Explain how the graph of y-5 = (x-3)2 can be obtained from ...



1. Explain how the graph of y-5 = (x-3)2 can be obtained from the graph of y = x2.

[pic]Shift the graph of y = x2 left 3 units and down 5 units

[pic]Shift the graph of y = x2 left 3 units and up 5 units

[pic]Shift the graph of y = x2 right 3 units and down 5 units

[pic]Shift the graph of y = x2 right 3 units and up 5 units

2. Give the equation for the circle with center C(3, -2) and radius 4.

[pic]x2+y2=52

[pic](x-3)2+(y-2)2=16

[pic](x+3)2+(y-2)2=42

[pic](x-3)2+(y+2)2=16

3. Given f(x) = 5x + 7 and g(x) = x2 + 7, find (g⋅f)(x).

[pic](g⋅f)(x)= 5x2 + 7

[pic](g⋅f)(x)= 5x2 + 42

[pic](g⋅f)(x)= (5x)2 + 14

[pic](g⋅f)(x)= 25x2 + 70x + 56

4. Find the point on the positive y-axis that is a distance 5 from the point P(3, 4).

[pic]A(0, 6)

[pic]B(0,8)

[pic]C(6,0)

[pic]D(8,0)

5. Give the center of the circle with equation x2+2x+y2-10y+22=0.

[pic]A(2, 4)

[pic]B(1, 5)

[pic]C(-1, 5)

[pic]D(-2, 4)

6. An object is projected upward from the top of a tower. Its distance in feet above the ground after t seconds is given by s(t)=-16t2+64t+80. How many seconds will it take to reach ground level?

[pic]1 second

[pic]4 seconds

[pic]5 seconds

[pic]8 seconds

7. The figure shows the graphs of y = f(x) and y = g(x). Express the function g in terms of f.

[pic]

[pic]g(x) = f(x - 2)

[pic]g(x) = -f(x + 2)

[pic]g(x) = 2 - f(x)

[pic]g(x) = 2 - f(x - 2)

8. From a square piece of cardboard with width x inches, a square of width x - 3 inches is removed from the center. Write the area of the remaining piece as a function of x.

[pic]f(x) = 6x - 9

[pic]f(x) = 6x + 9

[pic]f(x) = 2x2 - 9

[pic]f(x) = 2x2 - 6x – 9

9. Find the midpoint of the line segment from A(-2, 9) to B(4, 5).

[pic]C(1, 7)

[pic]D(3, 7)

[pic]P(4, 9)

[pic]Q(5, 9)

10. The figure shows the graph of a function that is ____.

[pic]

[pic]even

[pic]odd

[pic]both even and odd

[pic]neither even nor odd

11. If P(4, -5) is a point on the graph of the function y = f(x), find the corresponding point on the graph of y = 2f(x - 6).

[pic]A(1, 8)

[pic]B(2, -5)

[pic]C(6, 8)

[pic]D(10,-10)

12. If f(x) = x(x - 1)(x - 4)2, use interval notation to give all values of x where f(x) > 0.

[pic](-∞,0)∪(4,∞)

[pic](-∞,1)∪(4,∞)

[pic](-∞,1)∪(4,∞)

[pic](-∞,0)∪(1,4)∪(4,∞)

13. A rectangle is placed under the parabolic arch given by f(x) = 27 - 3x2 by using a point (x, y) on the parabola, as shown in the figure. Write a formula for the function A(x) that gives the area of the rectangle as a function of the x-coordinate of the point chosen.

[pic]

[pic]f(x) = 6(27 - 3x2)

[pic]f(x) = 27x - 3x3

[pic]f(x) = 54x - 6x3

[pic]f(x) = 162x - 6x3

14. If f(x) = x(x + 3)(x - 1), use interval notation to give all values of x where f(x) > 0.

[pic](-3, 1)

[pic](-3, 0) ∪(1,∞)

[pic](1, 3)

[pic](0, 1)∪(3,∞)

15. Find all roots of the polynomial x3 - x2 + 16x - 16.

[pic]1, 4, -4

[pic]-1, 4, -4

[pic]-1, 4i, -4i

[pic]1, 4i, -4i

16. Find a polynomial with leading coefficient 1 and degree 3 that has -1, 1, and 3 as roots.

[pic]x3 - 3x2 - x + 3

[pic]x3 - 3x2 + x - 3

[pic]x3 + 3x2 - x - 3

[pic]x3 + 3x2 + x + 3

17. Express the following statement as a formula with the value of the constant of proportionality determined with the given conditions: w varies directly as x and inversely as the square of y. If x = 15 and y = 5, then w = 36.

