Expert: Sherman D. - 2/15/2005

Question
4: Find the equation of the line that passes through the points:
( 2, 5) , ( -2, -1)

 3x - 2y + 4 = 0
 3x + 2y - 4 = 0
 3x + 2y - 4 = 0
 4x - 2y + 4 = 0
 4x - 2y - 4 = 0
 4x + 2y - 4 = 0

5: True or False: Two distinct nonvertical lines are parallel if and only if their slopes are equal.

 True
 False

6: True or False: Two nonvertical lines that are perpendicular have slopes that are positive reciprocals of each other. (In other words, m is slope where:m and 1/m are the related slopes)

 True
 False

7: Evaluate at f(7) and pick best answer: Remember (x +9)^(1/2)
"means" take the square root of (x + 9)

f(x) = (x + 9)^(1/2) / (x^2 - 81)

 1/4
-1/8
 3/4
 1/16

8: Select the domain of f(x) = square root of (144 – x^2) where x is between:

 -44 less than and equal to x and less than and equal to 44
 -12 less than x and less than 12
 -12 less than and equal to x and less than and equal to 12
 All real numbers

9: Identify the transformation of this function:
h(x) = (-(x + 7)^1/2) + 9

From the common function f(x) = x^1/2:
"square root of x"

 No reflection in x axis, horizontal shift to the left 7, and vertical shift up 9.
 Reflection in x axis, horizontal shift to the left 7, and vertical shift up 9.
 Reflection in x axis, horizontal shift to the right 7, and vertical shift up 9.
 Reflection in x axis, horizontal shift to the left 7, and vertical shift down 9.
 Reflection in y axis, horizontal shift to the right 7, and vertical shift up 9.

10: Evaluate the composition of (f o g)(x)
given:
f(x) = (x^2) - 1 , g(x) = 3x

 (3x^2) - 1
 (9x^2) – 1
 (6x^2) - 1
 (3x^3) - 3x

11: Choose the inverse function of:
f(x) = 7x - 2

 g(x) = 7 x + (2/7)
 g(x) = -7 x - 2
 g(x) = 1 /(7x - 2)
 g(x) = (1/7 x) + (2/7)
 There is no inverse function

12: True or False:
The inverse of a function is a function if and only if every horizontal line intersects the graph of the given function at no more than two points.

 True
 False

13: Choose the correct mathematical model that depicts the statement: “ d varies directly with the square of t”. Given: d= 64 and t = 4

 d= (1/16)t^2
 d = 4 (t)^2
 d= (1/8) t^2
 d = 64 t
 d = 24 t^2

14: True or False: Joint variation means y = k / x for some constant k.

 True
 False

15: True or False: The vertical line test for functions if and only if no vertical line intersects the graph at more than one points.

 True
 False

16: True or False: The zeros of a function are points where the function crosses the x axis. They are the solution(s) to the function.

 True
 False

17: Describe how the graph of:
g(x) = (x-5)^2
differs from the common graph of f(x) = x^2.

 Horizontal shift 5 to the right.
 Horizontal shift 5 to the left.
 Vertical shift 5 up.
 Vertical shift 5 down.

18: Describe the right-hand and left-hand behavior of the graph of the function: f(x) = (x^8) - 5(x^4) + 6

 Graph falls to the left and the right.
 Graph rises to the left and the right.
 Graph falls to the left and rises to the right.
 Graph rises to the left and falls to the right.

19: Divide by using long division:

((3x^3)+ 4x – 1) / ((x^2) + 1)

 x + 1
 (x^2) + 1
 (x^3) + 1
 3x + 1

20: Perform the operation and choose the best result.
(2 + 3i) * (2 – 3i)

 5
 25
 -25
 -5
 23

21: When the function f(x) = (3x^3) + (4x^2) – 5x – 2 is factored to:
f(x) = (x-1)(3x+1)(x + 2)
Indicate all of the zeros of the function.

 2, 1/3, 3
 1, -1/3, -2
 -1, 1/3, 2
 1, 3, 2

22: i^2 =

 1
 -1
 i
 -i

24: True or False:
The ratio test is used to find a list of possible rational zeros.

 True
 False

25: True or False: A Complex zero can occur as the only solution.

 True
 False

26: True or False: Descartes Rule of signs sheds more light on the number of real zeros of a polynomial.

 True
 False

Answer
4: Find the equation of the line that passes through the points:
(2,5) , (-2,-1)
m (-1 - 5)/(-2 - 2)
m = (-1 + (-5))/(-2 + (-2))
m = (-6)/(-4)
m = (3/2)

(2,5), m = (3/2)
5 = (3/2)(2) + b
5 = 3 + b
b = 2

y = (3/2)x + 2
2y = 3x + 4
-3x + 2y - 4 = 0
-(3x - 2y + 4) = 0
or
3x - 2y + 4 = 0

ANS : 3x - 2y + 4 = 0

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5: True or False: Two distinct nonvertical lines are parallel if and only if their slopes are equal.

