Algebra 31 how to do the problems

Solve. Find all real solutions.

(find x) Factor then once it's (x-2) set equal to 0, find x

Range (general)

(highest number) - (lowest number)

Mean

(sum of the number) / (number of numbers)

Synthetic Division: (x^3-4x^2+3) / (x-1)

1. Use factors of 3 and -1 to find what goes on side of box 2. put all of numbers on top of box 3. move 1st down, multiply by side number 4. put that answer under second number and add last number is w out x then x then x squared etc 5. may have to divide again 6. factor or quadratic formula the new equation 7. find 2 answers (positive and negative)

Frequency Table

A table that displays how many time a data value occurs within a data set.

Observational Study

Observe only, report without ay manipulation by researcher

Combinations

Order NOT important nCr n=total C=math, Prb, nCr r=parts

Probability of disjoint Events A or B happening equation

P(A or B)+ P(A) + P(B) no P(A and B) bc there's no chance of both happening

Random Assignment

Randomly assigning people to a certain treatment so there's no bias and the results are accurate

Random Selection

Randomly choosing people so there's no bias

Equation for Sum of Infinite Geometric Sequence/Series

S= a1 / 1-r S= sum a1= first term r= what you multiply by to get each next term

Sum of Finite Arithmetic Series Equation and what each of the letters mean

Sn: sum n: # of terms a1: first term an: last term ( ): average # of all terms

Making a box plot on graphing calculator

Stat, edit, put data in L1 2nd, y=, enter make sure on is highlighted 4th graph highlighted Zoom, Zoomstat clear y= trace

Multiplying with Exponents: x^10/x^4

Subtract 10-4 x^6

Total number of people on histogram

add up all bars

Adding and Subtracting polynomials (how to)

only add and subtract like terms

To find Tangent in triangle and on unit circle

opposite/adjacent separate on unit circle or (sin/cos)

to find Sine in triangle and on unit circle

opposite/hypotenuse second in point on unit circle

(2-6i) squared

(2-6i) (2-6i) times itself/ appears twice

(3rd root of 27)^2

(27)^2/3 or -3^2= 9

What does everything stand for in A=P(1+/-r)^t What is this equation used for?5

+ exponential growth w/out compounding bc no n - exponential decay w/out compounding bc no n Used for non-banking problems or no compounding problems A: Amount you end up with P: Principle (starting amount) r: rate (convert to decimal) T: time (years)

Arithmetic Sequence

+ or - by same number every time

i^2

-1

Probability of picking one color NOT putting it back and picking the same color again. (Equation and steps to figure it out) Independent or Dependent?

... Dependent

Parts of a box and whisker plot

1. lowest value 2. lower quartile/ median of 1st half of data (25% of data from 1-2) [Q1] 3. median (50% of data on each side) 4. upper quartile/ median of 2nd half of data [Q3] 5. highest value (75% of data from 2-5) * outliers width of box is interquartile range-middle 50% of data (Q3-Q1)

Order of translating a graph

1. make original table of x values: -1, 0, 1 and y values: ex: 1/3, 1, 3 if growth factor is 3 2.multiply y values by A 3. add or subtract to x values (move left or right) 4. add or subtract to y values (move up or down)

Find the inverse of an expression: y= (x-3)^2

1. replace every x with a y and y with x 2. solve for y 3. replace y with f^-1 to find inverse

Find the inverse of an expression: y= cubed root of (2x-7)

1. replace every x with a y and y with x 2. solve for y 3. replace y with f^-1 to find inverse

Trig: find period of a graph

2B/pi= period

Degree to Radian ex: 30 degrees

30 * pi/180 adding pi to top

Summation Notation

6= end Sigma sign= sum i=1 is start 4i= rule

What does everything stand for in A=P(1+r/n)^rt What do you use this equation for?

A: Amount you end up with P: Principle (starting amount) r: rate (convert to decimal) n: # of times/ how often you compound in a year T: time (years) Used for banking problems

What does everything stand for in A=Pe^rt What do you use this equation for?

A: Amount you end up with P: Principle (starts amount) e: irrational number w/ no pattern going on forever so "continuous" r: rate in decimal form t: time (years Used for continuously compounded interest If r is positive, the function is exponential growth If r is negative, the function is exponential decay

Graphing Polynomials: identify end behavior and what determines it

Describes how the values of a function behave at each end of the graph. Degree and Leading coefficient determine it.

