Math Test 1470 #1 2.1-2.7 Name __________________

1. For the quadratic function

!

f (x) = " x 2 +10x + 4 , complete the square and determine which way the function opens, its vertex, and the equation of its line of symmetry.

2. A farmer has 288 feet of fencing and wants to build two identical pens for his prize-winning pigs. The pens will be arranged as shown. Determine the dimensions of a pen that will maximize its area.

y

x

3. The height, h(x), of a punted rugby ball is given by

!

h(x) = "164

x 2 +12x + 3

where x is the horizontal distance in feet from the point where the ball is punted. How far, horizontally, is the ball from the kicker when it is at its highest point? The ball hit the ground. How far from the kicker did the ball hit the ground?

4. For each of the following functions, pick the letter that describes the general shape of the function.

a. Because the degree is even and the leading coefficient is negative, the graph falls to the left and falls to the right.

b. Because the degree is even and the leading coefficient is positive, the graph rises to the left and rises to the right.

c. Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right.

d. Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right.

i.

!

f (x) =x 4 "17x 2 +16

20 ______________

ii.

!

g(x) =3x 5 "12x 3 "31 ______________ iii.

!

h(x) =" 2x 4 + 4x 3 "26 ______________

iv.

!

i(x) ="89(x 3 + 4x 2 "7x+1) ______________

5. Let

!

P(x) = x 3 + 4x 2 " x"4 . Write

!

P(x) as the product of factors. Then, list the zeroes of

!

P(x) .

6. Let

!

P(x) = x 3 + 4x 2 + 4x+16 . Write

!

P(x) as the product of factors. Then, list the zeroes of

!

P(x) .

7. Simplify

!

i"(9 + 4i) and write the answer in standard form.

8. Simplify

!

(4 + i)(2 +12i) and write the answer in standard form.

9. Simplify

!

6 + 3i5 + 4i

and write the answer in standard form.

10. Consider the function

!

f (x) =4x 2 + 33x + 355x 2 + 39x + 28

. Find the function’s domain and

identify any horizontal and vertical asymptotes.

Domain : __________ Vertical asymptotes : __________ Horizontal asymptotes: __________

11. Suppose a species of sea turtle is confined to an island and has a current

population of 71 turtles. If the population,

!

P , modeled by

!

P(x) =71(2 + 0.03t)2 + 0.06t

, t "0 where t is the time in years, how many turtles are

predicted to be on the island in 5 years? Round to the nearest integer. What is the limiting size of the population as time increases

!

(t"#)? Round to the nearest integer.

12. Find the solution set to the inequality

!

x " 6x +10

#" 7.

Xtra - Credit

13. A Norman window has the shape of a rectangle surmounted by a semicircle as in the figure below. If the perimeter of the window is of the window is 40 ft. express the area, A, as a function of the width, x, of the window.

y

x

Math Test 1470 #2 3.1-4.5 Name __________________

14. If $5000 is invested at 9% interest, find the amount after 7 years if the interest is compounded

a. Quarterly b. Monthly c. Continuously

15. Write

!

5ln x " 12ln y + 6ln z as a single logarithm

16. Answer the following

a. True or False:

!

ln xln y

= ln(x" y )

b. Write the equation in logarithmic form

!

7"2 =149

17. Solve for x:

!

5x = 41

18. Solve for x:

!

log2(x) + log2 (x " 3) = 2

19. Answer the following: a. Express 330° in radian measure

b. Express

!

5"9

in degree measure

20. Solve for x in the figure below.

x

35

m!BAC = 20.00°

B

AC

21. Find sec(θ) given that the terminal side of θ lies in Quadrant III and tan(θ) = 6.

22. Identify the following in the equation y = 3cos(2! (x !! )5

)+1

a. Amplitude b. Period c. Horizontal shift d. Vertical shift

23. An observer in a lighthouse 250 feet above sea level spots a ship off the shore. If the angle of depression to the ship is 5°, how far out is the ship.

Math Test 1470 #3 4.6-5.3 Name __________________

24. Given that the

!

tan" =32

, find the other five trigonometric functions of

!

" .

25. Given that the

!

sin" =x3

, find the other five trigonometric functions of

!

" .

26. The height of an outdoor basketball backboard is 12

!

12

feet, and the backboard

casts a shadow 17

!

13

feet long. Find the angle of elevation of the sun.

27. A jet leaves Reno, Nevada and is headed toward Miami, Florida at a bearing of

100°. The distance between the two cities is approximately 2472 miles. How far north and west of Miami is Reno? When the jet makes the return flight to Reno from Miami, what should the bearing?

28. A ship leaves port at noon and has a bearing of S 29° W. the ship sails at 20 knots. How many nautical miles south and how many nautical miles west will the ship have traveled by 6:00 PM? At 6:00PM the ship changes course to due west. Find the ship’s bearing and distance from port of departure at 7:00PM.

