AP Calculus and Calculus Honors Summer Review Packet
Teacher
Semester
Name: - - - - - - - - - - - This review packet is worth 65 points on your marking period grade!!
All students (honors/ap spring/fall semester) MUST submit their packet by September 7th
to either their math teacher or their study hall teacher. Packets handed in late will NOT
receive any points.
1. Determine whether the following equation is an identity, a conditional equation or
neither.
7(x + 7) -7 = 1lx + 45
A) Neither
B) Identity
C) Conditional equation
2. Solve the following equation.
12+7y + 10+2y =20
y
y
A) y=O
B) Y = 2
C) none of these
D) y =-2
E) infinitely many solutions
3. Solve the following equation.
4x-7
---=
1
9x -17 2
A) none of these
B) all real numbers
C) x=3
D) x=4
E) x =-3
Page 1
4. Solve the following quadratic equation for x by factoring.
7x 2 +28x = 0
A) x=-4
B) x=4
C) no real solutions
D) x=0,x=4
E) x = O,x =-4
5. Solve the following quadratic equation by extracting the square roots.
(6x-8t = 2
A)
B)
C)
D)
E)
x = -1.57;x = -1.10
x=1.57;x=1.10
x=-1.57;x=1.10
x=1.57;x=-1.10
x = 2.00; y = -2.00
6. Solve the following quadratic equation by extracting the square roots.
2
13x = 57
A) x = 2.09
B) x = -2.09; x = 2.09
C) no real solutions
D) x = 27.22;x = -27.22
E) x=4.38; x=-4.38
7. Use the Quadratic Formula to solve the following equation.
5x 2 +8x- 2 = 0
A) x = 9.40;x = -11.00
B) x = -9.40;x = 11.00
C) x = 0.22;x =1.82
D) x = -0.22;x =-1.82
E) x=0.22;x=-1.82
Page 2
8. Determine whether x = -5 satisfies the following inequality.
x+9
0:::;--:::;2
5
A) False
B) True
9. Solve the following inequality.
x+8
-4<--<6
6
A)
B)
C)
D)
E)
-32<x<30
-35<x<30
-35<x<28
-32<x<28
None ofthe above
10. Solve the following inequality.
x 2 -18x + 77 :::; 0
A) x:::; 11
B) x :::; 7 or x 11
C) 7 :::;x:::; 11
D) 2:$x:S;11
E) 2 :$ x :s; 16
11. Solve the following inequality.
17x-121 < 4
A)
12
12
--<x<-
7
7
4
4
B) --<x<7
7
16
x<7
8
16
D) -<x<C)
7
7
E) None of the above
Page 3
/
,
1
12. Find the distance between the points given below.
(%'%} (- ,±)
A)
B)
C)
D)
E)
5.01
0.25
5.15
5.25
1.25
13. Find the x- and y- intercept(s) of the graph of the equation given below.
y = 6x 3 -30x 2
A) x-intercept(s):
y-intercept(s):
B) x-intercept(s):
y-intercept(s):
C) x-intercept(s):
y-intercept(s):
D) x-intercept(s):
y-intercept(s):
E) x-intercept(s):
y-intercept(s):
(0,0)
(0,2.24)
(0,0), (2.24,0)
(0,0)
(5,0)
(0,0)
(0, 0), (5,0)
(0,0)
none
(0,0)
14. Find the x- andy-intercepts of the graph of the equation given below.
l
A)
B)
C)
D)
E)
2
+x = 64
x-intercept(s):
y-intercept(s):
x-intercept(s):
y-intercept(s):
x-intercept(s):
y-intercept(s):
x-intercept(s):
y-intercept(s):
x-intercept(s):
(-8,0), (8, 0),(0,0)
(0, -8), (0, 8)
(8,0), (0,0)
(0, 8)
(-8,0), (8, 0)
(0, -8), (0, 8)
(0, 8), (0,0)
(8,0), (0,0)
(0, -8), (0, 8)
y-intercept(s): (-8,0), (8, 0)
Page 4
15. Find the slope ofthe line passing through the points given below.
(-8, 3), (-1, -1)
A) 0.29
B)
C)
D)
E)
-0.57
0.57
-1.75
1.75
16. Determine whether the lines L1 and L2 passing through the pairs of points are parallel,
perpendicular or neither.