Stem and Leaf Plots

Key= 9]0= 90

Multiplying with Exponents: x^-2

Move expression to denominator 1/x^2

Multiplying with Exponents: 1/x^-2

Move expression to numerator x^2/1

Multiplying with Exponents: (x^2)^6

Multiply bc exponent raised to another exponent x^12

In how many ways can you go to at least 3 games? Multiply or add?

OR bc you could go to 3 or 4 or 5 etc. Add (opposite of what you think because not both As)

Permutations

ORDER use when there is ranking nPr n=total P=math, Prb, nPr in calc r=how many rankings, order

Probability of Event A or Event B happening equation

P(A or B)= P(A) + P(B) - P(A and B) and= can't have middle piece count twice

Equation for Sum of Finite Geometric Sequence/Series. What each letter stands for.

Sn= a1 (1-r^n/1-r) Sn= sum of n number of terms a1= first term r= what you multiply by n= number of terms

Solve. Check for extraneous solutions.

Solve problem then plug in all answers to see if they work. If they don't then they aren't an answer only circle ones that work as final answers

U in probability

Union, together so if A U B it's A and B

Skewed

a graph that has one side sketched out further than the other

Symmetric

a graph that has the same shape on it's left and right sides

Sample

a part of the population from which data is collected and generalized to whole population ex: 40 cars of faculty chosen

Margin of Error

a survey may have a margin of error of plus or minus 3 percent at a 95 percent level of confidence. These terms simply mean that if the survey were conducted 100 times, the data would be within a certain number of percentage points above or below the percentage reported in 95 of the 100 surveys.

Range of a Sequence

actual terms of the sequence a1, a2, a3, a4,......an

Multiplying with Exponents: x^3 * x^4

add bc two different exponents x^7

to find Cosine in triangle and on unit circle

adjacent/hypotenuse first in point on unit circle

Probability of picking yellow out of red, blue and yellow

amount of yellow out of total number??????

Geometric Sequence/Series Rule Equation and what the letters stand for.

an= a1 * r^n-1 n= number of terms a1= first term an= last term r= what you multiply by

If degree is even and leading coefficient is positive What is end behavior? How many curves?

as x +infinity, f(x) is +infinity as x -infinity, f(x) is +infinity odd number of curves: 1,3,5 etc. possible W shape

If degree is odd and leading coefficient is positive What is end behavior? How many curves?

as x +infinity, f(x) is +infinity as x -infinity, f(x) is -infiinity even number of curves: 2,4,6 etc.

If degree is odd and leading coefficient is negative What is end behavior? How many curves?

as x +infinity, f(x) is -infinity as x -infinity, f(x) is +infinity even number of curves: 2,4,6 etc.

If degree is even and leading coefficient is negative What is the end behavior? How many curves?

as x +infinity, f(x) is -infinity as x =infinity, f(x) is -infinity odd number of curves: 1,3,5 etc.

Identify the number of solutions/zeroes (includes all, not just rational/real) y=2x^5-7x^3-x^2+x-4

at most 5 bc highest exponent

(16)^3/4

bottom of fraction goes first (4th root of 16) to the 3rd power

Statistic

data collected from sample ex: $18,300 (average of 40)

Outliers

data points that are very far from the rest of the numbers

Dependent event

depends on what happened before ex: picking a blue candy, eating it, and picking a blue candy again (lower chance of getting a blue)

Independent event and examples

doesn't matter what happened before ex: rolling a dice, starts over each roll

Leading coefficient is negative, graph ends ________

down

Degree is odd, graph has _______ curves

even number of

Equation of sin/cos graph What do all the letters mean?

f(x)= A sin or cos (Bx)+D A= amplitude (up to) B= period (horizontal) D= whole graph up or down

Original Equation for Exponential Growth and Decay Functions

f(x)=ab^x

Fill in 2 way table

fill in known information then calculate the rest

Functions: find f(2)

find 2 on x-axis of given graph and answer is where line is on y-axis

Functions: find f(x)=3

find 3 on y-axis where is the line on x-axis (sometimes more than 1 answer)

Transforming graphs: -f(x)

flip over x-axis

Increasing, decreasing, constant

follow graph left to right and answer ex: [-5, 3] (-3, -1] (-1,0] (0, infinity)