[pic]

|w |=3 |x |

| | |[pi|

| | |c] |

| | |y2 |

[pic]

|w |=12 |x |

| | |[pi|

| | |c] |

| | |y2 |

[pic]

|w |=36 |x |

| | |[pi|

| | |c] |

| | |y2 |

[pic]

|w |=60 |x |

| | |[pi|

| | |c] |

| | |y2 |

18. Find the third degree polynomial whose graph is shown in the figure.

[pic]

[pic]f(x) = x3 - x2 -2x + 2

[pic]

|f(x) = |1 |x3 |-|1 |x2 |-|x+2 |

| |[p| | |[p| | | |

| |ic| | |ic| | | |

| |] | | |] | | | |

| |4 | | |2 | | | |

[pic]

|f(x) = |1 |x3 |-|1 |x2 |+ |2x+2 |

| |[p| | |[p| | | |

| |ic| | |ic| | | |

| |] | | |] | | | |

| |4 | | |4 | | | |

[pic]

|f(x) = |1 |x3 |-|1 |x2 |-|x+2 |

| |[p| | |[p| | | |

| |ic| | |ic| | | |

| |] | | |] | | | |

| |2 | | |2 | | | |

19. The period of a simple pendulum is directly proportional to the square root of its length. If a pendulum has a length of 6 feet and a period of 2 seconds, to what length should it be shortened to achieve a 1 second period?

[pic]1 foot

[pic]1.5 feet

[pic]2 feet

[pic]3 feet

20. The figure shows the graph of:

|f(x) = |6x-10 |

| |[pic] |

| |2x-a |

Find the value of a.

[pic]

[pic]2

[pic]3

[pic]4

[pic]6

21. The figure shows the graphs of f(x) = x3 and g(x) = ax3. What can you conclude about the value of a?

[pic]

[pic]a < –1

[pic]–1 < a < 0

[pic]0 < a < 1

[pic]1 < a

22. Find the horizontal asymptote of the rational function:

|f(x)= |8x-12 |

| |[pic] |

| |4x-2 |

[pic]y = 1/2

[pic]y = 3/2

[pic]y = 2

[pic]y = 4

23. Find the quotient and remainder of f(x) = x4 - 2 divided by p(x) = x - 1.

[pic]x3 + x2 + 1; -1

[pic]x3 + x2 + x + 1; -1

[pic]x3 + x + 1; -1

[pic]x3 - x2 - x - 1; -1

24. Identify the rational function whose graph is shown in the figure.

[pic]

[pic]

|f(x) = |3x+5 |

| |[pic] |

| |x+1 |

[pic]

|f(x) = |x+5 |

| |[pic] |

| |x+3 |

[pic]

|f(x) = |3x-5 |

| |[pic] |

| |x-1 |

[pic]

|f(x) = |x+5 |

| |[pic] |

| |x+1 |

25. Find the polynomial f(x) of degree three that has zeroes at 1, 2, and 4 such that f(0) = -16.

[pic]f(x)=x3-7x2+14x-16

[pic]f(x)=2x3-14x2+28x-16

[pic]f(x)=2x3-14x2+14x-16

[pic]f(x)=2x3+7x2+14x+16

26. Find the vertical asymptote of the rational function:

|f(x) = |3x-12 |

| |[pic] |

| |4x-2 |

[pic]x = 1/2

[pic]x = 3/4

[pic]x = 2

[pic]x = 4

27. The figure shows the graph of y = (x - 3)(x - 5)(x - a). Determine the value of a.

[pic]

[pic]3

[pic]4

[pic]5

[pic]7

28. The table shows several values of the function f(x) = -x3 + x2 - x + 2. Complete the missing values in this table, and then use these values and the intermediate value theorem to determine (an) interval(s) where the function must have a zero.

|x |–2 |–1 |0 |1 |2 |

|f(x) |16 |  |  |  |–4 |

[pic](0, 1)

[pic](1, 2)

[pic](0,1)∪(2,∞)

[pic](-∞)∪(2,∞)

29. For the following equation, find the interval(s) where f(x) < 0.

|f(x) = |1 |

| |[pic] |

| |x2-2x-8 |

[pic](-4, 2)

[pic](-2, 4)

[pic](2, 4)

[pic](2, 8)

30. Find the quotient and remainder of f(x) = x3 - 4x2 + 5x + 5 divided by p(x) = x - 1.

[pic]x2 + 2x + 2; 7

[pic]x2 - 3x + 3; -5

[pic]x2 - 3x + 2; 7

[pic]x2 - 2x + 3; -5

31. The figure shows the graph of the polynomial function y = f(x). For which of the values k = 0, 1, 2, or 3 will the equation f(x) = k have complex roots?

