Download Math 095 – Formulas

Math 095 Reference Sheet
Order of Operations: (Please Excuse My Dear Aunt Sally)
Parentheses
Exponents
Multiplication
Done at the same time - left to right
Division
Addition
Done at the same time - left to right
Subtraction
Slope Formulas:
𝑚 = 𝑠𝑙𝑜𝑝𝑒
Q
𝑚=
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
𝑝𝑜𝑖𝑛𝑡𝑠 = (𝑥1 , 𝑦1 ) 𝑎𝑛𝑑 (𝑥2 , 𝑦2 )
or 𝑚 =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 (∆𝑦)
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 (∆𝑥)
}
}
𝑦 −𝑦
or 𝑚 = 𝑥2 −𝑥1
2
1
Horizontal Line:
𝑦 = 𝑚𝑥 + 𝑏
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
Vertical Line:
What Can Slope and Intercepts Tell You About lines?
One Line
Two Lines
Y - Intercept formula:
Point – Slope formula:
𝑦=𝑏
𝑥=𝑎
Slope:
Positive
Negative
Zero
Undefined
Same
Same
Opposite Reciprocal
Different
y-Int:
Any
Any
Any
None
Different
Same
Same or Different
Same or Different
Increasing
Decreaseing
Horizontal
Vertical
Parrallel
Coinciding
Perpendicular
Intersecting
Solving Linear Equations:
1. Deal with any parentheses in the problem.
2. Combine like terms on each side of the equal sign.
3. Move terms with the needed variable on one side of
the equal sign everything else to the other side.
4. Isolate the needed variable.
Interest:
𝐴=𝑃+𝐼
Simple
𝐼 = 𝑃𝑟𝑡
First, Outside, Inside, Last
(𝑎1 𝑥 + 𝑏1 )(𝑎2 𝑥 + 𝑏2 ) = 𝑎1 𝑎2 𝑥 2 + (𝑎1 𝑏2 + 𝑎2 𝑏1 )𝑥 + 𝑏1 𝑏2
Factoring Special Forms
𝐴2 + 2𝐴𝐵 + 𝐵2 = (𝐴 + 𝐵)2
𝐴2 − 2𝐴𝐵 + 𝐵2 = (𝐴 − 𝐵)2
𝐴2 − 𝐵2 = (𝐴 + 𝐵)(𝐴 − 𝐵)
3
𝐴 + 𝐵3 = (𝐴 + 𝐵)(𝐴2 − 𝐴𝐵 + 𝐵2 )
𝐴3 − 𝐵3 = (𝐴 − 𝐵)(𝐴2 + 𝐴𝐵 + 𝐵2 )
P=Principle
r = rate (decimal)
A=Future Amount
t = time
n = # of times compounded per year
Continuously
Compound
Compounded
𝑟 𝑛𝑡
𝐴 = 𝑃 (1 + )
𝑛
𝐴 = 𝑃𝑒 𝑟𝑡
Step 3:
Step 4:
Step 5:
PT
P2
Use with 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
The “AC” Method:
Step 1:
Step 2:
Mixture Problems: A “Visual” Way
P1
Binomial Expansion:
You may know it as a four letter word starting with “F”
Multiply “a” and “c”
Find factors of your new number that will add to
equal “b”.
Rewrite “bx” using your two new numbers.
Factor by grouping.
Move your “common” binomial out, and make your
other two terms a new binomial
Quadratic Equation:
A1
A2
AT
𝑥=
Discriminant:
𝑨𝟏 𝑷𝟏 + 𝑨𝟐 𝑷𝟐 = 𝑨𝑻 𝑷𝑻
Variation:
“k” the constant of variation
Varies with the Square
Directly
𝑦 =𝑘∙𝑥
𝑦 = 𝑘 ∙ 𝑥2
Inversely
Page 1 of 2
𝑦=
𝑘
𝑥
𝑦=
𝑘
𝑥2
+
0
2 real solutions
2 complex solutions
1 real solution
−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎
The Vertex of a Quadratic:
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
The vertex form:
𝑏 2 − 4𝑎𝑐
A is the amount and P is the percent. One of these will be missing
Use with 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘
The vertex is the point (ℎ, 𝑘)
(ℎ, 𝑘) = (−
𝑏
𝑏
, 𝑓 (− ))
2𝑎
2𝑎
Distance, Rate, and Time
𝑑 = 𝑟𝑡
Rev 2.0
4/17/2015
Math 095 Reference Sheet
Square
Rectangle
Triangle
Parallogram
Trapazoid
Circle
Perimeter
𝑃 = 4𝑠
𝑃 = 2𝑙 + 2𝑤
𝑃 = 2𝑎 + 2𝑏
𝐴 = 𝑠2
𝐴 = 𝑙𝑤
𝑃 =𝑎+𝑏+𝑐+𝐵
1
𝐴 = ℎ(𝑏 + 𝐵)
2
𝐶 = 2𝜋𝑟
Area
Cube
Rectangler Solid
𝑃 = 𝑎+𝑏+𝑐
1
𝐴 = 𝑏ℎ
2
Right Circular
Cylinder
Cone
Right Pyramid
Sphere
Volume
𝑉 = 𝑠3
𝑉 = 𝑙𝑤ℎ
𝑉 = 𝜋𝑟 2 ℎ
1
𝑉 = 𝜋𝑟 2 ℎ
3
1
𝑉 = 𝑙𝑤ℎ
3
4
𝑉 = 𝜋𝑟 2
3
Surface
Area
𝑆 = 6𝑠 2
𝑆 = 2𝑙𝑤 + 2𝑙ℎ + 2ℎ𝑤
𝑆 = 2𝜋𝑟ℎ + 2𝜋𝑟 2
𝑆 = 𝜋𝑟√𝑟 2 + ℎ2 + 𝜋𝑟 2
𝑤 2
𝑙 2
𝑆 = 𝑙𝑤 + 𝑙√( ) + ℎ2 + 𝑤 √( ) + ℎ2
2
2
𝑆 = 4𝜋𝑟 2
𝐴 = 𝑏ℎ
Pythagorean Theorem:
Between Two Points
(𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 )
The hypotenuse “c” is the
longest side and it is
opposite the right angle!
