Let's not fail algebra 2

Function notation

F(x)=

Naming Polynomials

Make sure like terms are combined Degree/Exponent: Use highest degree 0= Constant (ex: y=5) 1= Linear (ex:y=5x) 2= Quadratic (ex: y=5^2) 3= Cubic 4= Quartic 5= Quintic Number of terms 1= monomial 2= binomial 3= trinomial ≥4= polynomial

What's are other words for solution?

zero, root, x=intercept

Transformations

|a| > 1 Stretch |a| < 1 Compress -a Reflect over x-axis x+h Left x-h Right +k Up -k Down

linear inequality

< or > = dotted line ≤ or ≥ = solid line Diving by a negative flips the direction of the sign.

Complex Number

A combination of a real and an imaginary number A ± i Real number must come first Graphed on a complex number plane where the real number is the X coordinate and the imaginary is the Y Ex: 3-2i = (3, -2) -i = (0,-1) [same as 0-i] 7= (7, 0) [same as 7+0i] Absolute value= distance from origin on a complex plane, | a+bi | = √(a^2 + b^2) Ex: | 6 + 3i | 6^2 + 3^2= 45 √45 = 3√5 | 6 + 3i | = 3√5 Add & Subtract- combine like terms as you would like i was a variable: Ex: (4-3i) - (-5+2i) 9-5i i= √-1 i^2= -1 i^3= -√-1 i^4= 1 i^5= √-1 i^6= -1 i^7= -√-1 i^8= 1 Etc. Pattern repeats every 4 To find i^#, divide exponent by four and use remainer as the exponent. i^4 aka no remainder. Ex: 2^27 27/4= 6 r 3 i^3= -2√-1 √-1 = i -2i Ex2: i^16 16/4=4 r 0 i^16=1 Multiplying, multiply like you would if i was a variable, but using info about i^# Ex: (4 + √-9)(-1 - √-4) (4+3i)(-1-2i) -4 -8i -3i -6i^2 -4 - 11i +6 2 - 11i Imaginary solutions com in pairs ex: √-9 = ±3i DORITO

Scatter plot

A graph with coordinates plotted plotted to show a relationship between two sets of data.

Axis of Symmetry

A line that divides a figure evenly down the middle, expressed as x=

Standard form of a linear equation

Ax + By = C A must be positive Coefficients are integers (fraction/whole number/terminating decimal) Given standard form, plug in 0 for x to get y-intercept. Plug in 0 for y to get x-intercept. Use the slope formula with the intercepts to get the slope. Convert to y=mx+b

Task/Time

Common denominator.

Interest (exponential)

Compound Interest: A(t)= a (1+ r/n) ^ tn N= number of times compounded per year (If it is only compounded once annually, you can use a(b)^x Compounded Continuously A(t)= P x e^rt Without turning to natural log, you have to solve by graphing (one side into y1 the other in y2; in y=)

Reciprocal function

Domain/ HA from H Range/ VA from k

Turning points

Highest power - 1 Ex: x^3 3-1=2 There are two turning points

Fundamental Theorem of Algebra

Highest power= number of solutions (including multiplicity) Multiplicity= number of times a factor occurs. Ex: x(x-2)^3 0 has a multiplicity of 1 2 has a multiplicity of 3 Although there are two solutions, the highest power is x^4.

Radical Equations

Isolate radical/rational exponent Raise both sides to the power of the index/ reciprocal of exponent Check for extraneous Ex:Radical 3√(2x-3) = 2 2x-3 = 8 Solve Ex: Rational Exponent (x+1)^2/3 = 4 x+1= 4 ^3/2 x+!= ±√64 Solve

Piecewise Function

Look at restrictions to find where to plug in Ex: F(x)= x+5, x<-2 -x+4, x≥-2 F(3) 3≥-2 so you plug into the second equation -3+4=1

Quartic factoring special case

When you have an ax^4 ± bx^2 ± C, you can use the ac method, but where it is (x^2 ± #)(x^2 ± #) Ex: x^4 - 3x^2 -4 - 4 * 1 = -4 -4 + 1 = -3 (x^2-4)(x^2+1)

Exponent laws

a^m * a^n = a^(m+n) a^m ÷ a^n = a^(m-n) (a^m)^n = a^mn (ab)^m = a^m b^m a^-m= 1 / a^m (a/b)^m = a^m / b^m

factor by grouping

factor by using pairs Ex: x^3 - 2x^2 -5x + 10 Group x^3 and -2x^2 x^2 (x-2) Group -5x+10 -5(x-2) Factors are (x-2)(x^2-5) Some cannot be factored Ex: 3x^3 - 9x^2 -2x + 4 Group 3x^3 and -9x^2 3x^2 (x-3) Group -2x+4 -2 (x-2) (x-3) ≠ (x-2)

angle of elevation/depression

the angle formed by a horizontal line and the line of sight to an object above (below) the horizontal line.

