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]