Post on 18-Dec-2021
Magellan Mambou - Algebra 3 CP & PreCalculus - ARHS Page 1
Name: ____________________________________________ Class: _____________ Date: __________
Part 1
Consider the function f(x) and g(x) graphed below.
1. Find f (g (β2)).
2. Find g (f (2)).
Consider the function graphed below.
3. What is the average rate of change for the function below from
x = 0 to x = 3.
Express your answer as a fraction simplest form.
Boundedness, symmetry, domain and range.
4. For the function π(π₯) = |π₯ + 1| , graphed beside, which of the
following statements are true?
I. b (x) is bounded below
II. b (x) has a domain of ,
III. b (x) has even symmetry
A) I, II, and III B) II, and III only
C) I, and III only D) I, and II only
Consider the function π¦ = β5π₯ + 10 β 3.
5. What is the domain of the function?
A) (2, + β) B) [β2 , +β) C) [β7/5 , +β) D) [2 , +β)
Magellan Mambou - Algebra 3 CP & PreCalculus - ARHS Page 2
Free Response: Write your response on a separate piece of paper. Show all steps to receive full credit.
7. Given the function π(π₯) = β2π₯ + 33
β 17. Find πβ1(π₯).
8. Prove that the functions f (x) and g (x) are inverses of each other by composing the functions.
π(π₯) =(π₯β1)2+5
2 and π(π₯) = β2π₯ β 5
2+ 12
9. Solve the following equations:
3(5π₯ β 6)2 = 75
10. Which of the following gives the zeros of the graph and their multiplicity?
A) x = 1 (multiplicity 1); x = 3 (multiplicity 2)
B) x = 1 (multiplicity 2); x = 2 (multiplicity 1)
C) x = 1 (multiplicity 3); x = 3 (multiplicity 1)
D) x = 1 (multiplicity 2); x = 3 (multiplicity 3)
E) x = 1 (multiplicity 1); x = 2 (multiplicity 3)
6. Find the inverse of the function. π¦ = β5π₯ + 10 β 3
A) π¦ = (π₯ + 3)2 β 2 B) π¦ =π₯2β1
5 C) π¦ =
(π₯β3)2+10
5 D) π¦ =
(π₯+3)2β10
5
Consider the function π(π) = βπππ + πππ β πππ .
11. What is the degree of π(π₯)?
A) β3 B) 5 C) β2 D) 4
12. Using limit notation, describe the end behavior of π(π₯)
Magellan Mambou - Algebra 3 CP & PreCalculus - ARHS Page 3
Consider the function π(π₯) =2π₯2β5π₯+3
π₯2+3π₯β4
14. What is the domain of h (x)?
15. What is the equation of vertical asymptotes of
g(x)?
Write the unique equation in standard form of the polynomial in the following conditions.
16. Zeros: -2 , 3 and 1/2, leading coefficient: 2
17. Zeros: -1 , and 1+2i , leading coefficient: 1
Part 2
1. If you invest $2,000 at an annual interest rate of 13% compounded daily, calculate the final amount
you will have in the account after 5 years. Round your answer to 2 decimal places.
2. A sum of $1000 was invested for 5 years, and
the interest was compounded semiannually. If
this sum amounted to $1534.49 at the given
time, what was the interest rate? Round your
answer to 4 decimal places
3. A woman invests $6500 in an account that
pays 6% interest per year compounded
continuously. How long will it take for the
amount to be $8000?
Round your answer to 2 decimal places
A) limπ₯βββ
π(π₯) = ββ, limπ₯β+β
π(π₯) = ββ B) limπ₯βββ
π(π₯) = +β, limπ₯β+β
π(π₯) = ββ
C) limπ₯βββ
π(π₯) = ββ, limπ₯β+β
π(π₯) = +β D) limπ₯βββ
π(π₯) = +β, limπ₯β+β
π(π₯) = +β
13. Factor π(π₯) and list all zeros with their multiplicity (5 pts)
A) π₯2(π₯ + 1)(β3π₯ + 2); π₯ = 0 ; Multiplicity: 2, π₯ = β1 ; Multiplicity: 1, π₯ = 2/3 ; Multiplicity: 1
B) π₯2(π₯ β 1)(β3π₯ β 2); π₯ = 0 ; Multiplicity: 2, π₯ = 1 ; Multiplicity: 1, π₯ = β2/3 ; Multiplicity: 1
C) π₯2(π₯ β 1)(β3π₯ + 2); π₯ = 0 ; Multiplicity: 2, π₯ = 1 ; Multiplicity: 1, π₯ = 2/3 ; Multiplicity: 1
D) π₯2(π₯ + 1)(β3π₯ β 2); π₯ = 0 ; Multiplicity: 2, π₯ = β1 ; Multiplicity: 1, π₯ = β2/3 ; Multiplicity: 1
Magellan Mambou - Algebra 3 CP & PreCalculus - ARHS Page 4
4. The present value of a sum of money is the amount that must be invested now, at a given rate, to
produce the desired sum at a later date. Find the present value of $2000 if interest is paid at a rate of
6% per year compound semiannually, for 8 years. Round your answer to 2 decimal places.