Quadratic Formula Number of solutions

x = -b ± √(b² - 4ac) / 2a The numbers under the radical (Discriminant) determines the number of real solutions b^2 - 4ac > 0 2 real b^2 - 4ac = 0 1 real b^2 - 4ac < 0 No real

Point Slope Form

y-y1=m(x-x1) Point slope form is left unsimplified If given multiple points, pick either one Used to find parallel/perpendicular lines that pass through a point. Parallel=same slope. Perpendicular= opposite reciprocal.

Transformation Form Linear Quadratic/Vertex Reciprocal Square Root Cube Root Exponential Logarithmic

y=a(x)+k y= a (x-h)^2 + k F(x)= a/x-h + k y=a √x-h) + k y= a 3√x-h) + k y= A(b)^x-h +k y= a logb (x-h) + k

Quadratic Systems

Linear-Quadratic Can have 0, 1, or 2 intersections aka solutions Solving will give you ___ x values, plug in for Y Ex: y= -x^2+5x+6 y= x+6 x+6 = -x^2 +5x +6 x^2 + x + 6 =5x + 6 x^2 - 4x = 0 x (-x + 4) x= 0 or 4 Plug into one of the equations 0+6= 6 4+6= 10 Intersect at (0,6) & (4,10) Ex2: y= -x^2 - x + 6 y= x+3 x+3= -x^2 -x +6 x^2 + 2x -3 AC method 3 x -1= -3 (AC) 3 + -1= 2 (B) (x+3)(x+1) X= -3 or 1 y= x +3 -3 +3= 0 1 + 3 =4 Solutions: (-3,0) & (1,4) Quadratic system, solve same way, set equations equal to each other System of inequalities, graph both equations. Overlapping region.

Natural Log and e

Ln and E are inverses. Natural Log= ln, same properties as a log Ln is just a log with a base of e Ln= condense, isolate ln, rewrite as an exponential. Ex: ln(x)=2 e^2=x solve with calc e= isolate e, cancel out with ln Ex: e^x-2 = 12 x-2 = ln 12 Calc

normal distribution

Mean is the peak. Empirical Rule: 1 stand deviation from the mean includes 68% of data, 2 is 95%, etc.

Finding percentiles

Multiply decimal form times the total number of terms. That gives the place the value is in a data set. Ex: 41, 54, 61, 65, 67, 73, 74, 77, 82, 98 60th percentile= .6 * 10= 6 The 6th term is 73. 73 is the 60th percentile.

Vertical Line Test

On a graph, if a vertical line passes through more than one point, the relation is not a function

Zero Product Property

You can set factors equal to 0 Ex: (x+2)(x+4) x+2=0 and x+4=0

Pascal's Triangle

a triangular array of numbers in which those at the ends of the rows are 1 and each of the others is the sum of the nearest two numbers in the row above

Graphing Radicals

Square root: starting point= vertex Cube root: center cluster= vetex

no radicals in the denominator

Square the denominator. EX: 2/√5 * √5/√5 = 2√5/25

Extrapolation vs Inter

Extrapolation: Predicting number not listed on table Interpolation: Prediction for number inside table (literally just reading the table)

Find polynomial given solutions

Factor theorem: x=a gives the factored form (x-a) Turn zeroes into factored form an multiply. Imaginary solutions, isolate the imaginary number and square both sides. Ex: x= 2-3i x-2= -3i (x-2)^2= 9 x^2 -4x + 4 =9 (x^2 - 4x -5)

Rational Root Theorem

Factors of the last term / Factors of the first term Must be in standard form. Lists ALL possible rational solutions.