L) : (-4,-3), (-5,-4)
L2 : ( 0 , - 3 ) , (-2,-5)
A) Neither parallel nor perpendicular
B) Parallel
C) Perpendicular
17. Find the slope and x-intercept (if possible) of the line with equation given below.
y
= -5x+25
A) Slope:-5
Intercept: ( -5, 0)
B) Slope: 5
Intercept: (0,5)
C) Slope: 5
Intercept: (5, 0)
D) Slope:-5
Intercept: (5,0)
E) Slope:-5
Intercept: (0, -5)
Page 5
18. Find the slope-intercept fonn of the equation of the line that passes through the given
point and is perpendicular to the given line.
(i3' 3
A)
'
3x + 61J = 5
.I'
C)
D)
1
2
B)
E)
2
1
2
1
1
2
1
2
19. Depreciation A business purchases a piece of equipment for $80,000. After 20 years
the equipment will have to be replaced. Its value at that time is expected to be $10,000.
Write a linear equation giving the value V of the equipment during the 20 years it will be
used.
A) V = -3500.00x -80000.00
B) V = 3500.00x + 80000.00
C) V = -80000.00x - 3500.00
D) V=80000.00x-3500.00
E) V = -3500.00x + 80000.00
20. Determine whether the equation below represents as a function of x.
x2 + = 9
A) Function
B) Not a function
21. Determine whether the equation below represents as a function of x.
= 21 x - 21
A) Not a function
B) Function
Page 6
22. Evaluate the function at the specified value of the independent variable and simplify.
f(t)=lOt-9
A) lOt-4
B) lOt-4l
C) lOt -14
D) lOt +59
E) lOt-59
f(t -5)
23. Evaluate the function at the specified value of the independent variable and simplify.
9X- 3 x<O
{ 2x-3 X20
A) -13
f(x) =
B)
C)
D)
E)
f(5)
-17
7
42
4
24. Find the domain of the following function.
2
f(x)=7x +4x+8
A) --4 <x < 8
B) x> 0
C) x>-7
D) All values of x
E) x< 7
25. Find the domain ofthe following function.
w-5
u(w)=-b
w2 - 9
A) The set of real
B) The set of real
C) The set of real
D) The set of real
E) The set of real
numbers
numbers
numbers
numbers
numbers
such that w;t:. 9.
such that w ;t:. 3.
such that w;t:. 3 and w;t:. -3.
such that w;t:. 3 , w;t:. -3, and
such that w ;t:. 5.
w;t:.
Page 7
J
'0
Ju,
t-==-.J
,
5.
26. Find the domain of the following function.
g(z) = A) The set of real numbers such that z ?: +2.
B) The set of real numbers such that z -2.
C) The set of real numbers such that z > -2 .
D) The set of real numbers such that z ?: -2.
E) The set of real numbers such that z < -2 .
27. Use the Vertical Line Test to determine whether y is a function ofx.
'I
'I
-'I
A) Function
B) Not a function
Page 8
J uJ
L8;.
,.,
.
L-J
28. Find the zeros ofthe function given below by factoring.
f(x)
= 64x
2
-
4
x
A)
1
1
X=-- X=-
B)
16'
16
1
x =-- x = 0
D)
1
X=- x=O
4'
1
1
C) x=-4"'x =4"
E)
16'
1
X=- x=O
4'
29. Find the zeros of the function given below by factoring.
f(x) = x 3 - 6x 2 - 25x + 150
A) x=o , x=6 , x=5
B) x = 6,x = 5
C) x = -6 , x = 5, x = -5
D) x = 0, x = -6 , x = -5
E) x = 6 , x = 5 ,x = -5
30. Find the linear function
f
that has the values given below.
j(-3) =21, j(3)=-9
2
15
A) f(x)=--x-1
1
B) f(x)=-5x+6
1
6
C) f(x)=-sx+
s
D) f(x) = -5x
E) f(x) = -3x + 21
Page 9
/
31. Determine whether the function given below is .even, odd, or neither.
f(x) =5x 8 -4x 6 - 4
A) Neither
B) Even
C) Odd
32. Determine whther the function given below is even, odd, or neither.
f(x) = -3x 4
A) Odd
B) Neither
C) Even
-
2x 3 - 6
33. Use a graphing utility to graph the function given below and visually determine the
intervals over which the function is increasing, decreasing, or constant.
f(x) = x-Jx - 2
A) Increasing on x > 2
B) Increasing on x > 0
C) Constant
D) Decreasing on x < 2
E) Increasing on x < 2
34. Use a graphing utility to graph the function given below and visually determine the
intervals over which the function is increasing, decreasing, or constant.
