PreCalc and Trig

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Name: _____________________________

MTH129 Fall 2018 - FINAL EXAM A

Show all work neatly on paper provided. Label all work. Place final answers on the answer sheet.

PART I: Omit 1 complete question. Place an “X” on the problems & answer space you are omitting.

1. Find the inverse of the following functions:

a. 𝑓(𝑥) = 2𝑥 − 3

b. 𝑓(𝑥) = 3𝑥 +1

𝑥−2

2. If 𝑓(𝑥) = 𝑥 2 − 2𝑥 + 3 and 𝑔(𝑥) = −3𝑥 + 4, find the following: a. (𝑓°𝑔)(𝑥) b. (𝑓°𝑔)(2)

3. Find the domain for the following expression:

a) √𝑥 + 5 𝑏) 7𝑥 2 + 3𝑥 − 1 𝑐) 𝑥 2+4

𝑥 2−9

4. Find the radian measures of the angles with the given degree measures.

a) 81° Find the degree measures of the angles with the given radian measures.

b) 13𝜋

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5. Solve the following equations:

a) (5t) = 20 b) 6000 = 40(15)t

6. Expand the following logarithmic expressions:

a. log(𝐴𝐵2 )

b. ln( 4

√3 )

7. Describe how the graph of each function can be obtained from the graph f

a. 𝑦 = 𝑓(𝑥) − 8 b. 𝑦 = 𝑓(𝑥 + 4) − 5

8. A real number t is given 𝑡 = 2𝜋

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a. Find the reference number for t. b. Find the terminal point P(x,y) on the unit circle determined by t c. The unit circle is centered at __________________ and has a radius of _________________

PART II: Omit 1 complete question. Place an “X” on the problems & answer space you are omitting.

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1. A sum of $7,000 is invested at an interest rate of 4 1

2 % per year, compounding monthly. (round all answers to

the nearest cent)

a. Find the amount of the investment after 2 1

2 years.

b. How long will it take for the investment to amount to $12,000? c. Using the information in part (a), find the amount of the investment if compounded quarterly.

2. When a company charges price p dollars for one of its products, its revenue is given by

𝑅 = 𝑓(𝑝) = 500𝑝(30 − 𝑝) a. Create a quadratic function for price with respect to revenue.

b. What price should they charge in order to maximize their revenue?

c. What is the maximum revenue?

d. What would be the revenue if the price was set at $10?

e. Sketch a rough graph – indicate the intercepts and the maximum coordinates.

3. The charges for a taxi ride are an initial charge of $2.50 and $0.85 for each mile driven.

a. Write a function for the charge of a taxi ride as a linear function of the distance traveled.

b. What is the cost of a 12 mile trip?

c. Find the equation of a line that passes through the following points: (1,-2) , (2,5) Express in 𝑦 = 𝑚𝑥 + 𝑏 form d. Graph part ( c )

4. a. Divide the following polynomial and factor completely.

𝑃(𝑥) = 3𝑥 4 − 9𝑥 3 − 2𝑥 2 + 5𝑥 + 3; 𝑐 = 3 b. Given polynomial−𝑥 2 + 5𝑥 − 6, state the end behavior of its graph. c. Using the polynomial on part ( c ), would this graph have a maximum or a minimum? State why.

5. An item cost $1.00 in 2000. The same item $1.37 in the year 2014. Let’s assume that the growth of the dollar

is following an exponential pattern. (Round rates to the 6th decimal place; money to the nearest cent)

a. Find an exponential function that will model the data given above.

b. How much would the item cost in 2010?

c. When would the item cost $1.50?

6. Given rational function 𝑟 = 𝑥2−𝑥−2

𝑥2−2𝑥−3 , determine the following:

a. x intercept (s)

b. y – intercept(s)

c. vertical asymptote(s)

d. horizontal asymptote(s)

7. a. Determine if the following point lies on the unit circle and state why it does or why it does not: ( 4

5 , −

3

5 )

b. For the given value 𝑡 = − 3𝜋

4 : Find the values of sin t , cos t and tan t

c. For part ( b) state what quadrant the point land in.

8. a. A 20 foot ladder leans against a building so that the angle between the ground and the ladder is 72°.

How high does the ladder reach on the building? (draw a sketch to help you)

b. 5. Use the Law of Cosine and the Law of Sine to find the remailing parts of the triangle:

𝛼 = 130°; 𝑏 = 5; 𝑐 = 7

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FORMULAS:

𝑎2 + 𝑏2 = 𝑐 2 𝑥 = − 𝑏

2𝑎 𝑥 =

−𝑏±√𝑏2−4𝑎𝑐

2𝑎

𝐴 = 𝑃(1 + 𝑟

𝑛 )(𝑛∗𝑡) 𝑦 = 𝑦0 + 𝑚(𝑥 − 𝑥0) 𝑡𝑎𝑛𝜃 =

𝑠𝑖𝑛𝜃

𝑐𝑜𝑠𝜃

𝑠𝑖𝑛2𝜃 + 𝑐𝑜𝑠2𝜃 = 1 SOH CAH TOA

𝑦 = 𝑚 = ∆𝑦

∆𝑥 𝑆 = 𝑟𝜃

𝑐𝑖𝑟𝑐𝑙𝑒 𝑟2 = (𝑥 − ℎ) 2 + (𝑦 − 𝑘)2

𝑛(𝑡) = 𝑛0𝑒 𝑟𝑡

𝑎2 = 𝑏2 + 𝑐 2 − 2𝑏𝑐 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼

𝑎 =

𝑠𝑖𝑛𝛽

𝑏 =

𝑠𝑖𝑛𝛾

𝑐

𝑦 − 𝑦0 = 𝑚(𝑥 − 𝑥0) 𝑓(𝑥) = 𝑎(𝑥 − ℎ) 2 + 𝑘