Download Essential Algebra 2 Formula Cheat Sheet and more Cheat Sheet Algebra in PDF only on Docsity!
Essential Formulas for Algebra 2 Final Exam
Laws of Exponents
Multiply Powers of the Same Base = Adding Exponents (a
m
)(a
n
) = a
m + n
Divide Powers of the Same Base = Subtracting Exponents
n
m
a
a
= a
m −−−− n
Power Rule = Multiplying Exponents
(a
m
)
n
= a
m ×××× n
Zero Exponent = 1 a
0
= 1
Distribution of Exponent with Multiple Bases
(ab)
n
= a
n
b
n
n
b
a
=
n
n
b
a
Negative Exponent = Reciprocal
a
−−−−n
=
n
a
n
m
b
a
−−−−
−−−−
=
m
n
a
b
Distribution of Negative Exponent with Multiple Bases
(ab)
− −−
−n
= a
− −−
−n
b
− −−
−n
=
n n
a b
n
b
a
−−−−
=
n
a
b
=
n
n
a
b
Properties of Radicals
Distribution of Radicals of the Same Index
(where a ≥≥≥≥ 0 and b ≥≥≥≥ 0 if n is even)
n
ab = (((( ))))(((( ))))
n n
a b
n
b
a
n
n
b
a
Power Rule of Radicals = Multiplying Exponents
m n
a =
(((( m n))))
a
××××
Reverse Operations of Radicals and Exponents
n n
a = a (if n is odd)
n n
a = | a | (if n is even)
a b a b a b a b
a b a b a b a b
−−−− ≠≠≠≠ −−−− ÷÷÷÷ ==== ÷÷÷÷
× ××
= ×
× = ××
≠ + ≠≠
n
m
a
n m
a
The index of the radical is the
denominator of the fractional exponent.
Special Products (A + B)(A −−−− B) = A
2
−−−− B
2
(A + B)
2
= A
2
+ 2AB + B
2
(A + B)
3
= A
3
+ 3A
2
B + 3AB
2
+ B
3
(A −−−− B)
2
= A
2
−−−− 2 AB + B
2
(A −−−− B)
3
= A
3
−−−− 3 A
2
B + 3AB
2
−−−− B
3
Special Expressions
Difference of Squares A
2
−−−− B
2
= (A + B)(A −−−− B)
Perfect Trinomial Squares A
2
+ 2AB + B
2
= (A + B)
2
Perfect Trinomial Squares A
2
−−−− 2 AB + B
2
= (A −−−− B)
2
Sum of Cubes A
3
+ B
3
= (A + B)(A
2
−−−− AB + B
2
Difference of Cubes A
3
−−−− B
3
= (A −−−− B)(A
2
+ AB + B
2
Quadratic Formula:
a
b b ac
x
2
4
2
− −−
± − ±±
− ± −−
−
====
Discriminant = b
2
−−−− 4 ac
When Discriminant is Positive, b
2
−−−− 4 ac > 0 →→→→ Two Distinct Real Roots
When Discriminant is Zero, b
2
−−−− 4 ac = 0 →→→→ One Distinct Real Root
(or Two Equal Real Roots)
When Discriminant is Negative, b
2
−−−− 4 ac < 0 →→→→ No Real Roots
Note the pattern:
i
1
= i i
2
− 1 i
3
−i i
4
i
5
= i i
6
= −−−− 1 i
7
= −−−−i i
8
i
9
= i i
10
Pattern repeats every 4
th
power of i.