Completing the Square Find C Term If A=1 If there is not a perfect square When A≠1 Standard to vetex form

Find C Term: (B/2) ^2 Ex: x^2 +10x + ___ 10/2=5 5^2=25 x^2 +10x + 25 If A=1, factor as (x+B/2) ^2 Ex: x^2 + 6x + 9 (x+3)^2 If there is not already a perfect square, subtract C from both sides. Find new C and add it to both sides. Ex: x^2 -4x + 2= 18 x^2-4x=16 -4/2= -2 -2^2=4 x^2-4x+4=20 (x-2)^2=20 x-2= ± √20 (IF YOU ADD RADICAL ±) x-2= ±2√5 x= ±2√5 +2 If A is not one and you cannot GCF with the other side, multiply the new C term by the A term = to get what you add to the other side Ex: 2x^2+4x=13 A≠1, Cannot divide 13 by A term (no GCF for 2x^2+4x-13) 2(x^2 + 2x + C) = 13 + ___ 2/2 ^2 = 1 2(x^2 + 2x + 1) = 13 + ___ (C) 1 x (A) 2 = 2 2(x^2+2x+1)=13+2 (divide on both sides) x^2 + 2x + 1 = 15/2 Solve Standard to Vertex Form Ex: x^2 - 8x + 11 x^2-8x= -11 8/2 ^2 =16 x^2-8x+16 = -11 + 16 (x-4)^2 = 5 (X-4)^2 - 5 (Vertex Form)

GCF

Pulling out the highest number able to divide to coefficients Ex: 3x^2 +6x +12 3(x^2 +3x +4)

Calculator, find factors tip

Put the number over x, and look in table Ex: Factors of 48 48/x

Unit Circle

Radius of 1 Cos=X Sin= Y Tangent= y/x

Slope

Ratio of vertical to horizontal change between two points (Y2-Y1)/(X2-X1) Rise/Run Positive +#/# Negative -#/# 0= 0/# Undefined- #/0

Parent Function Linear Quadratic Reciprocal Square Root Cube Root Exponential Logarithmic

Simplest form of a function y=x y=x^2 y=1/x y=√X y=3√ x (cube root) y= #^x y=log b (x)

Rational Function

f(x)= p(x)/q(x) (Basically just an equation over another equation) Continous: no breaks, jumps or holes, D:All Real Numbers, nothing can make the denominator=0 (note it's all REAL, using imaginary could make the denominator 0) Discontinuous: 2 sub categories Non removable: vertical asymptote, makes denominator 0 Removable: holes: values cancel out Ex: (x+3)(x+2) / (x+2) X+2 cancels out, meaning there is a hole an x=-2 X+3 remains, meaning the graph is just y=x+3 but with a hole at (-2,y) Calculator does not show holes/asymptote lines in graphing, only shown in table as "ERROR" where the hole is. The calc graphs the function correctly, but you must add in the asymptote and hole where it belongs. Find X-int= set the numerator equal to zero Y-int= plug in 0 for x Find the intercepts after canceling out Always get into factored form before simplifying/multiply/dividing When dividing, remember to take restriction of both the numerator and denominator of the divisor. (aka before and after keep,change,flip) Add/Subtract, common denominator. Remember to simplify at the end. When solving a ration equation, check for extraneous solutions.

Rational Exponent

Fraction as an exponent a^1/2= n√(a*m)= (n√a)m Given a decimal as an exponent, convert it to fraction Ex: 3.5 = 7/2 Cannot have a negative exponent, flip over the fraction bar. All exponent laws apply. Used to multiply/divide when radicals do not have like indexes Unless otherwise specified, final answer is always written as a radical

Fractions where both the numerator and denominator are under the radical,

√(1/2 )is the same as √1/√2

Perfect Square Trinomial

A and C are both positive and a perfect square B is double the square root of A and the square root of B multiplied. Ex: 4x-24x+36 A and C are positive. ±2x and ±6 (pick any value)= 2x * -6= -12 = -12 * 2 = B So, (2x-6)^2

Periodic data

Cycle: one complete pattern Identifiable 3 ways: peak to peak, trough to trough, point to point Period: length of one cycle (kinda like wavelength) Amplitude: always positive, maximum-minimum divided by 2

Function Operations

(f+g)(x)= f(x) + g(x) (f-g)(x)= f(x) - g(x) (f*g)(x)= f(x) * g(x) (f/g)(x)= f(x) / g(x) Denominator≠0 (rational functoin) Just add/subtract/multiply/divide f and x May have domains- radicals/fractions Function Composition (g o f)(x)= g(f(x)) (f o g)(x)= f(g(x)) Plug the 2nd equation into the first equation Ex:(f o g)(x)= f(g(x)) plug g(x) into the x values of f(x) Ex2: g(f(-3)) Solve f(x) using -3, then plug that answer into g(x)