f(x)=x 2-16x+69
A) Increasing
B) Decreasing
C) Decreasing for x > 8, increasing otherwise
D) Increasing for x> 8, decreasing otherwise
E) Constant
Page 10
or
35. Describe the transformation from a common function that occurs in the function given
below.
f(x)=1-x 2
A) An upward shift of y = x 2 of 1 unites) and then a reflection in the x-axis
B) Reflection of y
= x 2 in the x-axis and a downward shift of 1 unites)
C) Reflection of y
= x 2 in the y-axis and an upward shift of 1 unites)
D) Reflection of y = x 2 in the x-axis and an upward shift of 1 unites)
E) Reflection of y = x 2 in the y-axis and an upward shift of 1 unites)
36. Describe the transformation from a common function that occurs in the function given
below.
f(x)
= -Ix-51 +8
A) A horizontal shift of y = Ixl of 8 unites) to the right, a reflection in the x-axis, and
then a vertical upward shift of 5 unites)
B) A horizontal shift of y = Ixl of 5 unites) to the right, a reflection in the x-axis, and
then a vertical upward shift of 8 unites)
C) A vertical shift of y = Ixl of 5 unites) to upward, a reflection in the x-axis, and then
a horizontal shift of 8 unites) to the right
D) A horizontal shift of y = Ixl of 8 unites) to the right, a vertical upward shift of 5
unites), and then a reflection in the x-axis
E) A horizontal shift of y = Ixl of 5 unit(s) to the right, a reflection in the y-axis, and
then a vertical upward shift of 8 unites)
Page 11
Jl_
37. Describe the transformation from a common function that occurs in the function given
below.
f(x) = .Jx+4-9
A) A horizontal shift of y
=E
of 9 units to the left and then a vertical downward
shift of 4 units
B) A horizontal shift of y =
E
of 4 units to the left and then a reflection in the x-axis
C) A vertical shift of y = E of 4 units to upward, a reflection in the x-axis, and then a
horizontal shift of 9 units to the right
D) A horizontal shift of y = E of 4 units to the right and then a vertical downward
shift of9 units
E) A horizontal shift of y = E of 4 units to the right, a reflection in the x-axis, and
then a vertical downward shift of 9 units
38. Write an equation for the function that is described below.
The shape of f(x)
A)
(X-2)3_3
B)
(X-2)3 +3
C)
(X+2)3+3
=x
3
,
but moved two units to the left and three units upward.
D) (-X+2)3_3
E)
(X+2)3_3
39. Write an equation for the function that is described below.
The shape of f(x)
A)
-lx+31
B)
-Ixl +3
D)
IX- 3
-lx-31
E)
\x+3J
C)
= lxi, but moved three units to the left
1
Page 12
f
.J
and reflected in the x-axis.
40. Given the functions f and g below determine (f - g)( x) .
f(x)=x 2 , g(x)=2x+3
A)
x 2 -2x+3
B)
x 2+2x-3
C)
_x 2 -2x-3
D)
x 2-2x-3
E)
_x 2 -2x+3
41. Oiven th e following functions determine
= x 2 + 2,
f(x)
A)
B)
g(x)
J(x) and iIs domain.
= .J8x + 9
1
;x>2
j(x)= J8;+9
x +2
g 8x+9
8
[f
2
g
C)
-f J (x) = J8;+9
2
[g
x +2
D)
[fJ(x)=
E)
g 8x+9
8
g
8x+9
; X
:;t
-2, x :;t 2
9
42. Given the functions f and g below determine (fg )(x) .
f(x)
A)
B)
C)
1
= -2'
X
g(x)
1
=-6
x
6
x
1
x4
1
]?
x-
Page 13
,
r
I
I
43. Given the functionsjand g below find (go j)(x) and its domain.
j(x) = .Jx-49, g(x)
=x2
A) (goj)(x)=.)x 2 - 4 9 ; x>7 or x<-7
B) (g 0 j)(x) = x - 49; all real numbers
C)
(g 0 f)(x) = x - 49; x> 49
D) (goj)(x)=.)x 2 - 4 9 ; x>7
E) (goj)(x)=x-49; x >'] or x<-7
44. Giventhefunctionsjandgbelowfind (jog)(x).
j(x)=2x+2, g(x)=4x+3
A) 8x+ 11
B) 8x-8
C) 8x+8
D) -8x-8
E) -8x+ 11
45 . Given the functions j and g below find (g 0 j)(x) .
