Dr. Adler SPSU Math 1113 Cheat Sheet: Page 1
SPSU Math 1113: Precalculus Cheat Sheet
§5.1 Polynomial Functions and Models (review)
Steps to Analyze Graph of Polynomial
1. y-intercepts: f (0)
2. x-intercept: f(x) = 0
3. f crosses / touches axis @ x-intercepts
4. End behavior: like leading term
5. Find max num turning pts of f: (n – 1)
6. Behavior near zeros for each x-intercept
7. May need few extra pts to draw fcn.
§5.2 Rational Functions
Finding Horizontal/Oblique Asymptotes of R
where degree of numer. = n and degree of denom. = m
1. If n < m, horizontal asymptote: y = 0 (the x-axis).
2. If n = m, line =
is a horizontal asymptote.
3. If n = (m + 1), quotient from long div is ax + b and line y = ax
+ b is oblique asymptote.
4. If n > (m + 1), R has no asymptote.
§7.6 Graphing Sinusoidals
Graphing y = A sin (ωx) & y = A cos (ωx)
|A| = amplitude (stretch/shrink vertically)
|A| < 1 shrink |A| > 1 stretch A < 0 reflect
Distance from min to max = 2A
ω = frequency (stretch/shrink horizontally)
|ω| < 1 stretch |ω| > 1 shrink ω < 0 reflect
period = T =
§7.8 Phase Shift =
y = A sin (ωx – φ) + B
y = A cos (ωx – φ) + B
§8.1 Inverse Sin, Cos, Tan Fcns
y = sin
-1
(x)
Restrict range to [-π/2, π/2]
y = cos
-1
(x)
Restrict range to 0,
y = tan
-1
(x)
Restrict range to −
,
§8.2 Inverse Trig Fcns (con’t)
y = sec
-1
x where |x| ≥ 1 and 0 ≤ y ≤ π,
y ≠
y = csc
-1
x where |x| ≥ 1 and −
≤ y ≤
,
y ≠ 0
y = cot
-1
x where -∞ < x < ∞ and 0 < y < π
§8.3 Trig Identities
tan=
cot=
csc=
#
sec=
#
cot=
#
%&
Pythagorean: sin
2
θ + cos
2
θ = 1
tan
2
θ + 1 = sec
2
θ cot
2
θ + 1 = csc
2
θ
§8.4 Sum & Difference Formulae
cos'α±β+=cosα cos β ∓ sinα sin β
sin'α±β+=sinα cos β ± cosα sin β
tan'α±β+=
%&'.+±%&'/+
#∓%&'.+%&'/+
§8.5 Double-Angle & Half-Angle Formulae
sin '2α+= 2sin α cosα cos '2α+=cos
α−sin
α
cos '2α+= 1 − 2 sin
α = 2 cos
α−1
tan'2α+=
%&'.+
#2%&
3
'.+
sin
4
=±5
#2'4+
cos
4
=±5
#6'4+
tan
4
=±5
#2'4+
#6'4+
=
#24
4
=
4
#64
§9.2 Law of Sines
7
=
8
=
9
:
§9.3 Law of Cosines
a
2
= b
2
+ c
2
– 2bc cos A
c
2
= a
2
+ b
2
– 2ab cos C b
2
= a
2
+ c
2
– 2ac cos B
§9.4 Area of Triangle
K =
#
;< sin = =
#
>< sin ? =
#
>; sin @
Heron’s Formula
A =
#
'>+;+<+
K = CA 'A − >+'A − ;+'A − <+
§9.5 Simple & Damped Harmonic Motion
Simple Harmonic Motion
d = a cos(ωt) or d = a sin(ωt)
Damped Harmonic Motion
D'E+=>
F
−
'
;E
+ '
2G
+⁄
cos
I5J
−
3
KL
3
EM
where a, b, m constants:
b = damping factor (damping coefficient)
m = mass of oscillating object
|a| = displacement at t = 0
= period if no damping
§10.1 Polar Coordinates
Convert Polar to Rectangular Coordinates
x = r cos θ y = r sin θ
Convert Rectangular to Polar Coordinates
If x = y = 0 then r = 0, θ can have any value
else N=CO
+
=
P
Q
R
Q
S
tan
−1
T
U
tan
−1
T
U
+
2
⁄
− 2
⁄V
Q
X
YN Q
XZ
Q
XX
YN Q
XXX
O=0, >0
O=0, <0
§10.3 Complex Plane & De Moivre’s Theorem
Conjugate of z = x + yi is ]^ = x + yi
Modulus of z: |]|=√] ]^ =CO
+
Products & Quotients of Complex >bs (Polar)
z
1
= r
1
(cos θ
1
+ i sin θ
1
) z
2
= r
2
(cos θ
2
+ i sin θ
2
)
]
#
]
=N
#
N
cos'
#
+
++asin'
#
+
+
b
c
b
3
=
d
c
d
3
cos'
#
−
++asin'
#
−
+ z
2
≠ 0-
De Moire’s Theorem
z = r (cos θ + i sin θ)
]
e
=N
e
cos 'f++asin'f+ n ≥ 1
ghijklm nhhop n ≥ 2, k = 0, 1, 2, …, (n – 1))
]
q
=√N
rcos
e
+
q
e
+ a sin
e
+
q
e
s
where k = 0, 1, 2, …, (n – 1)
