SAT Math

proportions and precents - finding a percent from the data given strategy

"at random" = out of all the surgeons on the chart (HAD TO USE THE TOTAL AS THE DEOM) general rule: find criteria in the question then use the key terms to single out one value on the chart (use fingers - one finger on one term and another on another term, finger meet in the middle at one term)

statistics - frequency

# of times an event occurred in an experiment use ratios

(SUPER IMPORTANT) general test taking reminders... so far 3

(MAJOR PROBLEM IS SPEND TOO MUCH TIME ON Qs) save as much time as possible - general rule: if you've spent 15 seconds on a question and can't solve it: circle the question and skip it. SUPER QUICK DOUBLE CHECK ANSWERS BEFORE MOVING ON (DURING YOUR FIRST RUN THROUGH - but stay on track with time) must MEMORIZE (so use flashcard and active recall) THE MATH REFERENCE SHEET (a tool not a crutch) HIGHLIGHT or UNDERLINE or WRITE DOWN WHAT THE Q ACTAULLY WANTS YOU TO FIND (when you first read the question) - DO NOT SKIM SHORT QUESTIONS!! write down ALL your work, don't do it your head (prevent you from skipping steps, eliminate carless errors) MAKE SURE TO CLEARLY WRITE DOWN NUMBERS and NEGATIVE SIGNS **on psat: don't leave any Qs blank, don't get into the habit of doing that, just circle the Q/ flag it (bc you want to max out your score!!) *with extra time: - go back to flagged questions, - check the grid in question first, - then check go back to the beginning (quickly check by plugging in answer into the original question, key word in your answer choice compare to the question psat 7: missed out on a lot of points for math bc we got three wrong on no calc section (aim to get completely right) use calculator to eliminate careless errors and save time (even just basic and general shortcuts can save HUGE AMOUNTS of time, difference between 1400 and 1500)

exponent rule - (xy)^a distributing, applies to (x/y)^a

(xy)^a = x^a * y^a split base, raise to the same power (x/y)^a = (x^a)/(y^a)

circle equation

(x-h)² + (y-k)² = r²

prep - trig

*if given that sina=cosb, then they are complementary angles and b=(90-a) - substitute and solve *CARLESS ERROR: solved for the right answer, but marked the wrong answer *if given cos K=3/5 and not given the exact values for all the side length (use the ratio 3:4:5 from cosK) - separate each triangle and NEATLY WRITE IN THE SIDE LENGTHS GIVEN AND RATIO (5x,4x,3x) (5y,4y,3y) - determine the x variable from the sides given - determine more side lengths - solve for the question (ex in doc!) *when asked to find the area of a rectangle from 5 triangles within the rectangle - keep the values neatly next to their corresponding side - quickly jot a special right triangle and their side length for reference - solve accordingly (ex. in doc) *check comparison between special right triangles and trig special right triangles in the unit circle (ex in doc) https://docs.google.com/document/d/10hUNKAWHNEII5Qdhrjvj_gu9tiVvz3CO1foV_J5xBf4/edit?usp=sharing

prep - review of exponents

- A whole, positive exponent tells you how many times to multiply a base by itself. - A negative exponent tells you to take 1 divided by the base to the positive version of the exponent. - Anything non-zero to the 0th power is 1, and anything to the 1st power is itself. - A fractional exponent means you raise the number to the numerator, then use the denominator as a root.

triangle congruency theorems 5

- SSS, SAS, ASA, AAS, HL - no bad words...ASS

line of the best fit

- a line that reflects the general pattern in a graph - middle of all the data points

solving an equation with an incomplete data table

- add rows for the totals - work from there and complete the data table

solving linear equations strategy - how many solutions?

- although fastest way is to graph the equations (y-int, x-int, open up or down, etc) - can be rewritten into x=a: one solution - variable can be eliminated, left with a=b: no solution - can be written into x=x: infinitely many solutions

remainder theorem part 1

- let's say we have p(x) - if p(a) = 0 then (x-a) is a factor of p(x)

triangle reminders

- triangles that share a side have congruent sides - angle belongs to the side opposite to it - just bc two angles in two different triangles open up to the same side length DOES NOT MEAN THEY ARE CONGRUENT ANGLES, remember similar and congruent triangle knowledge

interpreting statements from the graph

- underline key point in the statement - within the answer choices, draw the words' corresponding symbol next to it (helps compare faster)

finding the center point from the circle equation

- use completing the square to go from x^2−10x to (x−h)^2term-110 ex.) x^2−10x+y^2+24y=1, find the center point

absolute value - equation relating the distance between points

- use when question asks you about two points on a number that are both d units away from coordinate a - The coordinates of the points at a *distance d units* from the point with *coordinate a* on the number line are the solutions to the equation - (makes sense the difference between two points should be 3)

inequalities strategy - writing our own system of linear inequalities based on the word problem

- write the inequalities to match the word prob, rewrite to match answer choices *answer choices are just rewritten versions (add, sub, etc to move around values)!

sum of the measures in degrees of the angles of a triangle is...

