Unit 01 • Algebraic Tools for Operating with functions 1.1
Reviewing Exponent Laws
• a2=1a-m
1.2
Rational Exponents
• a1n= na, when n is a natural number || if n is odd, a can be any real number.
and
• a-m= 1am
• amn= nam = (na) m. 1.3
Solving Ex Exponential Equation Half Life: •
1.4
Add, Subtract, Multiply, • a2 - b2 = (a - b)(a + b) • a2 + 2ab + b2 = (a + b)2 Divide Polynomials • a3 - b3 = (a - b)(a2 + ab + b2)
• a3 + b3 = (a + b)(a2 - ab + b2) Unit 02 • Quadratic Functions and equations 2.1
Complex Number System
• i2=-1
2.2
Max or Min of a Quadratic Function By completing Square
• y= a( x-h) 2+ k Minimum
2.3
Solving Quadratic Equation
2.4
Tools for Operating with >
3.4
• i=-1 if a> 0, if a < 0,
Maximum Quadratic Formula : x=-b±b2-4ac2a Complex Numbers
Reflections of of functions • y=
-x
• y= -f(x)
re ec n y
s
reflect in x-axis
K is K is
Unit 03 • Transformations of Functions
Unit 03 • Special Formula
Verti
Y =
a>1, expand vertically a<1, compress vertically b>1, compress horizontally by a factor of
- a f {-b {-b ( x- h) } -1/bk
b<1, expand horizontally by a factor of 1/b
Horizo k>0, Graph goes upward k<0, Graph goes Downward
Reflect in x
Reflect in y
h>0, then moves to the right h<0, then moves to left
Unit 04 • Trigonometry 4.1
Basic Formulas
• SOH CAH TOA
• csc θ= 1sinθ
• sec θ
=
1cosθ
• cot θ =
1tanθ
• sin2θ + cos2 θ= 1 • tan2θ+1= sec2θ 4.3
Sine and Cosine Law
• a/sinA= b/sinB
• 1+cot2θ=csc2θ • tan θ= sinθcosθ
•when measure of two angles and any sides are given •measure of two sides and
a2=b2+c2-2bccos A • a2=b2+c2-2bccos
• cosA= b2+c2-a22bc
• when measure of two sides and the contained angle • the measure three sides
4.4
The Sine Law: The Ambiguous case
Obsicute Triangle (A≥90®) If a≤b ( no triangle) If a>b ( one solution) Acute Ac ute Tria Tr iabg bgle le ( A<90 A< 90 ®) a≥b ( one triangle) a
Unit 05 • Trigonometric Functions 5. 1
Radians and Angles measure
• Number of Radians = arc lengthradius
or θ=
a/r • To change radian measure to degree measure, multiply the number of radians by (180π)°
• To change degree measure to radian measure, multiply the number of degrees by (π180)
5. 2
Trigonometric ratios of any Angle
• CAST law : S Sine T Tangent
A All C Cosine +
• 30-60-90 triangle ( hyp=2, Adj= 1, Oppo= 3 ) • 45- 45-90 Triangle ( hyp= 2, , Adj= 1, Oppo= 1)
Unit 05 Special Formula for Sine and Cosine Graph
If A>1, graph expand vertically If A<1, graph compressed vertically
Y = A sin k (x-θ)+b (x-θ)+b
If k>1, Graph compressed horizontally by 1/k If k<1, Graph expanded horizontally by 1/k
b is vertical
Θ is Phase shift, relative to sin(kx) K is period = A=
Unit 06 • Sequences and Series 6. 2
Arithmetic Sequnces
a+ (n-1)d a+3d,..., …, )
6. 3
Geometric Sequence
• tn=arn-1
6. 5
Arithmetic Series
• Sn=n2[2a+n-1d ]
6. 6
Geometric Series
• Sn=a(rn-1)r-1
• tn=
( a, a+d, a+2d, (a, ar, ar 2,ar3,…,…,..)
Unit 07 • Compound Interest and Annuities 7.2 Compound Interest
• A= P(1+i) n A= Amount P= Principal Invested i = Interest rate per compounding period • n= yN n= Number of compounding period r= Interest per annum N= number of compounding period/year • PV= A(1-i)n Y = number of years PV= Present value
• i= r÷N
7.4 Present value
Or • PV= A(1+i) -n
7.5 7.5 Amoun mountt in in Ord Ordin inar ary y Annuity
•
7.6 Prese resent nt Valu Value e of of an an ordinary Annuity
• PV=
A= R[1+i n-1]i
R [1-1+i -ni]
