Name: MAT104 College Algebra with Trigonometry I Practice Final Exam … · MAT104 Practice Final Exam - Page 12 of 19 40. A marketing firm would like to estimate the sales of a product

MAT104 Practice Final Exam - Page 1 of 19

Name:

MAT104 – College Algebra with Trigonometry I Practice Final Exam with Answer Key

Directions: The purpose of this packet is to provide you with the general scope and level of

difficulty of the class exam. Complete each of the problems in this packet (be sure to show all of

your work to earn credit) and check your answers with those provided at the end of the packet.

You are encouraged to work in groups, see me for help during my office hours or seek help at

our Tutoring Center. Remember that all answers should be exact (unless otherwise noted), fully

simplified and in proper form.

1. Evaluate 13 216x y for 8 and 4x y

2. Determine the domain of 1

5 8

x

x

. Place your answer in set notation or interval notation

3. Multiply 2(5 7)(3 8 5)x x x

4. Divide using long division or synthetic division 4 2( 5 8 3) ( 3)x x x x .

b. Indicate if ( 3)x is a factor of the polynomial 4 2( 5 8 3)x x x


5. Factor completely over the integers: 264 9a

6. Factor completely over the integers: 38 27x

7. Perform the operation and simplify: 2 2

2 2

6 9 7 12

2 2 6 2 2

x x x x

x x x x

8. Perform the operation and simplify: 2 2

2 1

4 3 10

x x

x x x

9. Perform the operation and simplify: 5 4 3 15

10. Simplify:

22

4

4

6

x y

xy

11. Simplify: 4 63 64x y


12. Find all real number solutions: 3(2 5) 1 2( 5)x x

13. Find all real number solutions and check your result: 3 4

3 3

x

x x

show check here

14. Find all real number solutions: 4 210 9 0x x

15. Find all real number solutions and check your result: 2 2x x

show check here


16. Find all real number solutions by completing the square: 2 2 6x x

17. Perform the operation in the complex number system: 3 (2 5 ) 2 (3 4 )i i i i

18. Simplify: 66i

19. a. Use the discriminant to determine the nature of the solutions of 22 4 4 0x x in the

complex number system.

b. Solve in the complex number system: 22 4 4 0x x


20. Solve 15 2

1 53

x . Express your answer in interval notation and graph the solution

set.

graph

interval notation

21. Solve the inequality 3 22 24 2x x x and express your solution using interval notation.

22. Find the midpoint of the line segment connecting the points (-5, -2) and (2, -3)

23. Find the distance between the points (1, -1) and (-5, -3)


24. Find the intercepts of 4 3 15x y and graph it using the intercepts.

x-intercept(s): y-intercept(s):

25. Find the equation of the line containing the points (-5, 6) and (-3, -4). Place your final

answer in slope-intercept form.

26. Find the equation of the horizontal line that passes through the point (-3, 2).

27. Find the slope of the line parallel to 2 3 7x y .


28. Indicate whether the graph below is symmetric with respect to the x-axis, the y-axis,

and/or the origin. Justify your response.

29. Indicate if the graph above (see #28) is the graph of a function. Justify your response.

30. Find the standard form of the equation of a circle whose center is (-7, 0) and radius is 4.

31. Find each of the following for 2( ) 5f x x x and ( ) 4 3g x x

a. ( )( )f g x

b. ( 3)f

g

c. ( )( )f g x


32. a. Is the function 3 9

( )7

xf x

one-to-one. Explain.

b. Write the equation for the inverse of the function 3 9

( )7

xf x

.

33. Use the graph of the function f below to answer the following:

a. Determine the domain and range of f

domain:

range:

b. For what values of x does f(x)>0

c. Determine the interval where the function is increasing

d. Determine the interval where the function is decreasing

e. Determine the interval where the function is constant


34. Use your graphing calculator to graph 4 2( ) 4 37 9f x x x and to find any intercepts

and local maxima and minima. Round answers to two decimal places when appropriate.

[Include your viewing window with your sketch.]

local maxima: x-intercept(s):

local minima: y-intercept(s):

35. Consider the function ( ) 5 1f x x

a. Identify the base function.

b. Identify all transformation on the base function

c. Sketch f using the information above


36. Consider the function 4

( ) 65

h x x .

a. Indicate the slope and coordinate of the y-intercept of the linear function; graph

the function.

slope: y-intercept:

b. Determine the average rate of change for the function.