Distance
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
𝑖 = √−1
Complex Numbers:
Midpoint Formula
𝑥2 + 𝑥1 𝑦2 + 𝑦1
(
,
)
2
2
𝑎2 + 𝑏 2 = 𝑐 2
𝐴 = 𝜋𝑟 2
Complex Numbers Have a
Real Part and an Imaginary
Part
Real
𝑎
+
Imaginary
𝑏𝑖
Complex numbers have complex conjugates
Circles:
Center = (𝒉, 𝒌)
Radius =
𝒓
Absolute Value: The distance from zero
!!! Just think it makes things positive !!!
|𝑥 − 𝑎| = 𝑏
(𝑥 − 𝑎) = 𝑏
−(𝑥 − 𝑎) = 𝑏
OR
𝑥−𝑎 =𝑏
−𝑥 + 𝑎 = 𝑏
𝑥 =𝑎+𝑏
𝑥 =𝑎−𝑏
2
(𝑥 − ℎ) + (𝑦 − 𝑘) = 𝑟
Interval
Set Builder
𝑥<𝑎
(−∞, 𝑎)
{𝑥| 𝑥 < 𝑎 }
𝑥≥𝑏
[𝑏, ∞)
{𝑥| 𝑥 ≥ 𝑏 }
𝑎<𝑥<𝑏
(𝑎, 𝑏)
{𝑥| 𝑎 < 𝑥 < 𝑏 }
𝑎<𝑥≤𝑏
(𝑎, 𝑏]
{𝑥| 𝑎 < 𝑥 ≤ 𝑏 }
𝑎≤𝑥<𝑏
[𝑎, 𝑏)
{𝑥| 𝑎 ≤ 𝑥 < 𝑏 }
𝑎≤𝑥≤𝑏
[𝑎, 𝑏]
{𝑥| 𝑎 ≤ 𝑥 ≤ 𝑏 }
Page 2 of 2
2
𝑎 + 𝑏𝑖
𝑎 − 𝑏𝑖
When you multiply remember to treat 𝒊 like 𝒙
Inequalities:
Remember to change the direction of the
inequality when multiplying or dividing by
a negative.
Solution Center: “How to answer the question”
A number that makes the equation true
Solve for “x”
An ordered pair that makes the equation true
Find a point
The point(s) where a graph crosses the X Axis
X -Intercept
The point(s) where a graph crosses the Y Axis
Y-Intercept
Notation
2
𝑎≥𝑏
−1 ∙ 𝑎 ≤ −1 ∙ 𝑏
−𝑎 ≤ −𝑏
until you get 𝒊𝟐 which is – 𝟏.
𝑖= 𝑖
𝑖5 = 𝑖
2
𝑖 = −1
𝑖 6 = −1
3
𝑖 = −𝑖
𝑖 7 = −𝑖
4
𝑖 = 1
𝑖8 = 1
𝑥=
(𝑥, 𝑦)
(
,0)
(0,
)
Graph
Meter
Conversions
Gram
kilo hecto deca UNIT deci
1000 100
10
1
0.1
1 𝑚𝑖 = 5280 𝑓𝑡
1 𝑙𝑏 = 16 𝑜𝑧
1 𝑔𝑎𝑙 = 4 𝑞𝑡 = 16 𝑐
9
𝐹 = 𝐶 + 32
5
Rev 2.0
Liter
centi
0.01
milli
0.001
1 𝑚𝑖 = 1.61 𝑘𝑚
1 𝑙𝑏 = 0.45 𝑘𝑔
1 𝑔𝑎𝑙 = 3.79 𝐿
1 𝑠𝑞𝑓𝑡 = 0.09 𝑚2
4/17/2015