Proability

Fundamental counting principle: To find number of combinations, multiply. Ex: 3 pants and 2 shits= 2 * 3= 6 possible combos Repeating- multiply all possible values Non-Repeating- multiply max possible value, max-1, max-2, etc. Ex: 3 Letters Repeating: 26 * 26 * 26 Non-Repeating: 26 * 25 * 24 Experimental: When the event is actually tested Number of times situation occurs / trials Ex: A baseball player hits 20 out of 50 times at bat. What is the probability, they will get a hit next time at bat? 20/50= .4 or 40% Can answer as decimal, fraction, or percentage. Experimental, because the event actually occurs. Theoretical: Event does not actually occur Number of possible occurrences/total options Ex: Getting a 5 on a fair number cube 1/6- A fair dice has 1 2 3 4 5 6 There is one five out of six options Permutation: arrangement of items in a particular order; non-repeating. ORDER MATTERS N!: Given number of items Ex: Ways to put 12 folders in a line N!= 479001600 ways The same as 12*11*10....*1 nPr: All combinations possible but with just a few at a time; n=number of items, r= number being organized, ORDER. Ex: Ten people in a race and can score first, second, or third place, how many ways can the runers finish first, second, and third. 10 P 3= 720 ways 10 total runners, 3 possible places Permutation because there is first, second and third. Order matters. Combination: nCr= All combinations possible but with just a few at a time; n=number of items, r= number being organized. ORDER DOES NOT MATTER. Ex: Ten people in a race, how many ways can people place top 3? 10 C 3= 720 ways 10 total runners, 3 people can be top 3 Combination because it is just asking for top 3, non-specific; Order does not matters. Combinatorics: calculates theoretical probability for more complex questions Ex: Probability of being dealt exactly 2 sevens in a 5- card hand from a standard deck of cards. (4C2 * 48C3)/ 52C5 4 sevens in a deck, you want two 48 cards other than a seven, you want 3 52 cards in a deck, you have a hand of 5 Geometric probability: area desired/ total area Ex:Probability a dart will hit a 24 inch square on a 374- inch dartboard. 24/374

binomial expansion

Given (a+b)^n, go the the nth row of pascal's triangle. (Remember, it starts at 0) Write the pascal triangle numbers vertically. Write the A term with exponents highest to lowest (n to 0) Write the B term with exponents lowest to highest (0 to n) Rewrite as a polynomial- in final answer do not write to the 1st power, and anything to the power of 0=1 (can be ignored) Ex: (3x-2y)^5 5th row of pascal's triangle: 1, 5, 10, 10, 5, 1 1 (3x)^5 (-2y)^0 = 1 * 243x^5 * 1=243x^5 5 (3x)^4 (-2y)^1 = 5 * 81x^4 *-2y= -810x^4y . *+*+* 10 (3x)^3 (-2y)^2 = 10 * 27x^3 * 4y^2= 1080x^3y^2 10 (3x)^2 (-2y)^3 = 10 * 9x^2 * -8y^3= -720x^2y^3 5 (3x)^1 (-2y)^4 = 10 * 3x * 16y^4= 240xy^4 1 (3x)^0 (-2y)^5 = 1 * 1 * 32y^5= -32y^5 Expanded as: 243x^5 - 810x^4y + 1080x^3y^2 - 720x^2y^3 + 240xy^4 - 32y^5 *+*+* Example: you would enter in calc without variable, 5 (3)^4 (-2)^1 = -810, then carry over the variables- which are just the variable to the power of the number outside of the parentheses. x^4 and y^1 Question will often ask for one specific term, to which case you would only have to find part of the expansion. Ex: For the above problem, if it asked for only the 4th term, you would only have to multiply out the 4th term, (row 3) 1 (3x)^5 (-2y)^0 5 (3x)^4 (-2y)^1 10 (3x)^3 (-2y)^2 10 (3x)^2 (-2y)^3 = 10 * 9x^2 * -8y^3= -720x^2y^3 Also, the exponents will always add up to n. Ex: (a+b)^3 1 a^3 b^0 3+0=3 3 a^2 b^1 2+1=3 3 a^1 b^2 1+2=3 1 a^0 b^3 0+3=3

Find the power of a polynomial

Given a table, find the difference of consecutive y values, until all the values are the same. The power can be found by finding how many times you had to subtract. Ex: x y -3 -43 -2 -10 -1 1 0 2 1 5 2 22 3 65 -43 -10 1 2 5 22 65 -33 -11 -1 -3 -17 -43 (1st- linear) -22 -10 2 14 26 (2nd- quadratic) -12 -12 -12 -12 (3rd- cubic) The function is cubic. (4th- quartic, 5th quintic, etc.)