37. Quahog Long Distance charges Peter a monthly fee of $7.00 plus $0.06 per minute.

a. Find a linear function that relates cost, C, to total minutes, x.

b. Determine how much it will cost Peter if he uses 120 minutes of long-distance

over the period of a month.

c. If Lois demands that Peter keep the bill under $80, how many minutes can he use

the phone next month?


38. Suppose that the quantity supplied, S, and the quantity demanded, D, of Adirondack

Gourmet Chocolates are given by the following functions:

( ) 800 100

( ) 200 500

S p p

D p p

where p is the price of a chocolate.

a. Sketch the functions together so that their intersection can be viewed; remember

the implied restrictions on the variables. Include your viewing window.

b. Find the equilibrium price for the chocolates. What is the equilibrium quantity?

c. Determine the prices for which the quantity demanded is greater than the quantity

supplied.

39. Quahog Golf Company has determined the marginal cost C of manufacturing x Big

Bertha golf clubs may be expressed by 2( ) 4.9 617.4 19600C x x x .

a. How many clubs should be manufactured to minimize the marginal cost?

b. Describe what happens to the marginal cost as the number of Big Bertha golf

clubs manufactured increases from 70 to 80 clubs.


40. A marketing firm would like to estimate the sales of a product (in thousands of dollars),

S, given the amount spent on advertising (in thousands of dollars), A. The collected data

are given in the table below:

Advertising

Expenditure (A)

20 22 22.5 24 24 27 28.3

Sales (S) 335 339 315 343 341 350 351

a. Identify the dependent variable and the independent variable.

b. Sketch a scatterplot for the data; include your viewing window and scale.

c. Interpret the scatterplot by describing/identifying each of the following:

Form:

Outliers:

Direction:

Strength:

d. Find the equation of best fit for the data and sketch your best-fit equation over

your scatterplot on part (b)

e. Interpret the slope.

f. Predict sales if expenditures are $25,000. Is it advisable to use our model to make

this prediction?


41. An engineer collects data on the speed of a Ford Taurus, s, and its average miles per

gallon, m.

Speed (s) 30 35 40 40 45 50 55 60 65 65 70

Miles per gallon

(m)

18 20 23 25 25 28 30 29 26 25 25

a. Identify the dependent variable and the independent variable.

b. Sketch a scatterplot for the data; include your viewing window and scale.

c. Interpret the scatterplot by describing/identifying each of the following:

Form:

Outliers:

Strength:

d. Find the equation of best fit for the data and sketch your best-fit equation over

your scatterplot on part (b)

e. Over what values can we use our model to make predictions? Explain.

f. Predict the miles per gallon a Ford Taurus would get at a speed of 20 miles per

hour. Is it advisable to use our model to make this prediction?


42. Consider the quadratic function 2( ) ( 2) 2f x x .

a. Will the graph open up or down? Explain.

b. Find the intercepts, axis of symmetry and vertex

x-intercept(s): y-intercept(s):

axis of symmetry: vertex:

c. Graph 2( ) ( 2) 2f x x ; label the intercepts, axis of symmetry and vertex.

43. A person standing close to the edge on the top of a 160 ft. building throws a baseball

upward. The projectile of the baseball resembles the quadratic function 2( ) 16 64 160s t t t where ( )s t is the height above the ground in feet and t is seconds

after the ball is thrown. Round answers to two decimal places when appropriate.

a. What is the maximum height reached by the ball?

b. How many seconds until the ball hits the ground?


44. Convert to degrees: 5

6

45. Find the exact value of the 6 trigonometric functions of .

46. Find the exact value of the remaining five trigonometric functions of the acute angle ,

given that 2

sin2

.

47. Suppose that you are headed toward a plateau 50 meters high. If the angle of elevation to

the top of the plateau is 20 . How far are you from the base of the plateau?