ANS : True

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6: True or False: Two nonvertical lines that are perpendicular have slopes that are positive reciprocals of each other. (In other words, m is slope where:m and 1/m are the related slopes)

ANS : False

Perpendicular lines have opposite reciprocal slopes of each other. For ex: 2 and (-1/2)

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7: Evaluate at f(7) and pick best answer: Remember (x + 9)^(1/2)
"means" take the square root of (x + 9)

f(x) = (x + 9)^(1/2) / (x^2 - 81)
f(7) = ((7 + 9)^(1/2))/(7^2 - 81)
f(7) = (16^(1/2))/(49 - 81)
f(7) = (±4)/(-32)
f(7) = -1/8

ANS : -1/8

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8: Select the domain of f(x) = square root of (144 – x^2) where x is between:

f(x) = sqrt(144 - x^2)
y = sqrt(144 - x^2)
0 = sqrt(144 - x^2)
144 - x^2 = 0
-x^2 = -144
x^2 = 144
x = sqrt(144)
x = ±12

ANS : -12 less than and equal to x and less than and equal to 12

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9: Identify the transformation of this function:
h(x) = (-(x + 7)^1/2) + 9

From the common function f(x) = x^(1/2):

In the equation
y = n(x + a)^(1/2) + b
+n = no reflection
-n = reflection
+a = "a" units to the left
-a = "a" units to the right
+b = "b" units up
-b = "b" units down

ANS : Reflection in x axis, horizontal shift to the left 7, and vertical shift up 9.

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10: Evaluate the composition of (f o g)(x)
given:
f(x) = (x^2) - 1 , g(x) = 3x

(f o g)(x) = f(g(x))

f(x) = x^2 - 1
g(x) = 3x

f(g(x)) = f(3x)
f(3x) = (3x)^2 - 1
f(3x) = 9x^2 - 1
(f o g)(x) = 9x^2 - 1


ANS : (9x^2) – 1

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11: Choose the inverse function of:
f(x) = 7x - 2
y = 7x - 2
x = 7y - 2
x + 2 = 7y
y = (1/7)x + (2/7)
g(x) = (1/7)x + (2/7)

ANS : g(x) = (1/7 x) + (2/7)

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12: True or False:
The inverse of a function is a function if and only if every horizontal line intersects the graph of the given function at no more than two points.

ANS : True

Since a regular graph has to pass the vertical line test, the inverse would have to follow the horizontal line test.

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13: Choose the correct mathematical model that depicts the statement: “ d varies directly with the square of t”. Given: d = 64 and t = 4

d = ksqrt(t)
d = kt^2
Found at www.mathnstuff.com/math/spoken/here/1words/v/v3.htm

ANS : d = 4(t)^2

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14: True or False: Joint variation means y = k / x for some constant k.

ANS : False

y = kxz
Same site listed in #13.

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15: True or False: The vertical line test for functions if and only if no vertical line intersects the graph at more than one points.

ANS : True

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16: True or False: The zeros of a function are points where the function crosses the x axis. They are the solution(s) to the function.

ANS : True

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17: Describe how the graph of:
g(x) = (x-5)^2
differs from the common graph of f(x) = x^2.

ANS : Horizontal shift 5 to the right

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18: Describe the right-hand and left-hand behavior of the graph of the function: f(x) = (x^8) - 5(x^4) + 6

ANS : Graph rises to the left and the right.

For a graph, go to www.calculator.com. Here is what i can tell you, going to the negative side, it drops then rises and same goes for the positive side, drops then rises.

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19: Divide by using long division:
(3x^3 + 4x – 1) / (x^2 + 1)

Sorry, i can't do this one.

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20: Perform the operation and choose the best result.
(2 + 3i) * (2 – 3i)
(2 * 2) + (2 * (-3i)) + (3i * 2) + (3i * (-3i))
4 + (-6i) + 6i + (-9i^2)
4 + (-9(sqrt(-1))^2))
4 + (-9 * -1)
4 + (9)
4 + 9
13

ANS : 13
But as you can see, unless you meant to type 13 instead of 23, then i don't see an answer.

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21: When the function f(x) = (3x^3) + (4x^2) – 5x – 2 is factored to:
f(x) = (x-1)(3x+1)(x + 2)
Indicate all of the zeros of the function.

ANS : 1, -1/3, -2

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22: i^2 = (sqrt(-1))^2 = -1

ANS : -1

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24: True or False:
The ratio test is used to find a list of possible rational zeros.

ANS : True

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25: True or False: A Complex zero can occur as the only solution.

ANS : True

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26: True or False: Descartes Rule of signs sheds more light on the number of real zeros of a polynomial.

ANS : True

I think