Product of Conjugate Complex Numbers
(a + bi)(a −−−− bi) = a
2
− b
2
i
2
= a
2
− b
2
(a + bi)(a −−−− bi) = a
2
2
Midpoint of a Line Segment Distance of a Line Segement Slope
1 2 1 2
x x y y
M d =
2
2 1
2
2 1
x −x + y −y
2 1
2 1
x x
y y
m
−
−
=
Standard Equation for Circles
(x −−−− h)
2
+ (y −−−− k)
2
= r
2
P (x, y) = any point on the path of the circle
C (h, k) = centre of the circle
r = length of the radius
Summary of Types of Functions: (see page 226 of textbook)
Linear Functions f(x) = mx + b
Power Functions f(x) = x
n
where n > 1 and n ∈∈∈∈ N
Root Functions f(x) =
n
x where n ≥ 2 and n ∈∈∈∈ N
Reciprocal Functions f(x) =
n
x
where n ∈ ∈∈
∈ N
Absolute Value Functions Greatest Integer Functions
Domain: x ∈ ∈∈
∈ R
Range: f(x) ∈∈∈∈ R
f(x) = mx
Domain: x ∈ ∈∈
∈ R
Range: f(x) ∈∈∈∈ R
f(x) = mx + b
b
Domain: x ∈ ∈∈
∈ R
Range: f(x) ∈∈∈∈ R
f(x) = b
b
f(x) = x
2
Domain: x ∈∈∈∈ R
Range: f(x) ≥ 0
Domain: x ∈∈∈∈ R
Range: f(x) ∈∈∈∈ R
f(x) = x
3
f(x) = x
5
Domain: x ∈∈∈∈ R
Range: f(x) ∈∈∈∈ R
f(x) = x
4
Domain: x ∈∈∈∈ R
Range: f(x) ≥ 0
f(x) = x
Domain: x ≥ 0
Range: f(x) ≥ 0
Domain: x ∈∈∈∈ R
Range: f(x) ∈∈∈∈ R
f(x) =
3
x f(x) =
5
x
Domain: x ∈∈∈∈ R
Range: f(x) ∈∈∈∈ R
f(x) =
4
x
Domain: x ≥ 0
Range: f(x) ≥ 0
f(x) = | x |
Domain: x ∈∈∈∈ R
Range: f(x) ≥ 0
Domain: x ∈∈∈∈ R
Range: f(x) ∈∈∈∈ I
f(x) = [[x]] or int(x)
f(x) =
x
Domain: x ≠ ≠≠
Range: f(x) ≠≠≠≠ 0
Domain: x ≠ ≠≠
Range: f(x) ≠≠≠≠ 0
f(x) =
3
x
Domain: x ≠ ≠≠
Range: f(x) > 0
f(x) =
4
x
Domain: x ≠ ≠≠
Range: f(x) > 0
f(x) =
2
x
g(x) = f(x + h) + k
h = amount of horizontal movement h > 0 (move left); h < 0 (move right)
k = amount of vertical movement k > 0 (move up); k < 0 (move down)
Reflection off the x-axis
g(x) = −f(x)
All values of y has to switch signs but
all values of x remain unchanged.
Vertical Stretching and Shrinking
g(x) = af(x)
a is the Vertical Stretch Factor
a > 1 (Stretches Vertically by a factor of a)
0 < a < 1 (Shrinks Vertically by a factor of a)
Reflection off the y-axis
g(x) = f(−x)
All values of x has to switch signs but
all values of y remain unchanged.
Horizontal Stretching and Shrinking
g(x) = f(bx)
b is the Horizontal Stretch Factor
0 < b < 1 (Stretches Horizontally by a factor of 1/b)
b > 1 (Shrinks Horizontally by a factor of 1/b)
For Quadratic Functions in Standard Form of f(x) = a(x − −−
− h)
2
Vertex at (h, k) Axis of Symmetry at x = h Domain: x ∈∈∈∈ R
a = Vertical Stretch Factor
a > 0 Vertex at Minimum (Parabola opens UP) Range: y ≥ ≥≥
≥ k (Minimum)
a < 0 Vertex at Maximum (Parabola opens DOWN) Range: y ≤≤≤≤ k (Maximum)
| a | > 1 Stretched out Vertically | a | < 1 Shrunken in Vertically
h = Horizontal Translation (Note the standard form has x − h in the bracket!)
h > 0 Translated Right h < 0 Translated Left
k = Vertical Translation
k > 0 Translated Up k < 0 Translated Down
For Quadratic Functions in General Form: f(x) = ax
2
y-intercept at (0, c) by letting x = 0 (Note: Complete the Square to change to Standard Form)
x-intercepts at
2
a
b b ac
if b
2
− 4ac ≥ 0. No x-intercepts when b
2
− 4ac < 0
Vertex locates at x = −
a
b
y =
a
b
f
Minimum when a > 0 ; Maximum when a < 0
f(x) = One-to-One Function f
−−−− 1
(x) = Inverse Function
(x, y) (y, x)
Domain of f(x) →→→→ Range of f
− −−
− 1
(x)
Range of f(x) → →→
→ Domain of f
−−−− 1
(x)
Note: f
−−−− 1
(x) ≠≠≠≠
( ((
( ) ))
f x)
(Inverse is DIFFERENT than Reciprocal)
If R = 0 when
(((( ))))
(((( x b))))
P x
, then (x −−−− b) is a factor of P(x) and P(b) = 0.
P(x) = D(x) × Q(x)
P(x) = Original Polynomial D(x) = Divisor (Factor) Q(x) = Quotient
If R ≠≠≠≠ 0 when
( ((
( ) ))
)
(( (( x b))))
P x
, then (x −−−− b) is NOT a factor of P(x).