Inverse Function

If (x,y) is part of a relation, it's inverse relation has (y,x). If both a relation and it's inverse are functions, they are inverse functions. (Aka one-to-one functions, each x corresponds to one y, and each y corresponds to one x) Ex: x: 1 2 3 4 5 y: 6 3 9 4 1 Is a function Inverse relation: x: 6 3 9 4 1 y: 1 2 3 4 5 Also a function So they are inverse functions. On a graph, the inverse is reflected over the line y=x Equtation, switch the x and y. Only the variables Ex: y= x^2 -1 x=y^2-1 (Switch to standard form) y^2=x+1 y=±√x+1 Inverse written as f^-1 Ex: find f^-1 given f(x) Domain and Range will be opposite Ex: F(x), D: x=2 R: y>3 F^-1, D: x>3 R: y=2 Application: In real-world example, you cannot simply switch the X and Y. You must actually solve by isolating. Ex: Area of a circle A=pi * r^2 R=pi * a^2 (Incorrect, this is just switching) A/pi= r^2 √(A/pi)= R (Correct) If f and f^-1 are inverse functions, (f o f^1)(x)=x and vice versa. Ex: (f o f^-1)(2)=2 (f^-1 o f)(2)=2

Conjugate Root Theorem

If a + √b is a solution, a - √b is a solution. Also applies to complex numbers. a + bi & a - bi Quicker way to solve (a + √b)(a - √b): Square A and subtract the radicand Ex: (6-√12)(6+√12) 6^2= 36 36-12 24

Dividing Polynomials

If dividing leaves no remainder, the divisor and quotient the are factors of the dividend. If there are any terms missing, plug in a 0 to keep it in standard form. For both long division and synthetic. Ex: given 4x^3 + 12x^2 +8, use 4x^3 + 12x+2 + 0x +8 Long divide: find the number that you can multiply the first term of the divisor by to get the first term of the dividend. Multiply that number by the other terms of the divisor and subtract from the dividend. Repeat. Ex: 3x^3+9x^2+8x+4 ÷ x + 2 3x^2 * x= 3x^3 3x^2 * 2 = 6x^2 3x^3+9x^2+8x+4 - (3x^3+6x^2) 3x^2+8x+4 3x * x = 3x^2 3x * 2 = 6x 3x^2+8x+4 - (3x^2+6x) 2x + 4 2 * x = 2x 2 * 2 = 4 2x + 4 - (2x+4)= 0 3x^3+9x^2+8x+4 ÷ x + 2 = 3x^2+3x+2 So, 3x^3+9x^2+8x+4 can be factored as (x + 2)(3x^2+3x+2) Likewise, if there is a remainder, that means the divisor is not a factor of the dividend. Ex: 3x^4-4x^3+12x^2+8 ÷ x^2 + 1 = 3x^2-4x-9 r 4x-1 There is a remainder, so x^2 + 1 is not a factor of 3x^4-4x^3+12x^2+8 Synthetic division: For linear binomials only (x±a), write only the coefficients of the dividend. Use the solution of (x±a).. Bring the first term down. Multiply the solution by the first term, and add/subtract from the next term. Multiply the solution by that number and repeat. Ex: x^4 + x^3 -11x^2 -5x +30 ÷ (x-2) Solution: 2 Rewritten: 1 + 1 - 11 -5 +30 1 2*1=2 +2 1 + 1 - 11 -5 +30 1 3 2*3 = 6 +6 1 + 1 - 11 -5 +30 1 3 -5 2*-5=-10 -10 1 + 1 - 11 -5 +30 1 3 -5 -15 2*-15=-30 -30 1 3 -5 -15 0 The last number ^ is the remainder. There is no remainder, so (x-2) is a factor of x^4 + x^3 -11x^2 -5x +30. And 2 is a solution. Use the new coefficients to form the quotient: x^3 + 3x^2 -5x -15 Synthetic imaginary solutions: you can find imaginary solutions by using both the real solutions on the same dividend. It will give you a quadratic to put into the quadratic formula and find the imaginaries. Ex: x^4 + x^3 -7x^2 -9x - 18 Solutions: -3, 3 Rewritten: 1 + 1 -7 -9 -18 1 3*1=3 +3 1 + 1 -7 -9 -18 1 4 3*4=12 +12 1 + 1 -7 -9 -18 1 4 5 3*5=15 +15 1 + 1 -7 -9 -18 1 4 5 6 3*6=18 + 18 1 4 5 6 0 (<- no remainder) (Use other solutions with the same coefficients, order that you use the solutions in does not matter) 1 4 5 6 1 -3*1=-3 -3 1 4 5 6 1 1 -3*1=-3 -3 1 4 5 6 1 1 2 -3*2=-6 -6 1 1 2 Now turn to quadratic x^2 +x + 2 Solve with the quadratic formula. Remainder Theorem: If you are only looking for the remainder, plug in the divisor's solution into the dividend. The resulting number is the remainder. Ex: Remainder of 7x^2+4x-10 ÷ x-2 Solution: 2 7(2)^2 + 4(2) - 10 28 + 8 -10 26 (remainder) Keep in mind, no remainder= it is a factor. This is a quicker way to check.