4
j(x) = x 5 , g(x) = x 2
5
A) x 8
14
5
B)
x
C)
x5
2
5
D) x 2
8
E)
x5
Page 14
46. Resistance The resistance of a wire carrying an electrical current is directly
proportional to its length and inversely proportional to its cross-sectional area.
lf28-gauge copper wire (which has a diameter of 0.0126 inch) has a resistance of 65.2
ohms per thousand feet, what length of 28-gauge wire will produce a resistance of 26.08
ohms.
A) 80 feet
B) 400 feet
C) 8 feet
D) 4 feet
E) 40 feet
47. Free Fall Neglecting air resistance, the distance s an object falls varies directly as the
square of the duration t of the fall.
An
A)
B)
C)
D)
E)
object falls a distance 400 feet in 5 seconds. How far will it fall in 9 seconds?
2304 feet
144 feet
16 feet
9 feet
1296 feet
48. Let f(x) = x 2 - 4. Determine the vertex and any x-intercepts.
A) Vertex: (0, 4)
Intercept(s): (-2, 0), (2, 0)
B) Vertex: (0, --4)
Intercept(s): (2, 0)
C) Vertex: (0, -2)
Intercept(s): (--4, 0), (4, 0)
D) Vertex: (0,--4)
Intercept(s): (-2, 0), (2, 0)
E) Vertex: (4, 0)
Intercept(s): (-2,0), (2, 0)
Page 15
-r
J
j
I
/
..........
,
49. Determine the vertex and any x-intercepts of the function below.
f(x)=(x+7)2- 16
A) Vertex: (-7, -16)
Intercept(s): (-11,0),
B) Vertex: (-7, 16)
Intercept(s): (-11,0),
C) Vertex: (7, -16)
Intercept(s): (-11,0),
D) Vertex: (7, 16)
Intercept(s): (-11, 0),
E) Vertex: (-7, -16)
Intercept(s): (-11, 0),
(0, 0)
(-3,0)
(-3,0)
(-3, 0)
(-3, 0)
50. Find a quadratic function that has the given vertex and whose graph passes through the
given point.
Vertex: (-2, -19) Point: (3,81)
A) f(x)=--4x 2-16x+3
B)
C)
D)
E)
f(x)=4x 2+16x-3
f(x)=4x 2-16x-3
f(x)=4x 2+16x+3
2
f(x) = --4x + 16x-3
51. Determine the right-hand and left-hand behavior ofthe graphs ofthe polynomial
function given below.
f(x)
= 6x 3 -
6x
A) f(x) ---+ 00 asx ---+ -00, f(x) ---+ -00 asx ---+ 00
B) None of the above
C) f(x) ---+ -00 -00, f(x) ---+ -00 asx ---+ 00
D) f(x) ---+ -00 as x ---+ -00, f(x) ---+ 00 as x ---+ 00
E) f(x) ---+ 00 asx ---+ -00, f(x) ---+ 00 asx ---+ 00
Page 16
J
... Jl_
52. Determine the right-hand and left-hand behavior of the graphs of the polynomial
function given below.
f (x) = -3 - 2x - 7 x 2
A) f(x) -00 as x -00, f(x)
B) f(x) -00 asx -00, f(x)
C) None ofthe above
D) f(x) 00 asx -00, f(x) E) f(x) 00 asx -00, f(x) as x
asx -00
00
asx asx -00
00
00
00
00
00
53. Find all real zeros of the polynomial function given below.
f(x) = x 4 + 4x 3 - 45x 2
A) x=O,x=5,x=-9
B) x = 4 , x = 5, x = 9
C) x = , x = -5 , ..y = -9
D) x=4,x=5,x=-9
E) x = , x = -5 , x = 9
°
°
54. Use synthetic division to carry out the following division.
(x 3 +6x 2 +12x+20)-;-,(x+4)
4
A)
x 2 +2x+4--x+4
B)
x +2x+4+-x+4
4_
x 2 +2x_4+_
x+4
C)
D)
E)
2
4
2
4
x -2x+4--x+4
"
4
4
x 2--,-x+
+-x+4
Page 17
/
55. Use synthetic division to carry out the following division.
4x 4
x-I
3
2
4
x-I
A)
4x +4x +4x+4---
B)
2
4x3
-4x
+4x+4+--
C)
44x 3 +4x 2 -4x+4+-
D)
E)
4
x-I
x-I
3
2
4
4x +4x +4x+4+-x-I
4
2
4x 3
-4x
-4x-4+--
x-I
56. Consider the function given below:
f(x)
= x 4 +3x 3 -12x 2 -
20x+48.