29. Factor and simplify:

!

sec4 x " tan4 xsec2 x + tan2 x

30. Add and simplify:

!

cos"sin"

+sin"cos"

31. Verify the identity:

!

sec2 x tan2 x + sec2 x = sec4 x

32. Find all the solutions to the equation in the interval [0,2π).

!

tan2 x + tan x = 0

33. Find all the solutions to the equation in the interval [0,2π).

!

4cos2 x " 3 = 0

Math Test #4 1470 5.4-6.3 Name __________________

34. Find the exact value of the given expression using a sum or difference formula.

!

tan19"12

35. Write the given expression as the cosine of an angle.

!

cos60 cos55"sin60 sin55

36. Write the given expression as the sine of an angle.

!

sin25 cos55 " sin55cos25

37. Use the figure below to determine the exact value of the given function.

3

2

find tan 2!

!

38. Find the exact solutions of the given equation on the interval

!

[0,2" ) .

!

sin2x = cos x

39. Use the figure below to find the exact value of the given trigonometric expression.

36

15

find sin !

2

!

40. Use the half-angle formulas to determine the exact value of the given trigonometric expression.

!

tan 3"8

41. Use the product-to-sum formula to write the given product as a sum or difference.

!

8 sin "10cos "10

42. Use the sum-to-product formulas to write the given expression as a product.

!

cos9" # cos7"

43. Given C = 123°, a = 12.9, and c = 8.3, use the Law of Sines to solve the triangle (if possible) for the value(s) of b. If two solutions exist, find both. Round answer to two decimal places.

44. After a severe storm, three sisters, April, May, and June, stood on their front porch and noticed that the tree in their front yard was leaning 3° vertical toward the house. From the porch, which is 101 feet away from the base of the tree, they noticed that the angle of elevation to the top of the tree was 32° . Approximate the height of the tree. Round answer to two decimal places.

45. A vertical pole 39 feet tall stands on a hillside that makes an angle of 18° with the horizontal. Determine the approximate length of cable that would be needed to reach from the top of the pole to a point 73 feet downhill from the base of the pole. Round answer to two decimal places.

46. Find the magnitude and direction angle of vector

!

! v = 4i " 6 j . Round direction angle to nearest hundredth.

47. Find the component form of vector !v if !v = 4 and the angle it makes with the x-axis is 60°.

48. Three forces with magnitudes of 66 pounds, 78 pounds, and 132 pounds act on an object at angles of 130°, 220°, and 290°, respectively, with the positive x-axis. Find the magnitude and direction of the resultant force. Round answers to two decimal places.

49. A force of F pounds is required to pull an object weighing W pounds up a ramp inclined at θ degrees from the horizontal. Find F if W = 5000 pounds and θ = 26°.

Math Test #5 1470 6.5-8.5 Name __________________ 50. Find the exact value of the given expression

!

(" 3 " i)9

Find the exact value of the given expression

!

(" 3 " i)17

51. A health insurance company advertises on television, on radio, and in the local newspaper. The marketing department has an advertising budget of $42,000 per month. A television ad costs $1000, a radio ad costs $200, and a newspaper ad costs $500. The department wants to run 60 ads per month, and have as many television ads as radio and newspaper ads combined. How any of each type of ad can the department run each month? Write the system of equations, show all your work and solve using Gaussian elimination.

52. A bank teller is counting the total amount of money in a cash drawer at the end of a shift. There is a total of $2600 in denominations of $1, $5, $10, and $20 bills. The total number of paper bills is 235. The number of $20 bills is twice the number of $1 bills, and the number of $5 bills is 10 more than the number of $1 bills. Write a system of linear equations to represent the situation. Then write the equations in reduced row echelon form to find the number of bills of each denomination.

53. Evaluate the following expression:

!

"3(0 "37 2#

$ %

&

' ( +

"6 38 1#

$ %

&

' ( ) " 2

4 "47 "9#

$ %

&

' (

54. You invest in AAA-rated bonds, A-rated bonds, and B-rated bonds. The average yields are 6.5% on the AAA bonds, 7% on A bonds, and 9% on B bonds. You invest twice as much in B bonds as in A bonds. Let x,y, and z represent the amounts invested in AAA, A, and B bonds, respectively. The total investment was $500000 and the annual return is $38000. Write a system of linear equations to represent the investment, use the inverse of a matrix to solve for the amount invested in each type of bond.

55. Consider the following systems of equations. Find all possible solutions for each system.

!

x " 3y + z =12x " y " 2z = 2x + 2y " 3z = "1

!

x + y " 3z = "10x + y " z = 0"x + 2y + 0z =1

!

x " 2y + 3z = 9"x + 3y + 0z = "42x " 5y + 5z =17

56. Use Cramer’s rule to solve (if possible) the system of equations.

!

6x " 5y =17"13x + 3y = "76

57. Use determinants to find the area of the triangle with the following vertices.

!

(6,10) ("4,"5) (6,"1)

58. Use a determinant to determine if the following points are collinear.

!

(3,"5) (6,1) (4,2)

59. Use a determinant to find an equation of the line passing through the given points.

!

(10,7) ("2,"7)