Use the laws of logarithms to expand each expression.
5. Expand πππ4(2π₯π¦)
a. log4 2 β log4 π₯ β log4 π¦
b. log4 2 + log4 π₯ β log4 π¦
c. log4 2 + log4 π₯ + log4 π¦
d. log4 2 β log4 π₯ + log4 π¦
6. Expand log8 (π₯4
2π¦π§2)3
a. 12 log8 π₯ + 3 log8 2 β 3 log8 π¦ β 6 log8 π§
b. 12 log8 π₯ β 3 log8 2 + 3 log8 π¦ β 6 log8 π§
c. 12 log8 π₯ β 3 log8 2 β 3 log8 π¦ + 6 log8 π§
d. 12 log8 π₯ β 3 log8 2 β 3 log8 π¦ β 6 log8 π§
e. 3 log8 π₯4 β 3 log8 2 + 3 log8 π¦ + 6 log8 π§
7. Expand πππ2 (ππ3
π)
a. log2 π β 3log2 π β log2 π
b. log2 π β 3log2 π + log2 π
c. log2 π + 3log2 π + log2 π
d. log2 π + 3log2 π β log2 π
e. log2 π + log2 π β 3 log2 π
8. Expand πππ5(π4π βπ3
)
a. 4 log5 π β log5 π β1
3log5 π
b. 4 log5 π + log5 π β1
3log5 π
c. 4 log5 π β log5 π +1
3log5 π
d. 4 log5 π +1
3log5 π + log5 π
e. 4 log5 π + log5 π +1
3log5 π
Combine the following into a single logarithm.
9. Combine into a single log 4πππ4(π₯) +πππ4(π¦)
a. log4 π₯4π¦
b. log4 π₯π¦4
c. log4(π₯π¦)4
d. log π₯4π¦
e. log π₯π¦4
10. Combine into a single log 6 ln(π) + ln(π) β
4 ln(π)
a. lnπ π6
π4
b. lnπ6π
π4
c. lnπ6π4
π4
d. lnπ6π
π
Exponential and Logarithm Equations.
11. Solve the equation: 3π₯+4 =1
27
13. Solve the equation: log(3π₯ + 5) = log(5π₯ + 1)
Magellan Mambou - Algebra 3 CP & PreCalculus - ARHS Page 5
12. Solve the equation: log5(6π₯ β 5) = 3
Express your answer as a fraction in the
simplest form.
14. Solve the equation log(8π₯ β 4) = 2
Exponential and Logarithm Equations.
15. Solve the equation: ln(π₯) β ln(6) = 3
Round your answer to 4 decimal places.
16. Solve the equation ln(π₯ + 3)2 = 10
Round your answer to 4 decimal places.
17. Solve the equation 2 β 32π₯β1 + 7 = 61
18. Solve the equation: 5β3π₯ = 25
Express your answer as a fraction in the
simplest form.
19. Solve the equation 1
4log3(π₯) + 3 = 4
20. Solve the equation 32π₯+1 = 5π₯+2
Round your result to 4 decimal places
Exponential and Logarithm Equations.
21. Solve the equation:
log2(π₯) + log2(π₯ β 3) = 2
22. Solve the equation:
2 log4(π₯) β log4(π₯ β 1) = 1
23. Solve the equation:
log2(π₯ β 7) β log2(π₯ + 5) = 2
24. Solve the equation 4ππ₯ = 20 Round your answer to 4 decimal places.
25. A 15-g sample of radioactive decay iodine decays in such a given way that the mass remaining after
t days is given by π(π‘) = 20πβ(0.087π‘), where m(t) is measured in grams. After how many days is
there only 5 g remaining? Round your answer to the nearest whole day.
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Part 3
1. Show that the points below are on the unit circle. (Show work for full credit). All results must be
expressed as fraction simplest form or a radical simplest form. NO DECIMAL
a) π (β5
7,
β2β6
7)
b) π (4
3,
β3
5)
2. Find one positive and one negative co-terminal angle for each given angle. Express your answer in
term of π for radian angles, and decimal for degree angles.