[pic]

[pic]0

[pic]1

[pic]2

[pic]3

32. The polynomial f(x) divided x - 3 results in a quotient of x2+3x-5 with a remainder of 2. Find f(3).

[pic]-5

[pic]-2

[pic]2

[pic]3

33. Let f(x) = x3 - 8x2 + 17x - 9. Use the factor theorem to find other solutions to f(x) - f(1) = 0, besides x = 1.

[pic]-2, 5

[pic]2, -3

[pic]2, 5

[pic]2, 10

34. The electrical resistance R of a wire varies directly as its length L and inversely as the square of its diameter. A wire 20 meters long and 0.6 centimeters in diameter made from a certain alloy has a resistance of 36 ohms. What is the resistance of a piece of wire 60 meters long and 1.2 centimeters in diameter made from the same material?

[pic]24 ohms

[pic]27 ohms

[pic]30 ohms

[pic]48 ohms

35. The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3 + i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.

[pic]f(x)=(x+4)(x2+6x+10)

[pic]f(x)=(x-4)(x2-6x-9)

[pic]f(x)=(x-4)(x2-6x+10)

[pic]f(x)=(x-4)(x2-6x+9)

36. Given that (3x - a)(x - 2)(x - 7) = 3x3 - 32x2 + 81x - 70, determine the value of a.

[pic]1

[pic]3

[pic]5

[pic]7

37. Identify the exponential function of the form f(x)=a(2x)+b whose graph is shown in the figure.

[pic]

[pic]f(x)=3(2x)

[pic]f(x)=2x-3

[pic]f(x)=2(2x)-4

[pic]f(x)=2x-2

38. From the information in the table providing values of f(x) and g(x), evaluate (f • g)-1(3)

 

|x |1 |2 |3 |4 |5 |

|f(x) |5 |3 |5 |1 |2 |

|g(x) |4 |5 |1 |3 |2 |

[pic]1

[pic]2

[pic]4

[pic]5

39. For the function f(x) shown,

[pic]

find the domain and range of f -1(x).

[pic]Domain = [0, 6], Range [ 2, 5]

[pic]Domain = [0, 5], Range [ 2, 6]

[pic]Domain = [2, 5], Range [ 0, 6]

[pic]Domain = [2, 6], Range [ 0, 5]

40. Write the expression loga(y+5)+2loga(x+1) as one logarithm.

[pic]loga(y+2x+7)

[pic]loga(y+x2+7)

[pic]loga[2(y+5)(x+1)]

[pic]loga[(y+5)(x+1)2]

41. Solve loga(8x+5)=loga(4x+29)

[pic]4

[pic]5

[pic]6

[pic]8

42. For the function defined by f(x)=2-x2, 0≤ x, use a sketch to help find a formula for f-1(x).

[pic]f-1(x) = x2-2, x ≤ 2

[pic]

|f -1(x)= |1 |, 0 ≤ x |

| |[pic] | |

| |2-x2 | |

[pic]f-1(x) = - √2 + √x , 0 ≤ x

[pic]f-1(x) = √(2-x) , x ≤ 2

43. The figure shows the entire graph of the function f(x). If the graph of f -1(x) was sketched in the same figure, which of the following would give the best description?

[pic]

[pic]The graph of f -1(x) decreases from 5 to 2

[pic]The graph of f -1(x) decreases from 6 to 0

[pic]The graph of f -1(x) increases from 2 to 5

[pic]The graph of f -1(x) increases from 0 to 6

44. The amount A in an account after t years from an initial principle P invested at an annual rate r compounded continuously is given by A = Pert where r is expressed as a decimal. How many years will it take an initial investment of $1,000 to grow to $1,700 at the rate of 4.42% compounded continuously?

[pic]10 years

[pic]11 years

[pic]12 years

[pic]13 years

45. The population P of a certain culture is expected to be given by a model p=100ert where r is a constant to be determined and t is a number of days since the original population of 100 was established. Find the value of r if the population is expected to reach 200 in 3 days.