180

diameter of a circle

2r

# of radians of arc in a circle

2π

Circumference of a circle - MUST MEMORIZE (for speed and accuracy)!!!

2πr or πd

special right triangles - 30 60 90 - MEMORIZE

30: x 60: x√3, 90: 2x

# of degrees of arc in a circle

360

two most common Pythagorean triples

3:4:5 5:12:13

special right triangles - 45 45 90 - MEMORIZE

45 x, 45 x, 90 x√2

surface area formula for a cylinder

=2πr×h+2πr2 . This first term is the circumference of the circle times the height — basically, the area of the "tube" part of the cylinder. The second term is the area of the two circles at the end.

area of a circular sector

A = 1/2 * θ * r² OR area of entire circle * (central angle/360)

Area formula for a triangle

A=1/2bh (height is from tip to base, altitude)

surface area formula for rectangular solid

A=2lh+2lw+2hw Basically, it's just the sum of the areas of all six rectangles on the solid.

Compound Interest Formula MEMORIZE (faster, more accurate on tests)

A=P(1+r/n)^nt

area formula for a rectangle

A=lw

Area of a circle - MUST MEMORIZE (for speed and accuracy)!!!

A=πr²

factorization of cubes

A^3+B^3 = (A+B)(A²-AB+B²)

integer

All whole numbers (both positive and negative) and zero.

standard form and units

Ax+By=C both sides must have the same units

Getting the vertex point from an quadratic in standard form

Ax^2+bx+c and vertex is (h,k) H=-b/2a Find h and plug it into the equation to solve for k

adding data points into the data set = change in the mean

New values that are greater than the mean always increase the mean, new values that are less than the mean always decrease it.

center of a circle

Point inside a circle that is the same distance from every point on the circle

prep - dividing polynomials

Rational expressions work just like regular fractions, and can be simplified, expanded, added, and subtracted. The remainder theorem tells us that when p(a)=0, then (x−a) is a factor of p(x). It also tells us that p(a)=the remainder of(p(x)/(x−a)) We can perform long division on polynomials the same way we do with regular numbers.

Prep - scatter plots

Scatterplots show two variables as measured for many data points. Positive correlations go "up" to the right; negative correlations go "down." Strong correlations are closely grouped; weak correlations are spread out. The line of best fit is as close as possible to all the points on a scatterplot. specific question: "Another fast-food company in the province sells cheeseburgers for 128 pesos each. If the *price at this company is more than that predicted by the line of best fit*, what is the least number of franchises the company could have?" = the point (?,128) is above the line of best fit what could be the smallest x-value? (answer should model at actual data point on the graph) (ex in doc)

prep - system of equations with nonlinear equations

These can be solved by substitution, but it's usually the slow way. They will always be systems with one linear and one quadratic equation. They will always have 0, 1, or 2 solutions. The fastest way to solve Qs (that ask for the number of solution in a system of equation - quadratic and linear) is to draw them! *DO NOT SKIM THE QUESTION MAKE SURE TO HIGHLIGHT WHAT YOURE ACTUALLY SOLVING FOR!!!! - note when the system of equations has no real solution - note when an equation has y and x switch from y=mx+b

Inverse Trig Functions

Used to find the angle when sides are known. sin-1, cos-1, tan-1

Volume Formula of a square pyramid

V = (1/3) lwh (right angle from tip to base) (height is from tip to base)

Prep - function notation

When asked a question like this: if f(x)=3x^2-7 and f(x+n)=3x^2+24x+41, what is the value of n? - you can solve the question directly by substituting x+n into the function 3x^2-7 f(x+n)=3(x+n)^2−7 - expand f(x+n)=3(x^2+2nx+n^2)−7=3x^2+6nx+3n^2-7 3x^2+6nx+3n^2−7 = 3x^2+24x+41 - compare the same term from each equation and solve for n 6nx=24x SO n=4

prep - extraneous solutions

Whenever we have square root equations, we must check (plug back into the equation) each answer in the original equation. ex.) √(2x-5) = x-, what is x when r=5? MUST CHECK ANSWERS

exponent rule - x^a * x^b multiplication

addition x^(a+b)

negative association

as x increases, y decreases (opposite directions)

positive association

as x increases, y increases (positive direction)

isosceles triangles

at least two sides of the same length and two angles of the same measure.