(a) Determine the factors off
(b) List the zeros off
A) Factors: x - 2; x - 4; x - 2; x - 3
Zeros: 2 (twice); 4; 3
B) Factors: x + 2; x - 4; x + 2; x + 3
Zeros: -2 (twice); 4; -3
C) Factors: x - 2; x + 4; x - 2; x + 3
Zeros: 2 (twice); -4; -3
D) Factors: x - 2; x + 4; x - 2; x + 3
Zeros: -2 (twice); -4; -3
E) Factors: x - 2; x + 4; x - 2; x - 3
Zeros: 2 (twice); -4; 3
57. Write the complex number below in standard form.
2+1=9
A) -2 + 3i
B) 2 + 3i
C) 2 + 9i
D) 2 - 9i
E) 2 - 3i
Page 18
-r
j
.
"-'tI:C
(
Jl--
,
58. Perform the operation below and write the result in standard form.
(5+6i)+(-2-i)
A)
B)
C)
D)
E)
5 + 3i
-3 - 5i
3 + 5i
3 - 5i
-3 + 5i
59. Perform the operation below and write the result in standard form.
(3 + 7i)(3 - 6i)
A) 51 + 3i
B) 51 - 3i
C) 51 - 39i
D) 51 + 39i
E) -51 + 3i
60. Perform the operation below and write the result in standard form.
-1 +5i
9+6i
7 17
15 15
7 17.
B) - - - - I
15 15
7 17
A) - + - i
C)
--+-i
39 39
D)
---I
E)
-+-i
39 39
7
17.
39
39
7
17
Page 19
61. Use the Quadratic Formula to solve the following equation.
x2 A)
B)
C)
D)
E)
1Ox + 34 = 0
x = 8,x = 2
x = -S + 6i, x = -S - 6i
x=-S+3i,x=-S-3i
x=S+3i,x=S-3i
x=-8,x=2
62. Find all the zeros ofthe function given below. Use the Rational Zero Test to fmd the
first root.
f(x) = 2x 3 - sx 2 - 28x + IS
1
A) x=-S , x=-3 , x=2
B)
1
x=5,x=-3,x=2
C)
1
x=-S , x=3 , x=-2
D)
x
= S, x = -3 , x =--2
E)
x
= S, x = 3 , x =-2
1
1
Page 20
63. Find the domain of the function given below, and determine the equations of any
horizontal or vertical asymptotes.
f(x) = (
1
x-9
)2
A) Domain:
Vertical asymptote:
Horizontal asymptote:
B) Domain:
Vertical asymptote:
Horizontal asymptote:
C) Domain:
Vertical asymptote:
Horizontal asymptote:
D) Domain:
Vertical asymptote:
Horizontal asymptote:
E) Domain:
Vertical asymptote:
Horizontal asymptote:
x>9
x=9
y=
x :;r 9
x=9
y=
x :;r -9
x = -9
y=
x<9
y=9
x=
x :;r -9
y = -9
x=
°
°
°
°
°
64. Identify the intercepts and any the equations of any horizontal or vertical asymptotes.
3+x 2
A) Intercept(s):
Vertical asymptote:
Horizontal asymptote:
B) Intercept(s):
Vertical asymptote:
Horizontal asymptote:
C) Intercept(s):
Vertical asymptote:
Horizontal asymptote:
D) Intercept(s):
Vertical asymptote:
Horizontal asymptote:
E) Intercept(s):
Vertical asymptote:
Horizontal asymptote:
(0,0)
x=4
y=O
(0,0)
none
y=4
(0,0)
none
y=-4
(0,0)
x=3
y=4
(0,0)
none
x=4
Page 21
-r
J.-f.
j
La
65. Determine the equations of any horizontal, vertical, or slanted asymptotes.
2+10
j(x)=- 4x
x-10
A) Vertical asymptote:
Horizontal asymptote:
Slant asymptote:
B) Vertical asymptote:
Horizontal asymptote:
Slant asymptote:
C) Vertical asymptote:
Horizontal asymptote:
Slant asymptote:
D) Vertical asymptote:
Horizontal asymptote:
Slant asymptote:
E) Vertical asymptote:
Horizontal asymptote:
Slant asymptote:
x=-10
none
y=4x-40
x=-10
none
y = -4x-40
x= 10
y=O
none
x= 10
none
y=-4x-40
x= 10
none
y=4x-40
Page 22