[pic]0.231

[pic]0.549

[pic]1.098

[pic]1.50

46. The figure shows the graph of g(x)=ex and a second exponential function f(x). Identify the second function.

[pic]

[pic]f(x)=2+e-x

[pic]f(x)=2-ex

[pic]f(x)=-2+ex

[pic]f(x)=2+ex

47. A bacteria culture started with a count of 480 at 8:00 A.M. and after t hours is expected to grow to f(t)=480(3/2)t. Estimate the number of bacteria in the culture at noon the same day.

[pic]810

[pic]1920

[pic]2430

[pic]4800

48. The amount of a radioactive tracer remaining after t days is given by A = Ao e-0.058t, where Ao is the starting amount at the beginning of the time period. How many days will it take for one half of the original amount to decay?

[pic]10 days

[pic]11 days

[pic]12 days

[pic]13 days

49. Solve the equation 42x+1=23x+6.

[pic]-5

[pic]2

[pic]4

[pic]5

50. If a piece of real estate purchased for $75,000 in 1998 appreciates at the rate of 6% per year, then its value t years after the purchase will be f(t)=75,000(1.06t).  According to this model, by how much will the value of this piece of property increase between the years 2005 and 2008?

[pic]$14,300

[pic]$21,500

[pic]$37,800

[pic]$59,300

51. The amount A in an account after t years of an initial principle P invested at an annual rate r compounded continuously is given by A = Pert where r is expressed as a decimal. What is the amount in the account if $500 is invested for 10 years at the annual rate of 5% compounded continuously?

[pic]$750.00

[pic]$800.00

[pic]$814.45

[pic]$824.36

52. Find the number:

|log5 |(|1 |)|

| | |[p| |

| | |ic| |

| | |] | |

| | |5 | |

[pic]-5

[pic]-1

[pic]0.2

[pic]1

53. Given that loga(x)= 3.58 and loga(y)=4.79, find loga(y/x).

[pic]1.21

[pic]1.34

[pic]8.37

[pic]17.1

54. The decibel level of sound is given by:

|D= |10 |log |(|I |)|

| | | | |[pic] | |

| | | | |10-12 | |

where I is the sound intensity measured in watts per square meter. Find the decible level of a whisper at an intensity of 5.4 x 10-10 watts per square meter.

[pic]2.73 decibel

[pic]3.73 decibels

[pic]27.3 decibels

[pic]37.3 decibels

55. The amount of a radioactive tracer remaining after t days is given by A = Ao e-0.18t, where Ao is the starting amount at the beginning of the time period. How much should be acquired now to have 40 grams remaining after 3 days?

[pic]47.9 gm

[pic]48.8

[pic]61.6 gm

[pic]68.6 gm

56. Find the exact solution to the equation 3x+5=9x.

[pic]5/3

[pic]5/2

[pic]5

[pic]6

57. For the function defined by f(x) =5x - 4, find a formula for f -1(x).

[pic]f-1(x) = -5x+4

[pic]

|f -1|(x)= |1 |

| | |[pic] |

| | |5x-4 |

[pic]

|f -1|(x)= |x+4 |

| | |[pic] |

| | |5 |

[pic]

|f -1(x)= |x |+4 |

| |[p| |

| |ic| |

| |] | |

| |5 | |

58. Solve the equation ln(x + 5) - ln(3) = ln(x - 3).

[pic]2.5 [pic]4.5

[pic]5

[pic]7

59. The figure shows the graph of g(x) = log2 (x) and a second function f(x). Identify the function f(x).

[pic]

[pic]log2 (x + 2)

[pic]2 log2 (x)

[pic]2 + log2 (x)

[pic]log2 (2x)

60. The amount A in an account after t years from an initial principle P invested at an annual rate r compounded continuously is given by A = Pert where r is expressed as a decimal. Solve this formula for t in terms of A, P, and r.

[pic]

|t = ln |(|AP |)|

| | |[pic| |

| | |] | |

| | |r | |

[pic]

|t = ln |(|A |)|

| | |[pi| |

| | |c] | |

| | |rP | |

[pic]

|t =r ln |(|A |)|

| | |[p| |

| | |ic| |

| | |] | |

| | |P | |

[pic]

|t= |1 |ln |(|A |)|

| |[p| | |[p| |

| |ic| | |ic| |

| |] | | |] | |

| |r | | |P | |

61. Find an exponential function of the form f(x)=bax+c with y-intercept 2, horizontal asymptote y=-2, that passes through the point P(1,4).

[pic]f(x)=-2(2x)

[pic]f(x)=2(2x) -2

[pic]f(x)=2(1.5x)-2

[pic]f(x)=4(1.5x)-2

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