f(x) and g(x) graphed below. - Dorchester School District Two

Magellan Mambou - Algebra 3 CP & PreCalculus - ARHS

Name: ____________________________________________ Class: _____________ Date: __________

Part 1

Consider the function f(x) and g(x) graphed below.

1. Find f (g (–2)).

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 𝑏(𝑥) = |𝑥 + 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 𝑦 = √5𝑥 + 10 − 3.

5. What is the domain of the function?

A) (2, + ∞) B) [−2 , +∞) C) [−7/5 , +∞) D) [2 , +∞)


Free Response: Write your response on a separate piece of paper. Show all steps to receive full credit.

7. Given the function 𝑀(𝑥) = √2𝑥 + 33

− 17. Find 𝑀−1(𝑥).

8. Prove that the functions f (x) and g (x) are inverses of each other by composing the functions.

𝑓(𝑥) =(𝑥−1)2+5

2 and 𝑔(𝑥) = √2𝑥 − 5

2+ 12

9. Solve the following equations:

3(5𝑥 − 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. 𝑦 = √5𝑥 + 10 − 3

A) 𝑦 = (𝑥 + 3)2 − 2 B) 𝑦 =𝑥2−1

5 C) 𝑦 =

(𝑥−3)2+10

5 D) 𝑦 =

(𝑥+3)2−10

5

Consider the function 𝒈(𝒙) = −𝟑𝒙𝟒 + 𝟓𝒙𝟑 − 𝟐𝒙𝟐 .

11. What is the degree of 𝑔(𝑥)?

A) −3 B) 5 C) −2 D) 4

12. Using limit notation, describe the end behavior of 𝑔(𝑥)


Consider the function 𝑔(𝑥) =2𝑥2−5𝑥+3

𝑥2+3𝑥−4

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) lim𝑥→−∞

𝑔(𝑥) = −∞, lim𝑥→+∞

𝑔(𝑥) = −∞ B) lim𝑥→−∞

𝑔(𝑥) = +∞, lim𝑥→+∞

𝑔(𝑥) = −∞

C) lim𝑥→−∞

𝑔(𝑥) = −∞, lim𝑥→+∞

𝑔(𝑥) = +∞ D) lim𝑥→−∞

𝑔(𝑥) = +∞, lim𝑥→+∞

𝑔(𝑥) = +∞

13. Factor 𝑔(𝑥) and list all zeros with their multiplicity (5 pts)

A) 𝑥2(𝑥 + 1)(−3𝑥 + 2); 𝑥 = 0 ; Multiplicity: 2, 𝑥 = −1 ; Multiplicity: 1, 𝑥 = 2/3 ; Multiplicity: 1

B) 𝑥2(𝑥 − 1)(−3𝑥 − 2); 𝑥 = 0 ; Multiplicity: 2, 𝑥 = 1 ; Multiplicity: 1, 𝑥 = −2/3 ; Multiplicity: 1

C) 𝑥2(𝑥 − 1)(−3𝑥 + 2); 𝑥 = 0 ; Multiplicity: 2, 𝑥 = 1 ; Multiplicity: 1, 𝑥 = 2/3 ; Multiplicity: 1

D) 𝑥2(𝑥 + 1)(−3𝑥 − 2); 𝑥 = 0 ; Multiplicity: 2, 𝑥 = −1 ; Multiplicity: 1, 𝑥 = −2/3 ; Multiplicity: 1


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 𝑙𝑜𝑔4(2𝑥𝑦)

a. log4 2 − log4 𝑥 − log4 𝑦

b. log4 2 + log4 𝑥 − log4 𝑦

c. log4 2 + log4 𝑥 + log4 𝑦

d. log4 2 − log4 𝑥 + log4 𝑦

6. Expand log8 (𝑥4

2𝑦𝑧2)3

a. 12 log8 𝑥 + 3 log8 2 − 3 log8 𝑦 − 6 log8 𝑧

b. 12 log8 𝑥 − 3 log8 2 + 3 log8 𝑦 − 6 log8 𝑧

c. 12 log8 𝑥 − 3 log8 2 − 3 log8 𝑦 + 6 log8 𝑧

d. 12 log8 𝑥 − 3 log8 2 − 3 log8 𝑦 − 6 log8 𝑧

e. 3 log8 𝑥4 − 3 log8 2 + 3 log8 𝑦 + 6 log8 𝑧

7. Expand 𝑙𝑜𝑔2 (𝑎𝑏3

𝑐)

a. log2 𝑎 − 3log2 𝑏 − log2 𝑐

b. log2 𝑎 − 3log2 𝑏 + log2 𝑐

c. log2 𝑎 + 3log2 𝑏 + log2 𝑐

d. log2 𝑎 + 3log2 𝑏 − log2 𝑐

e. log2 𝑎 + log2 𝑏 − 3 log2 𝑐

8. Expand 𝑙𝑜𝑔5(𝑐4𝑏 √𝑎3

)

a. 4 log5 𝑐 − log5 𝑏 −1

3log5 𝑎

b. 4 log5 𝑐 + log5 𝑏 −1

3log5 𝑎

c. 4 log5 𝑐 − log5 𝑏 +1

3log5 𝑎

d. 4 log5 𝑐 +1

3log5 𝑏 + log5 𝑎

e. 4 log5 𝑐 + log5 𝑏 +1

3log5 𝑎

Combine the following into a single logarithm.

