FORMULAS:Probability and Statistics

1. Summation Notation Sigma means “sum”

Ex]

2. Measures of Central Tendencya) Mean - averageb) Median – middle number when put in orderc) Mode – the number that occurs most oftend) Range – difference between highest and lowest

3. Standard Deviation how spread the numerical data is from the mean.

S(x) = sample standard deviation

= population standard deviation

Normal Distribution

Probability

Permutations – an arrangement of objects in a specific order.

In general, the number of permutations of n things, taken n at a

time, with r of these things identical repeated is given by

** Combinations – are selections for which ORDER DOES NOT MATTER ***The combination of “n things taken r at a time” is denoted by :

1. A Bernoulli Experiment has only two outcomes (success or failure) Probability = nCr prqn-r

Where n = number of trials r = the number of successes

p = the probability of a successq = (1 – p) = the probability of a failure

2. At least r successes in n trials: add together for r, r + 1, r + …, n

3. At most r successes in n trials: add together for 0, 1, …, r

Binomial Expansion

1. Use the combination formula

(x + y)n = nC0xny0 + nC1xn-1y1 + nC2xn-2y2 + …. + nCnx0yn

2. Use Pascal’s Triangle 1

1 1 1 2 1

1 3 3 1 1 4 6 4 1 etc.

3. Using Pascal’s Triangle with the binomial expansion(x + y)0 1(x + y)1 1x + 1y(x + y)2 1x2 + 2xy + 1y2

(x + y)3 1x3 + 3x2y + 3xy2 + 1y3

(x + y)4 1x4 + 4x3y + 6x2y2 +4xy3 + 1y4

etc.

Regressions and Curve Fitting

1. Enter the data into lists (L1, L2)[STAT EDIT 1:Edit]

2. Look at the scatter plot[STAT PLOT, On Xlist L1, Ylist L2]

3. Fit a curve to the data. Choose from: [STAT, arrow over to CALC]

4: LinReg(ax + b) Linear5: QuadReg Quadratic9: LnReg Logarithmic0: ExpReg ExponentialA: PwrReg Power

4. Type “L1, L2, Y1” after the regression name.

5. To see the Correlation Coefficient (the r value) turn the Diagnostics on, by going to the catalog and choosing DiagnosticsON. GEOMETRY FORMUALS

Area of a TrapezoidA = ½h(b1 + b2)

Area of a CircleA = πr2

Circumference of a circle2πr or πd

Volume of a right circular cylinderV = πr2h

Volume of a right circular cone

V =

Parallel lines Perpendicular lines Equal slopes Negative and reciprocal slopes

LINES:Point – Slope form of a line:

1. Distance formula: - used to find length

2. Slope:

-used to show lines are parallel (same slope) - used to show lines are perpendicular

(negative reciprocal slopes)

3. Midpoint:

- used to show that diagonals bisect each other

SEQUENCES & SERIESSum of a Finite Arithmetic Series

Sum of a Finite Geometric Series

Completing the square:

1. All variable on one side All constants on the other2. Get x2 term to be 1x2

3. Complete square ( )2 = number4. Take of each side5. Left side = + Left side = 6. Solve

Example: Solve: 2x2 4x 14 = 0[Divide by 2] x2 2x 7 = 0[Variables on left] x2 2x = 7[complete square] x2 2x + 1 = 7 + 1 [square form] (x 1)2 = 8

[square root both sides]

[solve] x 1 =

[simplify] x = (ans)

CIRCLES:General form: and

1. Remember . . . Center(0, 0) and

radius is 3 center (3, -2) and radius is 4

Standard Form: 2. Given: Find the center and radius.

