft
C. 22 ft
D. 16 ft
21. A sheet of cardboard with a length of 20 inches and a
width of 10 inches has a square cut out of each corner so it
can fold up into an open top box. Which of the following
equations could be used to find the volume of the box?
A.
C.
B.
D.
22. Congruent squares with side length x, are cut from the
corners of a 12-inch by 16-inch piece of cardboard to form
an open box. Which equation models the surface area, y, of
the open box after corners are cut away?
A.
B.
C.
D.
?
16. 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.
23. A rocket is launched. The function that models this
situation is
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?
14 | P a g e
Exponent Rules
Product of powers:
Quotient of powers:
Negative exponents:
or
Power of power:
Power of a quotient:
Power of a product:
Zero exponents:
Exponent Form:
Radical Form:
1. Simplify
5. Which expression is equivalent to
A. 12x
B. 36x
C. 12
D. 24
2. Simplify
3. Simplify
6. Which expression is equivalent to
4. Which expression is equivalent to
A.
C
B.
D.
?
A.
C
B.
D.
?
7. Simplify
15 | P a g e
Exponential Functions
Exponential Growth:
Exponential Decay:
b=1+r
Compound Interest
b=1-r
Interest Compounded Continuously
Half Life
Solving Exponential Equations
because bases are same
Ex: Solve for x.
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:
Step 1: Isolate the exponential expression.
Step 2: Take logarithm of both sides. Remember the exponent gets moved to multiply by the log(base).
Step 3: Simplify & Solve.
x=0.5396
1. In 1950, a U.S. population model was
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.
C.
B.
D.
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?
16 | P a g e
5. An investment has a balance of $2,000 and earns
3.2% interest each year. If $150 is added at the end of
each year by the account holder and no money is
withdrawn from the investment, which represents a
function that can be used to calculate the investment
balance for successive years?
A.
B.
C.
D.
6. The value, V of a car can be modeled by the
function
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?
7. The function
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?
8. Over a 10-year period, two colleges raised their
per-course tuitions (
each year. The
tuitions can be modeled by the following equations:
College 1:
College 2:
In these equations, the tuitions are in dollars, and x
represents elapsed time in years (x=0 is the
beginning of the 10 year period). Based on the model,
at approximately what time during the 10-year
period were the two tuitions equal?
A. 5 yrs B. 6 yrs C. 7 yrs D. 8 yrs
9. If the equation
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
10. Suppose a hospital patient receives medication
that is used up in the body according to the equation
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.
11. As the value of x becomes negative and continues
to decrease, what happens to the value of y in the
equations
A. y becomes negative
C. y gets closer to 0
B. y gets closer to 1
D. y gets closer to x
12. Solve
13. A city’s population, P (in thousands), can be
modeled by the equation
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?
14. A new automobile is purchased for $20,000. If
gives the car’s value after x years,
about how long will it take for the car to be worth
half its purchase price?
15. Solve for x:
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Solving Advanced Equations
Direct Variation
“y varies directly with x” Solve:
Ex: y varies directly with x. Find y If y is 2 when x is 3
find y when x is 6.
y=4
Inverse Variation
“y varies inversely with x” Solve:
Ex: Suppose y varies inversely with x. Find x when y
is 7, if y is 14 when x is 2.
x=4
The solution ¼ is an extraneous solution because it is
a solution to the transformed equation, not to the
original equation.
Ex. Solve
Step 1: Get a common denominator, in this case
2(x-1) It will eliminate the denominators altogether.
Step 2: Simplify.
Step 3: Solve for x.
Direct/Inverse Variation (combined)
“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.
Step 4: Check solutions in the original equation and
check for extraneous solutions (or excluded values).
1=1
Solving Rational and Radical Equations.
Ex: Solve
Step 1: Subtract 1 from each side to isolate the radical
term.
The solution -1 is an extraneous solution because -1
is an excluded value.
Step 2: Square both sides to eliminate the radical.
1. Solve for x:
Step 3: Set the right side equal to 0.
Step 4: Solve for x (quadratic so use factoring,
graphing or quadratic formula)
Step 5: Check solutions in the original equation and
check for extraneous solutions.
2. Solve for x:
3. For the function
i. Sketch a graph
4=4
so x=1/4 is not a solution.
So x=2 is a solution.
ii. State domain and range
iii. Describe the end behavior as x approaches
positive infinity. (increase or decrease?)
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4. For the function
i. Sketch a graph
ii. State domain and range
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?
iii. Describe the end behavior as x approaches
positive infinity. (increase or decrease?)
5. Solve for x:
6. Solve for x:
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.
10. The number of rotations of a bicycle wheel varies
directly with the number of pedal strokes. Suppose
that in the bicycle’s lowest gear, 6 pedal strokes more
the cyclist about 357 inches. In the same gear, how
many pedal strokes are needed to move 100 feet?
11. 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
12. 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
13. 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
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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
sin
Right Triangle Trig
opposite
adjacent
, cos
, and tan
hypotenuse
hypotenuse
opposite
adjacent
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.
Law of Sines
Law of Cosines
Use for A-S-A or A-A-S Triangles
Use for SSS or SAS triangles
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.
k
9.5
72
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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?
9. In the right triangle LMN, LN=728 cm and LM=700
cm. What is the approximate measure of <NLM?
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
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.
10. In the diagram below, Triangle MPO is a right
triangle and segment PN = 24 ft. Round to tenth.
i. what is the length of segment MP?
ii. How much longer is segment MO than segment
NM?
iii. How far is point O from point N?
11. What is the amplitude of y=3sin(4x)?
8. Which of the following functions is graphed below?
12. Graph
and midline.
. Identify they amplitude
A. 3sin(x) B. 3cos(x) C. sin(3x) D. cos(3x)
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13. 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?
17. What is the length of segment RS in the triangle?
A. 8.7 ft
B. 15.0 ft
C. 17.3 ft
D. 20.0ft
14. 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?
15. Find the area of the oblique triangle.
16. What is the approximate length of segment HJ in
the diagram?
A. 292 cm
B. 265 cm
C. 219 cm
D. 196 cm
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