9. Combine into a single log 4𝑙𝑜𝑔4(𝑥) +𝑙𝑜𝑔4(𝑦)

a. log4 𝑥4𝑦

b. log4 𝑥𝑦4

c. log4(𝑥𝑦)4

d. log 𝑥4𝑦

e. log 𝑥𝑦4

10. Combine into a single log 6 ln(𝑎) + ln(𝑏) −

4 ln(𝑐)

a. ln𝑎 𝑏6

𝑐4

b. ln𝑎6𝑏

𝑐4

c. ln𝑎6𝑏4

𝑐4

d. ln𝑎6𝑏

𝑐

Exponential and Logarithm Equations.

11. Solve the equation: 3𝑥+4 =1

27

13. Solve the equation: log(3𝑥 + 5) = log(5𝑥 + 1)


12. Solve the equation: log5(6𝑥 − 5) = 3

Express your answer as a fraction in the

simplest form.

14. Solve the equation log(8𝑥 − 4) = 2


15. Solve the equation: ln(𝑥) − ln(6) = 3

Round your answer to 4 decimal places.

16. Solve the equation ln(𝑥 + 3)2 = 10

Round your answer to 4 decimal places.

17. Solve the equation 2 ∙ 32𝑥−1 + 7 = 61

18. Solve the equation: 5−3𝑥 = 25

Express your answer as a fraction in the

simplest form.

19. Solve the equation 1

4log3(𝑥) + 3 = 4

20. Solve the equation 32𝑥+1 = 5𝑥+2

Round your result to 4 decimal places


21. Solve the equation:

log2(𝑥) + log2(𝑥 − 3) = 2


2 log4(𝑥) − log4(𝑥 − 1) = 1


log2(𝑥 − 7) − log2(𝑥 + 5) = 2

24. Solve the equation 4𝑒𝑥 = 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 𝑚(𝑡) = 20𝑒−(0.087𝑡), 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.


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) 𝑝 (−5

7,

−2√6

7)

b) 𝑞 (4

3,

−3

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) 𝑡 = 190°

𝑡 =−3𝜋

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) 𝑡 = 210° b) 𝑡 = 108°

4. Convert the angle below from radians to degrees.

a) 𝑡 = −10𝜋3⁄ radians

b) 𝑡 =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) 𝑝 (√3

2, 𝑦) , Quadrant IV

b) 𝑞 (

−3

5, 𝑦) , Quadrant III


6. Find the reference number for each value of t. Express your answer as fraction simplest form in term

of 𝜋. NO DECIMAL

a) 𝑡 = 9𝜋15⁄ b) 𝑡 = −11𝜋

3⁄

7. If 𝐬𝐢𝐧(𝒕) = −𝟑

𝟕 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(𝑡) = −𝟑

𝟕 b) cos(𝑡) = c) tan (𝑡) =

d) csc (𝑡) = e) sec(𝑡) = f) cot (𝑡) =

8. If 𝐜𝐨𝐬(𝒕) = 𝟓

𝟐𝟏 and sin(𝑡) < 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(𝑡) = b) cos(𝑡) =5

21 c) tan (𝑡) =

d) csc (𝑡) = e) sec(𝑡) = f) cot (𝑡) =

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) = __________


Part 4 – 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 .


Part 4 – 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.


Part 4 – 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 “none”.

Vertical asymptote(s): _______________ Horizontal asymptote(s): __________________

X-intercept: _______________________ Y – 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


Formulas:

𝑐𝑠𝑐2(𝑡) = 1 + 𝑐𝑜𝑡2(𝑡) 𝑠𝑖𝑛2(𝑡) + 𝑐𝑜𝑠2(𝑡) = 1 𝑡𝑎𝑛2(𝑡) + 1 = 𝑠𝑒𝑐2(𝑡)


Compound Interest Formula:

Compound interest is calculated by the formula 𝑨(𝒕) = 𝑷 (𝟏 +𝒓

𝒏)

𝒏𝒕

Where:

𝐴(𝑡) = 𝐴𝑚𝑜𝑢𝑛𝑡 𝑎𝑓𝑡𝑒𝑟 𝑡 𝑦𝑒𝑎𝑟𝑠

𝑃 = 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 (initial amount)

𝑟 = 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟

𝑛 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑖𝑠 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑒𝑑 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟

𝑡 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠

For a simple interest, 𝑛 = 1 𝑎𝑛𝑑 𝑡 = 1 the formula becomes 𝑨(𝒕) = 𝑷(𝟏 + 𝒓)

For compounded continuously, 𝑨(𝒕) = 𝑷(𝒆)𝒓𝒕

Properties of Logarithm Functions

log𝑎 1 = 0 (We must raise a to the power of 0 to get 1)

log𝑎 𝑎 = 1 (We must raise a to the power of 1 to get a)

log𝑎 𝑎𝑥 = 𝑥 (We must raise a to the power of x to get 𝑎𝑥)

𝑎log𝑎 𝑥 = 𝑥 (log𝑎 𝑥 is the power to which must be raised to get x)

Laws of Logarithms

Let b be a positive number, with 𝑏 ≠ 1. Let R, S and C be any real number with, 𝑅 > 0, 𝐶 > 0, 𝑎𝑛𝑑 𝑆 > 0

Product Rule: log𝑏(𝑅𝑆) = log𝑏 𝑅 + log𝑏 𝑆

Quotient Rule: log𝑏(𝑅/𝑆) = log𝑏 𝑅 − log𝑏 𝑆

Power Rule: log𝑏 𝑅𝑐 = 𝑐log𝑏 𝑅

f(x) and g(x) graphed below. - Dorchester School District Two

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