Center (2, 3) and radius is 3

3. Find the equation of a circle whose endpoints of the diameter are (1, 3) and (7, 5).

[find center]

[find radius: distance from center to one point]

Equation of circle: (x4)2 + (y1)2 = 25 (ans)

Factoring Polynomials

1. Greatest Common Factor

ax + bx = x(a+b)

2. Difference of two perfect squares

a2 – b2 = (a+b)(a-b)

or 9x2 -25 = (3x-5)(3x+5)

3. Factor completely

-means you will need to factor more than once

x3+9x2+14x = x(x2 +9x +14)

= x(x+7)(x+2)

4. Trial and Error

2x2 – x – 6 = (2x + 3)(x – 2)

Multiplying/ Dividing Rational Expressions1. If dividing, take reciprocal of 2nd fraction and change to multiplication2. Factor completely3. Cross cancel4. Multiply across

–2 (answer)

Adding/ Subtracting Rational Expressions1. Find the least common denominator (give missing part to top AND bottom of each fraction)2. Add or subtract numerators, (keep the denominator!)3. Simplify (if necessary)

Example:

(answer)

Simplifying Complex Fractions1. Find the common denominator2. Multiply each part by the common denominator3. Simplify

Example:

3–x (answer)

Solving Fractional Equations1. Find the common denominator2. Multiply each part by the common denominator – this should eliminate all fractions3. Solve the new equation4. Check

Example:

2y+1 – 15 = y – 6 2y – 14 = y – 6

y = 8 (answer)

Number Systems: 1. Natural Numbers : {1, 2, 3, 4, 5, ..}

2. Whole Numbers : {0, 1, 2, 3, 4, ..}

3. Integers : {…-3, -2, -1, 0, 1, 2, 3…}

4. Rational Numbers: have the form where a and b are

integers. (These can be fractions, repeating decimals, or terminating decimals)

5. Irrational Numbers – non-repeating, non-terminating

decimals

6. Real Numbers: made up of the rationals and irrationals.

7. Pure Imaginary Numbers : in the form bi

8. Complex Numbers : have the form a + bi.

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Absolute Value Equations

1.

x+a = b and -x – a = b

Solve both and check!!!

Absolute Value Inequalities

x + a < b and -x – a < b

Solve both, put on a number line and test!

Radicals

1. k is the index, a is the radicand

2. Table of Perfect Numbers

Perfect 2 Perfect3

1 1 1

2 4 8

3 9 27

4 16 64

5 25 125

… … …

3. Product Law :

4. Quotient Law:

Radical Equations1. To solve, isolate the radical terms, raise both sides to the inverse

power, solve for x, and check for extraneous roots.

Ex)

Complex Numbers1. Have the form a + bi, where i is the imaginary unit and a

and b are real numbers.

2.

3. “Cycle of 4” i0 = 1

i3 = -i i1 = i

i2 = -1

**remainder of exponent divide by 4

4. Types:

a) a + bi imaginary number

b) bi pure imaginary number

c) a + 0i real number

5. Properties:

a) The conjugate of a + bi = a – bi

b) The additive inverse of a + bi = -a – bi

c) The multiplicative inverse of a + bi =

d) The additive identity of a + bi = 0 + 0i

e) The multiplicative identity of a + bi = 1 + 0i

Quadratic Equation1. To find the roots of a quadratic equation, factor or use the quadratic

formula.

2. The Discriminant is b2 – 4ac

3. The Nature of the Roots

Discriminat Type of Roots Negative b2-4ac < 0 imaginary

Zero b2-4ac = 0 real rational, =

Positive, Perfect Square b2-4ac>0 Real, rational

Positive, Not a perfect square b2-4ac >0 Real irrational

4. Sum of the Roots: {r1,r2} of ax2 + bx + c = 0

5. Product of the Roots: {r1,r2} of ax2 + bx + c = 0

Transformations1. Line Reflections:

rx-axis(x,y) = (x,-y)

ry-axis(x,y) = (-x,y)

ry=x(x,y) = (y,x)

ry = -x(x,y) = (-y,-x)

rorigin(x,y) = (-x,-y)

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2. Point Reflection through the Origin is the same as

rotation.