P(x) = D(x) × Q(x) + R(x)
The Remainder Theorem:
To find the remainder of
x b
P x
: Substitute b from the Divisor, (x −−−− b), into the Polynomial, P(x).
In general, when
x b
P x
, P(b) = Remainder.
To find the remainder of
ax b
P x
: Substitute
a
b
from the Divisor, (ax − −−
− b), into the Polynomial, P(x).
In general, when
ax b
P x
, (((( ))))
a
b
P = Remainder.
The Factor Theorem:
- If
x b
P x
gives a Remainder of 0, then (x −−−−b) is the Factor of P(x).
OR
If P(b) = 0, then (x −−−− b) is the Factor of P(x).
- If
ax b
P x
gives a Remainder of 0, then (ax −−−−b) is the Factor of P(x).
OR
If (((( ))))
a
b
P = 0, then (ax − −−
− b) is the Factor of P(x).
Rational Roots Theorem:
For a polynomial P(x), a List of POTENTIAL Rational Roots can be generated by Dividing
ALL the Factors of its Constant Term by ALL the Factors of its Leading Coefficient.
Potential Rational Zeros of P(x) =
ALLFactorsoftheLeadingCoefficient
ALLFactorsoftheConstantTerm
The Zero Theorem
There are n number of solutions (complex, real or both) for any n
th
degree polynomial function
accounting that that a zero with multiplicity of k is counted k times.
Graphs of Exponential Functions
x
y
f(x) = a
x
for a > 1
x
y
f(x) = a
x
for 0 < a < 1
Graphs of Natural Exponential Functions
x
y
f(x) = e
x
x
y
f(x) = e
−x
y = a
x
x = log
a
y
Simple Properties of Logarithms
log
a
1 = 0 because a
0
log
a
a = 1 because a
1
= a
x
a
a
log
= x because exponent and logarithm are inverse of one another
log
a
a
x
= x because logarithm and exponent are inverse of one another
Common and Natural Logarithm
Common Logarithm: log x = y 10
y
= x
Natural Logarithm: ln x = y e
y
= x
Exponential Laws Logarithmic Laws
(a
m
)(a
n
) = a
m + n
log
a
x + log
a
y = log
a
(xy)
n
m
a
a
= a
m −−−− n
log a
x −−−− log a
y = log a
y
x
(a
m
n
= a
m × n
log a
x
y
= y log a
x
a
0
= 1 log a
ππ ππ rad = 180
o
OR
180
ππππ
rad = 1
o
y = a sin k(x + b) + c y = a cos k(x + b) + c
| a | = Amplitude c = Vertical Displacement (how far away from the x–axis)
b = Horizontal Displacement (Phase Shift) b > 0 (shifted left) b < 0 (shifted right)
k = number of complete cycles in 2π Period =
k
k
o
Range = Minimum ≤≤≤≤ y ≤≤≤≤ Maximum
y = a sin [ ωωωω (t + b)] + c y = a cos [ωω ωω (t + b)] + c
| a | = Amplitude c = Vertical Displacement (distance between mid-line and t-axis)
b = Horizontal Displacement (Phase Shift) b > 0 (shifted left) b < 0 (shifted right)
ω ωω
ω = number of complete cycles in 2π Period =
Frequency =
Range = Minimum ≤≤≤≤ y ≤≤≤≤ Maximum
Note: sin
−
(x) ≠≠≠≠
sin( )
x
sin
−
(x) ≠≠≠≠ (sin x)
−
(sin x)
−
sin( )
x
= csc x
y = sin
−
x
Domain: −1 ≤ x ≤ 1 Range: −
≤ y ≤
sin(sin
−
x) = x for −1 ≤ x ≤ 1
sin
−
(sin x) = x for −
≤ x ≤
y = cos
−
x
Domain: −1 ≤ x ≤ 1 Range: 0 ≤ x ≤ π
cos(cos
−
x) = x for −1 ≤ x ≤ 1
cos
−
(cos x) = x for 0 ≤ x ≤ π
y = tan
−
x
Domain: x ∈∈∈∈ R Range: −
≤ y ≤
tan(tan
−
x) = x for x ∈ ∈∈
∈ R
tan
−
(tan x) = x for −
≤ x ≤
Some Basic Trigonometric Definitions and Identities (proven equations)
tan θ =
cos
sin
csc θ =
sin θθθθ
sec θ =
cos θθθθ
cot θ =
tan θθθθ
sin
cos
cos
2
θ + sin
2
θ = 1