Logs

If no base is stated, it is assumed to be ten Type log where base≠10, with math, alpha, math Product property: logb(mn) = logb(m) +logb(n) Quotient property: logb(m/n) = log(m) - logb(n) Power property: logb(m)^n = n logb(m) (Remember rational exponents in power property, so logb^1/2 = 1/2 logb) Condense, write as one log Expand, each part of the equation gets its own log

Radicals

If the index is odd, there is only one answer. Ex: 3 OR -3 If the index is even, there are 2 real/imaginary solutions. Ex: 3 AND -3 / 3i AND -3i Divide variable's exponent by the index. Ex: 4√a^16 = a^4 If the exponent is not divisible, leave the remainder under the radical. Ex: 3√a^7 = a^2 √a On calc, enter index with math, 5 Multiply/Divide n√a * n√b= n√ab (must have same index) Multiply radicals before you reduce/simplify. n√a ÷ n√b= n√(a/b) (must have same index) Simplify after Reduce radical using perfect squares/cubes/ etc. Ex: √24 √(4*6) 2√6 You can also factor out perfect squares Ex: √(9x+18) √[9(x+2)] 3√(x+2) Add/Subtract Simplify first a(n√x) ± b(n√x) = a ± b(n√x) respectively Basically just combining like terms, must have same index and radicand.

Probability of multiple events

Independent events: outcomes do not affect each other Ex: Roll a cube then spin a spinner Mutually Exclusive: Events cannot happen at the same time. Ex: Rolling an even number and an odd number. You cannot roll an even and an odd at the same time. Probability of one even OR the other occurring n(AuB): If the events are mutually exclusive, add the probabilities. Ex: 20% chance of snow, 50% percent chance of rain Probability of rain OR snow= 70%. It cannot snow and rain at the same time. If the events are not mutually exclusive, add the probabilities and subtract the amount that both occur. Ex: Probability of circle or green. 2 red circles, 2 green circles, 1 orange circle, 1 green square, 1 red square, 1 yellow triangle, 1 orange square 5/9 + 3/9 - 2/9 = 6/9= 2/3 5 circles + 3 green - 2 green circles Probability of both events occurring, multiply (AnB) Ex: 2/3 chance of getting a diet drink, 2/5 chance of getting fat-free chips. Probability of getting a diet drink AND fat-free chips. 2/3 *2/5= 4/15. Dependent: outcome of one will affect the other Ex: Draw one card, then pick a new one (without putting back the first card) Conditional probability: Probability that an event will occur given another event occurs (B|A); A and B/A Ex: 20 males. 13 females. 15 males don't want rain. 8 females don't want rain. Probability female doesn't want rain. No Rain | Female. 8/13 8 females do not want rain, out of 13 total females Tree Diagram.

Angles

Initial side- lies on x axis Terminal- side that moves Pay attention to if the angle goes clockwise (-) or counter clockwise (+) Coterminal angle, share the same terminal side, add up to 360. Add or subtract 360 to find coterminal angle. May have to do so more than once. ex: -130 + 360 = 230 425-360=65 Central Angle- angle where the vertex lies in center of circle Intercepted arc- part of the circle intercepted by central angle's rays Radian- measure of a central angle Degrees to radians: multiply by pi rads / 180 Radians to degrees: multiply by 180 / pi rads Find intercepted arc length: radius * central angle in radians

Simplifying Radicals

Largest perfect square that can go into the radicand. Ex: √32 16 is the square of 4 16 goes into 32 two times 4√2

End Behavior

Leading Term (highest exponent)= ax^n A is positive, and N is even= up up A is positive, and N is odd= down up A is negative, and N is even= down down A is negative, and N is odd= up down Written as: As x approaches -∞ the function is decreasing/increasing As x approaches ∞ the function is decreasing/increasing Approaches may be abbreviated with an arrow: As x --> ±∞, the function is increasing/decreasing

line of best fit (trend line)