3. Rotations

R90(x,y) = (-y,x) = R-270

R180(x,y) = (-x,-y) = R-180

R270(x,y) = (y,-x) = R-90

4. Translations

T(a,b) (x,y) = (x + a, y + b)

5. Dilations

Dk(x,y) = (kx, ky)

When 0 < k< 1, shape shrinks

When k > 1, shape enlarges

6. Glide Reflections – a composition of a line

reflection and a translation parallel to the

line of reflection.

7. Definitions:

Composition – two or more transformations

Isometry – transformation that preserves distance (line reflection,

rotation, point reflection, and translation)

Direct Isometry – preserves orientation, (translation, rotation, point

reflection)

Opposite Isometry – reverses orientation (line reflection, glide

reflection)

Relations and Functions1. A relation is any set of ordered pairs.

2. A function is a relation in which every element in the

domain corresponds to only one element in the range.

(Vertical line test to see if we have a function)

3. The domain is the set of first elements (x-value)

4. The range is the set of second elements (Y-values)

Special Relations and Functions1. Circle ax2 + ay2 = c

two squared terms

same coefficient and sign

2. Ellipse ax2 + by2 = c

two squared terms

different coefficient

same sign

3. Hyperbola

Case 1: ax2 -by2 = c

Two squared terms

Different signs

Case 2: xy = c

Aka. Inverse Variation

5. Vertical Parabola y = ax2 + bx + c

axis of symmetry:

when a > 0

when a < 0

6. Horizontal Parabola: x =ay2 + by + c

when a > 0

axis of symmetry y =

when a < 0

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Variation1. Direct Variation y = xc ( Graph is a straight line)

2. Inverse Variation xy = c (Graph is a hyperbola)

Other Functions1. Composition of Functions

**Performed right to left

2. Inverse Function f-1 (x)

Rule: Switch the x and y and solve for y. This is a reflection over

the line y = x.

Scientific Notation

1. 38.7 = 3.87 x 101

2. .0387 = 3.87 x 10-2

Exponent Laws1. (xa)(xb) = xa+b

2. xab+c = (xab) (xc)

3.

4. (xa)b = xab

5. (xyz)a = xayaza or (xbyc)a = xbayca

6. =

7. x0 = 1, (x 0)

8. x-a = or = xa

9. =

10. =

11. (xy)m = xmym

12. xy-1 =

13.

14. (xy)-1 =

15. x-1 + y-1 =

Exponential Functions1. y = ax

2. y =

Solving Equations with Fractional Exponents1. Isolate the variable with the fractional exponent

2. Raise both sides to the reciprocal exponent

3. Check.

Example:

Logarithms

1. y = logbx x=by

2. Log Laws:

Product: logb(AB) = logbA + logbB

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Quotient logb = logbA - logbB

Power logb = clogbA

Caution: There are no laws for addition or subtraction,

(ex: logb(A + B) can’t be performed! but logbA + logbB = lobb AB)

3. Solve for x: log2x + log2(x2) = 3

log2 x(x2) = 3 [condense]

23 = x(x2) [exponential form]

8 = x2 2x [solve]

0 = x2 2x 8

0 = (x4)(x+2)

x = 4 and x = 2(reject, can’t have negative logs)

ans: x = 4

4. Solve for x: log (x+3) log x = log 4

[condense]

4x = x+3 [cross multiply]

3x = 3 [solve]

ans: x = 1

example: y = log2 x

Exponential functions and logarithms are inverses

Ex: If y = 2x then f-1 is x = 2y and in log form y = log2 x

Since they are inverses they are ry=x

Solving Exponential Equations Using Logs1. Isolate the base raided to the variable.

2. Take the common log of both sides

Example:

Change of Base for Logs

Exponent Phrases you Should know!

1. A logarithm IS an exponent.

2. Fractional exponents represent ROOTS.

Ex.

3. Negative exponents represent FRACTIONAL

EXPRESSIONS. Ex. x-2 =

8

cb

a

Trigonometry1. Given:

2. Standard Position Angle – The initial ray is on the positive x-axis with the vertex at the origin and the terminal ray anywhere on the Cartesian graph.

