FORMULAS:
Probability and Statistics
1. Summation Notation
Sigma means “sum”
Ex]
2. Measures of Central Tendency
a) Mean - average
b) Median – middle number when put in order
c) Mode – the number that occurs most often
d) 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 success
q = (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) Linear
5: QuadReg Quadratic
9: LnReg Logarithmic
0: ExpReg Exponential
A: 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 Trapezoid
A = ½h(b1 + b2)
Area of a Circle
A = πr2
Circumference of a circle
2πr or πd
Volume of a right circular cylinder
V = 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 & SERIES
Sum of a Finite Arithmetic Series
Sum of a Finite Geometric Series
Completing the square:
1. All variable on one side
All constants on the other
2. Get x2 term to be 1x2
3. Complete square ( )2 = number
4. Take of each side
5. 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: (x-4)2 + (y-1)2 = 25 (ans)
10
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 Expressions
1. If dividing, take reciprocal of 2nd fraction and change to multiplication
2. Factor completely
3. Cross cancel
4. Multiply across
–2 (answer)
Adding/ Subtracting Rational Expressions
1. 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 Fractions
1. Find the common denominator
2. Multiply each part by the common denominator
3. Simplify
Example: 3–x (answer)
Solving Fractional Equations
1. Find the common denominator
2. Multiply each part by the common denominator – this should eliminate all fractions
3. Solve the new equation
4. 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.
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 / Perfect31 / 1 / 1
2 / 4 / 8
3 / 9 / 27
4 / 16 / 64
5 / 25 / 125
… / … / …
3. Product Law :
4. Quotient Law:
Radical Equations
1. To solve, isolate the radical terms, raise both sides to the inverse power, solve for x, and check for extraneous roots.
Ex)
Complex Numbers
1. 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 Equation
1. 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
Transformations
1. 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)
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 Functions
1. 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 Functions
1. 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
Variation
1. Direct Variation y = xc ( Graph is a straight line)
2. Inverse Variation xy = c (Graph is a hyperbola)
Other Functions
1. 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 Laws
1. (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, (x0)
8. x-a = or = xa
9. =
10. =
11. (xy)m = xmym
12. xy-1 =
13.
14. (xy)-1 =
15. x-1 + y-1 =
Exponential Functions
1. y = ax
2. y =
Solving Equations with Fractional Exponents
1. Isolate the variable with the fractional exponent
2. Raise both sides to the reciprocal exponent
3. Check.
Example:
Logarithms
1. y = logbxx=by
2. Log Laws:
Product: logb(AB) = logbA + logbB
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(x-2) = 3
log2 x(x-2) = 3 [condense]
23 = x(x-2) [exponential form]
8 = x2 - 2x [solve]
0 = x2 - 2x - 8
0 = (x-4)(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 Logs
1. 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 =
10
10
10
10
Trigonometry
1. 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.
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
2. The Right Triangle
Leg = half the hypotenuse times
Hypotenuse = leg times
Trig Reference Table
1. Table of Positive Trig Functions
2. Quotient Identities
3. Reciprocal Identities
4. Pythagorean Identities
5. Sum of Two Angles 6. Difference of Two Angles
sin(A + B) = sinA cosB + cosA sinB sin(A – B) = sinA cosB – cosA sinB
cos(A + B) = cosA cosB – sinA sinB cos(A – B) = cosA cosB + sinA sinB
tan(A + B) = tan(A - B) =
7. Double angles
8. Half angles
10
10
The Unit Circle
Inverse Trig
Arcsin means angle whose sine is
Ex] x = arc cos cos x =