Linear regression equation of a scatter plot

Logarithm/Exponential

Logs and exponentials are inverses (So they are reflected over y=x, HA of exponential=VA of log, flipped x and y's, etc) Exponent: b^y=x Log: logb(x)=y Circle trick Any log of x=0, anything to the 0 power=1 Any log with the same base and x=1, a number to the power of one gives the same number When the same base method cannot be used, you must solve an exponent by using a log. Isolate the exponent, and insert a log (always b=10) Must use calc Ex: 15^3x=285 3x log (15) = log(285) x = log(285) / 3 log (15) Ex2: 3(5)^x-2 = 612 5^x-2 = 204 x-2 log (5) = log(204) x-2= log (204) / log(5) Solve Solve log: condense, write as exponent. Check for extraneous- cannot take the log of a negative.

measures of central tendency

Mean: add all/total of numbers Median: middle number in data, splits data into two halves Lower/1st quartile: median of lower half Upper/3rd quartile: median of upper half Interquartile range: upper-lower Mode: most frequently occurring number Bimodal= has 2 modes More than 2 modes means the mode is not statistically useful No mode- no number occurs more than the others Outlier: value substantially different from the rest Range: greatest value - lowest value. Calc: stat, calc, 1 vars stat. X with bar over= mean

What are the four ways to represent a relation?

Ordered Pairs (x,y) Mapping Table Graph

Relation

Pairs of inputs and outputs

Odd Function

Plugging in a negative x also flips ALL of the original numbers. Symmetric over the origin.

Even Function

Plugging in a positive and a negative X value gives the same Y. Symmetric over the y-axis.

Function

Relation where each x-value corresponds with only one y-value

Trigonometry

S=O/H C=A/H T=O/A Reciprocal Functions Csc= H/O Sec= H/A Cot = A/O Reciprocal identity- Csc=1/sin Sec=1/cos Cot=1/tan Sin=1/csc Cos=1/sec Tan=1/cot Tangent Identity: Tan= sin/cos Cotangent Identity: Cot=cos/sin On graph, theta is closest to origin Find angles using inverses: sin^-1 o/h cos^-1 a/h tan^-1 0/a Pythagorean Identity: Cos^2+Sin^2=1 Verify trig identities: Pick the more difficult side (you cannot change both sides) and everything in terms of sin and cos. Prove that one side = the other. Simplify: condense to smallest possible form Remember to get common denominators. Remember, you can factor out trig functions. Ex: sin^2 - sin^2 * cos^2 sin^2 (1 - 1*cos^2) sin^2 (1- cos^2) Etc. When solving: FAN Sin & Tan: Only Q 1 & 4 Cos: only Q1 & 2 Look at restrictions, if not explicitly state, use fan. Unless specified, answer in radians. Use inverse. Ex: What values satisy the equation: 3 tan - 2 = tan 2 tan- 2 = 0 2 tan = 2 tan = 1 tan^-1 (1) (Where in on unit cicle, within tan fan, does y/x = 1) Pi/4 radians

samples and surveys

Sample: part of a population Convenience: selects easily and readily available Ex: standing outside the local mall Self-selected: members volunteer Ex: optional online survey Systematic: select at regular intervals Ex: survey every 5th person Random: All equally likely to be chosen, least bias Study methods: Observational: observe sample without interacting Controlled Experiment: affect one group, but not the other. Compare the end results of the groups Survey: Ask sample a set of questions Bias: Part of a population is over/underrepresented Combining 2 or more issues Double negative Overlapping answer choices Loaded: strong/harsh wording Leading: clearly suggests a particular answer

Standard deviation/Variance

Standard Deviation: shows how much data value deviate from the mean. Represented by sigma. Found in calc. Each standard deviation is one ± from the mean. Ex: Mean 6 Standard deviation 1 One standard deviation 5 6 7 Two standard deviations 4 5 6 7 8 Variance: signma^2 Calc doesn't show variance. Just enter standard deviation and square it.

Sereis

Sum of the terms in a sequence, written as #+#+#+# Infinite and finite Arithmetic Finite: Sn= n/2 (a1+an) Use explicit formula to find number of terms (n) if unknown Ex: 4+9+14+...+99 5n-1=99 100/5=20 plug into Sn Alternatively, last term-first term, divided by common difference, +1 99-4=95 95/5=19 19+1=20 Summation Notation: Σ, to the right is the explicit formula, top is upper limit= last number plugged into the formula, bottom is lower limit= first number plugged into formula; calc press math, 0 Plug into Sn N= upper-lower + 1 a1= plug lo\wer limit into explicit an= plug upper limit into explicit Convert to summation: find explicit formula, set it equal to the first term, then the last term Geometric Finite: Sn= [a1(1-r^2)] / 1 - r Use explicit formula to find number of terms (n) if unknown Ex: 3+6+12+...+3072 3 * 2^n-1 = 3072 2^n-1= 1024 Turn into a log to find n n-1 log (2) = log 1024 Solve for n plug into Sn Summation Notation: same as arithmetic, but with a1(r)^n for explicit Geometric Infinite: a1 / r-1 Converge: | r | < 1 Has a sum, gets infinitely close to a number, kind of like an asymptote ex: 1+1/2+1/4+... 1 ÷ 1 - 1/2 1 ÷ .5 2 Will get infinitely close to 2, but never reach it. Think: 1 + 1/2 + 1/4 + 1/8 + 1/16 ... Diverge: | r | > 1 Has no sum