3. Positive Angles open counter-clockwise: negative angles open clockwise.

= 135

= -45

4. Co-terminal Angles are angles that have the same terminal rays Ex: co-terminal

5. Trig ratios of co-terminal angles are equal. Ex: tan = tan ( )

Special Right Triangles

1. The Right Triangle Leg opposite = half the hypotenuse Leg opposite = half the hypotenuse times

3

1

2

60

30

45

452

1

1

2. The Right Triangle Leg = half the hypotenuse times Hypotenuse = leg times

Trig Reference Table

1. Table of Positive Trig Functions

all 6

sec

cos

cot

tan

csc

s in

2. Quotient Identities

3. Reciprocal Identities

4. Pythagorean Identities

5. Sum of Two Angles 6. Difference of Two Anglessin(A + B) = sinA cosB + cosA sinB sin(A – B) = sinA cosB – cosA sinBcos(A + B) = cosA cosB – sinA sinB cos(A – B) = cosA cosB + sinA sinB

tan(A + B) = tan(A B) =

7. Double angles

9

8. Half angles

10

s

r

The Unit Circle

(0,-1)

(-1,0)

(0,1)

(1,0)

(x,y) = (cos, sin)

Inverse Trig

Arcsin means angle whose sine is

Ex] x = arc cos cos x =

Ex] x = arc sin

Use 45 as a reference angle

Radians1. A radian is a measure of an angle that intercepts an arc equal to the length of the radius of the circle.

Central Angle (in radians) =

Cofunctions CO is abbreviation for COmplementary.

EX]

Ex]

Ex] sin(x + 10) = cos (3x) x + 10 + 3x = 90

x = 20

Solving Trig Equations

Ex] Find all for

2sin2 + 5 sin = 32sin2 + 5 sin -3 = 0

Degree 30 60 45 0 90 180 270 360

radian 0

sin 0 1 0 -1 0

cos 1 0 -1 0 1

tan 1 0 dne 0 dne 0

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Trig Graphs

1. y = a sin b , y = a cos bAmplitude - = maximum vertical height

The amplitude for tangent is undefined

Frequency = = # of full cycles from 0 to

Period = = length of one full cycle

2. Basic Graphs a) y = sin

b) y = cos

c) y = tan

3. If you need to write the equation of a trig function by looking at the graph, use the following formulas:

choose either a) y = a cos(bxh) + c or b) y = a sin(bxh) + c

b = frequency = the number of times the curve repeats from 0 to

h = horizontal shift

Trig Applications

1. Law of Cosines (SAS or SSS given) – Any side squared equals the sum of the squares of the other two sides minus two times the product of the other two sides times the cosine of the angle between those two sides.

C

B

A

c

b

a

a2 = b2 + c2 – 2bc cosA

2. Law of Sines ( ASA or AAS given) – Any side ratioed with the sine of the angle opposite it equals any side ratioed with the sine of the angle opposite it.

3. Area of a Triangle – The area of any triangle is equal to one half the product of any tow sides times the sine of the angle between those two sides.

4. Area of a Parallelogram K = ab sin C

5. The Ambiguous Case (SSA given)

There may be 0, 1, or 2 possible triangles. Solve for the unknown angle using the law of Sines and also see if the obtuse angle from Quadrant II (use the solved for angle as a reference angle) is a viable answer.

6. Forces

resultant force1st force

2nd force

remember: opposite sides of a parallelogram are consecutive angles are supplementary

7. Angles of elevation and depression

angle of depression

angle of elev ation

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Algebra 2/Trigonometry Reference Sheet

Area of a Triangle

Functions of the Sum of Two Angles

Functions of the Difference of Two Angles

Law of Sines

Sum of a Finite Arithmetic Series

Law of Cosines

Functions of the Double Angle

Functions of the Half Angle

Sum of a Finite Geometric Series

Binomial Theorem