a) π‘ = 190Β°
π‘ =β3π
7 radians
3. Convert the angle below from degrees to radians. Express your answer as a fraction in a simplest
form and in term of π
a) π‘ = 210Β° b) π‘ = 108Β°
4. Convert the angle below from radians to degrees.
a) π‘ = β10π3β radians
b) π‘ =7π
4 radians
Locating a Point on the Unit Circle
5. The given point is on the unit circle. Find the missing coordinate of the point given its quadrant. All
results must be expressed as fraction simplest form or a radical simplest form. NO DECIMAL
a) π (β3
2, π¦) , Quadrant IV
b) π (
β3
5, π¦) , Quadrant III
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6. Find the reference number for each value of t. Express your answer as fraction simplest form in term
of π. NO DECIMAL
a) π‘ = 9π15β b) π‘ = β11π
3β
7. If π¬π’π§(π) = βπ
π and t is in quadrant IV, Find the value of all six trigonometric functions at t. All
results must be expressed as fraction simplest form or a radical simplest form. NO DECIMAL
a) sin(π‘) = βπ
π b) cos(π‘) = c) tan (π‘) =
d) csc (π‘) = e) sec(π‘) = f) cot (π‘) =
8. If ππ¨π¬(π) = π
ππ and sin(π‘) < 0 Find the value of all five trigonometric functions at t. All results
must be expressed as fraction simplest form or a radical simplest form. NO DECIMAL
a) sin(π‘) = b) cos(π‘) =5
21 c) tan (π‘) =
d) csc (π‘) = e) sec(π‘) = f) cot (π‘) =
9. Find the exact value of each expression. Put your answer in the blank. All results must be expressed
as fraction simplest form or a radical simplest form. NO DECIMAL
a) 5
tan4
= __________ b) csc2
= ________ c)
5sin
3
= ________
d) 11
cos6
= __________ e) cot 270 = _______
f) sin 210
= __________
g) 3
cos4
= _________ h) sec6
= __________ i) csc (2π
3) = __________
Magellan Mambou - Algebra 3 CP & PreCalculus - ARHS Page 8
Part 4 β Level 1
1.1) The graph of 2
2 1 40.2f x x x x is shown below. For what values of x is 0f x
1.2) Evaluate 5
tan6
and write your answer in simplest radical form.
1.3) Identify the amplitude and period from the
graph.
Amplitude: ____________
Period: ________________
1.4) Determine the horizontal asymptote (if any) of the function 2 3 2
3 2 4 3
x xg x
x x
1.5) If 3
54 9P x x , find an equation for 1P x .
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Part 4 β Level 2
2.1) Graphically solve the following inequality for x: 3log 5 2x
2.2) Evaluate 11
sec6
and write your answer in simplest radical form.
2.3) Identify the amplitude, period, vertical shift, and phase shift of the following trigonometric
function:
5
4cos 6 93
y x
Amplitude: _______________ Vertical Shift: ___________________
Period: __________________ Phase Shift: _____________________
2.4) Determine the slant (or oblique) asymptote for the function 3
2
5 4 1
2 3
x xf x
x x
2.5) Prove that 4 23 1xf x and 3log 1 2
4
xg x are inverses using composite functions.
Magellan Mambou - Algebra 3 CP & PreCalculus - ARHS Page 10
Part 4 β Level 3
3.1) For what values of x will 2
2
3 4 40
2 1
x x
x x
3.2) Let 1 cos
2sin 22
xxf x and h (x) be the graph
given. Find h f
3.3) Rounded to the nearest hour, Los Angeles averages 14 hours of daylight in June, 10 hours in
December, and 12 hours in March and September. Sketch a graph that displays the information
from June of one year to June of the following year and write an equation to model the daylight
using cos (x).
3.4) Determine all asymptotes, intercepts, and holes of the rational function 2
2
6 15
14 19 3
x xf x
x x
If they do not exist, simply state βnoneβ.
Vertical asymptote(s): _______________ Horizontal asymptote(s): __________________
X-intercept: _______________________ Y β intercept (s): _________________________
Hole(s): __________________________ Slant (Oblique) asymptote(s): ______________
3.5) Rewrite the expression below as an algebraic expression (an expression with no trig in it).
tan arcsin7
x
Magellan Mambou - Algebra 3 CP & PreCalculus - ARHS Page 11
Formulas:
ππ π2(π‘) = 1 + πππ‘2(π‘) π ππ2(π‘) + πππ 2(π‘) = 1 π‘ππ2(π‘) + 1 = π ππ2(π‘)
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Compound Interest Formula:
Compound interest is calculated by the formula π¨(π) = π· (π +π
π)
ππ
Where:
π΄(π‘) = π΄πππ’ππ‘ πππ‘ππ π‘ π¦ππππ
π = πππππππππ (initial amount)
π = πΌππ‘ππππ π‘ πππ‘π πππ π¦πππ
π = ππ’ππππ ππ π‘ππππ πππ‘ππππ π‘ ππ ππππππ’ππππ πππ π¦πππ
π‘ = ππ’ππππ ππ π¦ππππ
For a simple interest, π = 1 πππ π‘ = 1 the formula becomes π¨(π) = π·(π + π)
For compounded continuously, π¨(π) = π·(π)ππ
Properties of Logarithm Functions
logπ 1 = 0 (We must raise a to the power of 0 to get 1)
logπ π = 1 (We must raise a to the power of 1 to get a)
logπ ππ₯ = π₯ (We must raise a to the power of x to get ππ₯)
πlogπ π₯ = π₯ (logπ π₯ is the power to which must be raised to get x)
Laws of Logarithms
Let b be a positive number, with π β 1. Let R, S and C be any real number with, π > 0, πΆ > 0, πππ π > 0
Product Rule: logπ(π π) = logπ π + logπ π
Quotient Rule: logπ(π /π) = logπ π β logπ π
Power Rule: logπ π π = πlogπ π