Download = + Ax By C + 0 A > y y m x x - = - x y - = 2 4 3 y x = - + 8 x y - = -

0Preparation for AP Calculus AB: Summer Assignment
Due: The first day of class if you are taking AP Calculus AB in the fall,
or the first day of spring semester if you are taking the class in the spring.
It is assumed you can complete the problems without a calculator.
Please complete all of the problems. Most of the material is a review of
Algebra 2, so it is expected that you can finish the problems before starting
the AP Calculus class. If you experience difficulties you can find assistance
online, or find examples on procedure in your Algebra 2 and Precalculus
notes .
I. Linear Equations
There are three accepted forms for writing the equation of a linear function.
Those forms are as follows:
y  mx  b
ii. Point-slope form
y  y1  m  x  x1 
Ax  By  C where A, B, C are integers, A  0
iii. Standard form
y  y1
other formulas: m  2
{change in y with respect to the change in x }
x2  x1
i. Slope-intercept form
Problems:
1) Write the equation of the line that passes through the points
point-slope form. Convert your answer to standard form.
 6,3 ,  3,1 in
2) Write the equation of the line that runs perpendicular to the line
and passes through the point
 1, 2  .
Write the equation in point-slope form.
2
y x4
3
4) Sketch the graph of the linear function 3x  4 y  8
3) Sketch the graph of the linear function
4 x  3 y  17
II. Quadratic Equations
There are three accepted forms for writing the equation of a quadratic function.
Those forms are as follows:
i. Standard form
ii. Vertex form
y  ax 2  bx  c
y  a  x  h   k {where  h, k  is the vertex}
y  a  x  xint  x  xint 
2
iii. x -intercept form
To determine the solution to a quadratic equation, one can use factoring or
the quadratic formula.
Problems:
5) Solve by factoring:
3x 2  4 x  7
6) Solve using the quadratic formula:
7) Solve using the quadratic formula:
12 x  7  2 x 2
2 x2  7  4 x
8) Sketch the graph of the following parabolas. Indicate the x  intercepts (if they
exist), the y  intercept, the axis of symmetry, and the vertex. Determine the
domain and the range for each function.
a)
b)
c)
d)
e)
f  x   x2  4x  5
f  x   4x  x2
f  x   x2  6x  9
f  x   x 2  6 x  10
1
2
f  x    x  4  5
2
III. Polynomials of degree greater than two
Review: For every cubic or quartic function in the form of
f  x   ax 3  bx 2  cx  d
4
3
2
or f  x   ax  bx  cx  dx  c ;
a) the y-intercept is determined by plugging in zero for
x.
b) if the lead coefficient a is positive, the graph ends UP on the right
c) if the lead coefficient a is negative, the graph ends DOWN on the right
Examples:
a>0
a<0
Problems:
9) Sketch the graph of the cubic
f  x   x3  6 x 2  8 x .
10) Sketch the graph of the cubic
11) Sketch the graph of the quartic
f  x    x3  x 2  9 x  9 .
f  x   x4  5x2  4 .
IV Factoring
Factoring is a skill, and we use factoring daily in Calculus to solve problems.
Please factor completely all of the following:
Problems:
i. Easy to factor:
12)
13)
14)
15)
16)
17)
18)
19)
20)
x 2  14 x  48
x 2  9 x  36
x 2  14 x  24
x 2  13x  48
x 2  21x  20
x 2  13x  30
x 2  12 x  36
x 2  11x  24
x 2  5 x  36
21)
x 2  x  30
ii. A little more difficult
5 x 2  33x  18
2
23) 7 x  24 x  20
2
24) 3x  13x  30
2
25) 5 x  26 x  24
2
26) 6 x  37 x  45
2
27) 8 x  2 x  1
2
28) 7 x  19 x  10
2
29) 3x  26 x  16
2
30) 7 x  13x  2
2
31) 10 x  23x  12
22)
iii. Difference of squares
x 2  25
2
33) x  100
2
34)1  x
2
35) 36 x  25
2
36) 100 x  1
2
2
37) x  y
2
38) 49 x  64
32)
iv. Factor out the GCF first and then factor again.
3x3  18x 2  48x
3
2
40) 6 x  24 x  24 x
3
41) 2 x  50 x
3
42) 3x  3x
39)
2 x 2  38 x  96
3
2
44)15 x  21x  6 x
43)
v. Factor by grouping
x3  2 x 2  6 x  12
3
2
46) 10 x  6 x  5 x  3
3
2
47) 2 x  x  2 x  1
3
2
48) 2 x  3x  8 x  12
45)
V. Trigonometry
49) Please fill in the associated values in the chart below. Angles are in radians.
0
    2 3 5 
6 4 3 2 3 4 6
7 5 4 3 5 7 11
6 4 3 2 3 4 6
sinx
cosx
tanx
Trigonometric identities you need to know/ Please memorize:
opposite
hypotenuse
adjacent
cos 
hypotenuse
opposite sin 
tan  

adjacent cos
sin 2   cos 2   1
sin  
tan 2   1  sec 2 
cot 2   1  csc 2 
1
sin 
1
sec 
cos
1
cos
cot 

tan  sin 
csc 
Problems:
Using the identities on the preceding page, simplify the following:
50)
sec x  csc x
1  tan x
51)
cot x 
52)
1
1

sec x  1 sec x  1
53)
cot 2 x
1
csc x  1
sin x
1  cos x
54) Sketch the graph of f
graph is __________.
55) Sketch the graph of f
graph is __________.
56) Sketch the graph of
 x   cos x on the interval 0, 2 .
The range for this
 x   sin x on the interval 0, 2 .
The range for this
f  x   tan x
on the interval
  
  2 , 2  .
The range for this
graph is __________.
57) Solve the following trigonometric equation for  on the interval
0,2  ;
2sin 2   1  0
58) Solve the following trigonometric equation for  on the interval
0,2  ;
59) Solve the following trigonometric equation for  on the interval
0,2  ;
sec sin   2sin   0
2sin  cos  tan   0
VI. Exponents and Logarithms
Problems:
3
2
43x  8  6  26
61) Solve: log 2  x  3  log 2 x  log 2  x  2  2
3
62) Solve: log x 27 
2
125 x  5
 125
63) Solve:
25 x
2
64) Simplify: log 8
2
60) Solve:
VII Graphing
Problems:
65) Sketch the graph of
66) Sketch the graph of
this function.
67) Sketch the graph of
68) Sketch the graph of
function.
69) Sketch the graph of
70) Sketch the graph of
71) Sketch the graph of
f  x  x .
State the domain and range of this function.
f  x   x  3  2 .
f  x  x .
State the domain and range of
State the domain and range of this function.
f  x   x  4  1.
State the domain and range of this
1
. State the domain and range of this function.
x
1
f  x   2 . State the domain and range of this function.
x
1
f  x 
. State the domain and range of this function.
x3
f  x 
72) Sketch the graph of
f  x 
x2
.
2
x  x6
function.
VIII Limits
Problems:
Determine the following limits if they exist.
x4
x 2  2x  8
x 5
74) lim x  25
x  25
x 3
75) lim
x 3  x  3
73)
lim x 4
2x 2  x 1
76) lim x 
x3 1
3x  4
77) lim x 
2 x
x3  8
78) Let f  x  
x2
a) lim x 2 f  x  =
b) lim x  f  x  =
State the domain and range of this