MAT104 – College Algebra with Trigonometry I Practice Final Exam Key

1. 2 17. 7 12i

2. 8 8 85 5 5

( , ) ( , )D x x 18. 1

3. 3 215 61 31 35x x x 19. There will be 2 complex solutions (not real)

b. 1x i

4. a. 3 23 4 4 R 15x x x

b. No, there is a remainder.

20. 9 0x ; interval notation: [0,9)

5. (8 3)(8 3)a a 21. ( 4,0) (3, )

6. 2(2 3)(4 6 9)x x x 22. 1 12 2

1 , 2

7. 2

( 1)( 3)

( 4)( 3)

x x x

x x x

23. 2 10

8. 2 2

( 5)( 2)( 2)

x

x x x

24. x-int: 34

3 ,0 ; y-int: (0,5)

9. 2 5 15 3

10. 2 10

9

4x y

11. 2 34xy x

12. 3x

13. 1 + checkx (x = 3 is extraneous)

14. 1, 3x x

15. 2 + checkx (x = 1 is extraneous) 25. 5 19y x

16. 1 7x 26. 2y



27. 2

3m 35. a. base function: y x

b. transformations: (1) reflection over the x-

axis, (2) horizontal shift left 5 units, (3)

vertical shift down 1 unit

28. It is symmetric with respect to the x-axis because

the part of the graph above the x-axis is a mirror

image of the part below it.

It is symmetric with respect to the y-axis because the

part of the graph to the left of the y-axis is a mirror

image of the part to the right of it.

It is symmetric with respect to the origin because the

graph is the same when viewed after rotation of 180

degrees.

29. No, it fails the vertical line test (VLT).

30. 2 2( 7) 16x y

36. a.

4

5m y-intercept: (0, -6)

b. average rate of change is 4

5

31. 2( ) 2 4f g x x x

2413

( 3)f

g

2( ) 9 9 4f g x x x

32. a. domain: [ 6,5] ; range: [ 4,4]

b. [ 6, 2.5) c. (0,2)

d. ( 6, 1) (2,5) e. ( 1,0)

33. a. Yes, it passes the horizontal line test.

b. 1 7 9( )

3

xf x

34. local maxima: (0,9)

local minima: ( 2.15, 76.56) and (2.15, 76.56)

1 12 2

ints: ( 3,0),( ,0),(3,0),( ,0); int: (0,9)x y

37. a. ( ) 7.00 0.06C x x

b. (120) 14.2C

It will cost Peter $14.20 to use 120 minutes

of long distance during the month.

c. Peter can use a maximum of 1216 minutes

next month.



38. a.

b. The equilibrium price is $0.60; the

equilibrium quantity is 380

c. The demand is greater than supply when

price is less than $0.60.

41. a. dependent: mpg

independent: speed

c. form: quadratic

outliers: none

strength: moderate

d. 20.017 1.935 25.341y x x

e. We can use the model to predict mpg for

speeds between 30-70 mph.

f. 6.36mpg ; it is not advisable to make this

prediction as 20mph falls outside 0f 30-70 mph

42. a. It will open up as the leading coefficient

is positive.

b. int: none ; int: (0,6)x y

axis of symmetry: 2 ; vertex: (2,2)x

39. a. 63 clubs should be manufactured to

minimize marginal cost.

b. The marginal cost increases.

40. a. dependent: sales

independent: advertising expenditure

c. form: linear

outliers: one at (22.5, 315)

direction: positive

strength: moderate

d. 2.747 273.289y x

e. For every $1 spent on advertising, there is a

$2.75 increase in sales .

f. $342 thousand ; it is advisable to make this

prediction as $25 thousand falls inside the

interval of $20 thousand to $28.3 thousand.

43. a. The maximum height reached by the ball

is 224 feet.

b. The ball will hit the ground 5.74 seconds

after it is thrown.


44. 150 46. exact value of trig. functions:

2sin csc 2

2

2cos sec 2

2

tan 1 cot 1

45. exact value of trig. functions:

7 8sin csc

8 7

15 8 15cos sec

8 15

7 15 15tan cot

15 7

47. You are 137.37 m from the base of the

plateau.

Name: MAT104 College Algebra with Trigonometry I Practice Final Exam … · MAT104 Practice Final Exam - Page 12 of 19 40. A marketing firm would like to estimate the sales of a product

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