Parabola trick

The parent function goes right one and up in odd number intervals: 1/1, 3/1, 5/5, 7/1, etc. If the function has a slope, multiply the slope by 1, 3, 5 etc.

X is the in/dependent variable Y is the in/dependent variable

X= Independent Y= Dependent

Rationalizing

You cannot have a radical in the denominator, for a square root, square the denominator and multiply the numerator by the denominator. If it is a cube root, you must figure out what to multiply the numbers by in order to get a perfect cube. Ex: 3√5y^2 must be multiplied by 3√5^2y May need conjugate to rationalize Ex: √3 + √5 * √3-5

AC Factoring Method

__+__=B __*__=AC If the A term is one, those are the factors. Ex: -3 and 2 = (x-3)(x+2) If the A term is not one, you must plug the factors back into the original equation and factor by grouping. EX: -3 and 2 2x^2 +x - 6 2x^2 +4x -3x -6

Exponential (basics)

a(b)^x Gets infinitely closer to HA asymptote but never touches a>1 and b>1, exponential growth a<1 and b<1, exponential decay b>1, b=growth factor b<1, b=decay factor b=1+r (growth r is positive; decay r is negative, basically just 1-r) R= rate of increase/ growth rate Rate of decay (expressed as decimal) Find R: Percentage to decimal (y2-y1) / y1 . ; aka (new-old) / old Half-life: initial amount (1/2) ^ time/halflife Ex: Half-life of 14.3. How much of a 50mg sample is left after 80 days. y=50(1/2) ^ 84 / 14.3 Same Base method: find a common base raised to a power, the bases cancel out. Ex: 27^3x=81 (3^3)^3x = 3^4 (3's cancel, leaving power to a power 3*3x and the four) 9x=4 Solve.

Difference of Two Squares

a^2 -b^2 = (a+b)(a-b) Ex: 25x^2 - 49 5x & 7 ( 5x+7)(5x-7)

Difference of two cubes

a^3 ± b^3 (a±b)(a^2±ab+b^2) Same Opposite Always Positive

Sequence

list of numbers: #,#,#,# A(n) is expressed where n is a subscript. Arithmetic: D=common difference/slope explicit: slope(n) ± ___ Find ___ by plugging in 1 for n, and find difference between the first term Ex: 10, 8, 6, 4 -2n+___ -2+__12__=10 so a(n)= -2n+12 recursive: A(n)= a(n-1) + d two missing middle terms: plug in a known term, and use a(n)= a + (n-1)d Ex: 100, ___ , ___ , 82 a(4)= 100 + (4-1)d 82=100 + 3d -18= 3d d= -6 Arithmetic Mean: x+y / 2, used to find a missing middle Ex: 15, ___ , 45 15+45=60 60/2=30 (aka average the terms) Geometric: R=Common Ratio (divide=multiply by a fraction) Explicit: a(n) = a * r^n-1 Recursive: a(n)= a(n-1) * r Missing middle terms: plug into the explicit formula Ex: 2, ___ , ___ , -54 a(4)= 2 * r^4-1 -54= 2 * r^3 -27= r^3 r=-3 Geometric Mean: √(xy) Ex: 3, ___ , 12 3 * 12 = 36 √36= 6

Correlation

strength of a scatter plot's relationship The closer R is to 1 or -1 the stronger the correlation (Linear uses R, but the other regressions use R^2) Correlation: Weak/Strong & Positive/Negative To enter the regression into y=: stat, edit, calc, choose regression, scroll to store regEQ, vars, y-vars, enter, enter It will now show how the regression runs through the points (This is especially useful when you need to find the y given x and vice versa. 2nd Window to set table to indpt/depend ask mode.) Find the best regression by running the highest one, if it is a lower one, it will just shows 0 for the values. Ex: If you ran quadratic on linear: A=0 B=2 x=3 0x^2 + 2x +3 = 2x+3 Two points= linear Three points= linear or quadratic Four points= linear, quadratic, or cubic Etc. There are degree+1 points.

Standard Form of a Parabola

y=ax^2+bx+c

Slope-intercept form

y=mx+b