Box and Whisker: range

highest - lowest values not including outliers

Subsets

if every element of set A are also an element of set B then A is a subset of B

Conditional Statements Equation and what are they?

if-then statement in which p is a hypothesis and q is a conclusion probability of B happening under the condition that A has already happened P(B]A) "probability of B given A" P(B]A)=P(A intersection (upside down U) B) / P(A)

Experiment

imposing treatment to see effect of it on the response

Upside down U

intersection of 2 sets. A ___ B is both but not only A or only B

Transforming graphs: f(x+1)-3

left 1, down 3

Compliment of set A

line over A or A to power of C all elements of the universal set that are NOT A

Sequence

list of #s in a pattern

Find probability from 2 way table ex: pirates and male

look at both categories and see where they overlap, that's the answer

Arithmetic Sequence/Series Equation and what the letters stand for.

m= what you are +or - by y= first term x= relative number that corresponds to the term (usually 1 if using first term) b= solve for b

Odds

measures the chances in favor or against an event occurring "may the ODDS be ever in your FAVOR/.Z"

Median

middle number of a data set

Trig: Vertical Shift/Slide

midline

5 number summary

min, Q1, median, Q2, max

Fundamental Counting Principle

multiply all numbers of ways events could occur 3 types of jackets, 6 types of shirts, 8 types of ties, 4 types pants 576 outfit combinations

Multiplying polynomials (how to)

multiply all terms by all other terms

Multiplying with Exponents: x^0

no x's but still coefficient of 1 so 1

Graphing Polynomials: identify degree

number of curves odd # of curves= even degree

Degree is even, graph has _______ curves

odd number of

Radian to Degree ex: pi/6

pi/6 * 180/pi cancelling out pi so one in numerator one in denominator

Given f(x)=2x-11 find: f(3)

put 3 in for x

Domain of a Sequence

relative position of each term 1, 2, 3, 4,.......n understood unless told otherwise that it starts at 1

Transforming graphs: f(x-2)

right 3

In how many ways do can you go to this and that? Multiply or add?

says AND so multiply

Given f(x)=2x-11 find: f(x)=25

set equal to equation/ put in as y 2x-11=25

What is difference between set and element in probability venn diagram

set is in entirety of brackets element is each number in bracket

What does square root -12 equal

square root 12i

P(green]red)

take 1 red out 1st what is P(green) now

1st step of factoring

take out GCF

Co-terminal angles

terminal ray that ends in the same spot ex:............................................

Population

the entire group in interest ex: all faculty with cars

Mode

the number which appears most often in a list of numbers

Series

the terms of a sequence added together

Standard Deviation

the typical distance of numbers from the mean in a data set

Quartiles

the values that divide a list of numbers into quarters

Summation form and then find sum of series

to find some, first find values of sequence using rule example: 4=19 5=28 6=39 7=52 8=67 add to get 205

Survey

type of observational study that gets data by asking people questions

Leading coefficient is positive, graph ends _______

up

Transforming graphs: f(x)+2

up 2

How to find which quadrant an angle is in

use unit circle

Theoretical Probability

want/total ex: # of outcomes in event A/total # of outcomes

Trig: How does period affect graph

what it goes up to on x-axis

Parameter

what you want to know about the population sometimes number, many times just words ex: average car value ($)

When is an equation showing exponential growth?

when the b/growth factor is over 1

When is the equation showing exponential decay?

when the b/growth factor is under 1

when is it plus or minus

when you've taken the square root

Geometric Sequence

where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ex: 2, 6, 18 with common ratio of 3

Graphing Polynomials: identify x-intecepts

where line crosses x-axis

Graphing Polynomials: how to find y-intecepts

where line crosses y-axis

Find all zeros of a function on graph

where line touches x-axis

Quadratic Formula

x = -b ± √(b² - 4ac)/2a

Domain

x values [if definite] (if infinite or open circle)

Factoring cubes

x^3+27= x^3 + 3^3 a=x b=3 a^3 + b^3 = (a + b) (a^2 − ab + b^2 ).

Range of a graph. What is the difference between ( ) and [ ]?

y values [if definite] (if infinite or open circle)

Translating Exponential Growth Graphs Equation and what each letter stands for

y=ab^x-h +k a: stretch, shrink, flip b: growth factor h: right(-) or left(+) k: up(+) or down(-) [asymtote]