perfect square trinomial

a² + 2ab + b² = (a + b)² a² - 2ab + b² = (a - b)² ex.) x²-18x+81 (1)²x² and (9)² - check to see if 1st and last terms are perfect squares (2)(1)(9)x = 18x - check to see if 2ab = the original 2nd term (use absolute value) (x-9)² - account for any negatives

circle theorems

central angles/360 degrees = sector area/circle area = arc length/circumference

right triangle trig and complementary angles

cosx=sin(x-90) sinx=cos(x-90) Or cosx=sin(x-(π/2)) sinx=cos(x-(π/2))

distance formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

degrees to radians

degrees x π/180

writing exponential equations - changing the exponent's time unit

ex.) 10(1.07)^t (where t=year) to 10(1.07)^h/2 (h=half years) general rule when changing the exponent's time unit: - create an equation to describe the relationship between the time units (ex. - 30 months = 5/2 years, 1 hour = 2 half-hours) - plug in the equivalent statement (in the units you want to change to) into the answer choices' exponents WHICHEVER EQUALS 1 IS CORRECT tip - larger time unit to a smaller time unit = make the exponent smaller - smaller time unit to a larger time unit = make the exponent larger

rational expression strategy (fractions and sums)

find common denominator multiply to get common denominator then cancel ex.) 4/(x-3) - 5/2(x+8) common denominator: 2(x+8)(x-3)

data analysis - based on the results of the survey, which of the following statements must be true? general rule

general rule: if the question directly states a percent, then ASSUME IT IS AN EXACT PERCENT ex.)"If another 1,000 adults selected at random from the city were surveyed, 78 percent of them would report they are satisfied with the quality of air in the city."

general rule for identifying the graph from equation probs

identify key point on the graph (if we calculate the y-intercept (at x=0) first, we can quickly eliminate choices)

remainder theorem part 2

if p(a)=k, then k is the remainder when p(x) is divided by (x-a)

trig strategy

label side H/O/A

radian to degrees

radian x 180/π

must save 5 min of review time for each section

reading - at question 26 @ 30 min mark writing - at question 22 @ 15 min mark math no calc - at question 10 @ 10 min mark math ok calc - at question 19 @ 25 min mark

length of circular arc

s=rθ OR circumference * (central angle/360)

given variable a and b within a system of linear equations, find the values of a and b when the system has exactly two real solutions

substitute get x alone on one side of the equation and plugged in the answer choices for a and b

exponent rule - x^a/x^b division

subtraction x^(a-b)

right angles in rectangles and square

there is a right angle at each corner

structuring expressions - using A as a variable

use A as a variable, substitute (w-3)^5 - (w-3)^2 and A=w-3 A^5 - A^2

write two system of equations from word problem strategy

use the variables given in the equation (variable = first letter of the word) ex. novel = "n" as variable DO NOT USE X ANDY INSTEAD (more to remember, more errors) ex. - English to equation - 12 grams of a chemical are added to a metal. The amount A, in grams, of the chemical remaining during a reaction with a metal plate decreases by 0.5 g per second. ----> A=12-.5s - If instead the plate were dissolved, the amount B, in grams, of chemical remaining would decrease by half of itself every 4 seconds. (Again, initial amount is 12 grams.) ----> A=12(.5)^(s/4) *create a variable for the amount your are solving for* and create the rest of the equation around that value - do not over complicate your equations and variables (ex. "Ken earned $8 per hour for the first 10 hours he worked, so he earned a total of $80 for the first 10 hours he worked. For the rest of the week, Ken was paid at the rate of $10 per hour." = y=10x+(10*8)) *break down the info in the question by underlining info to help write the equation (bracket amounts, circle relationship (ex. sum, product, etc) (ex in doc)

must be able to multiply squared binomials quickly

what is the variable squared, what is the variable and constant and 2 multiplied together, what is the constant squared ex.) (m-3)² (m)² * (2*m*(-3)) * (-3)²

If 3x-y=12, what is the value of 8^x/2^y

write the equation in question to find the relationship (linear equation) between the two variables, 8^x/2^y=2^(3x)/2^y=2^(3x-y) substitute in the given equation answer: 2^12

exponent rule - (x^a)^b exponents

x^(ab) multiplication

difference of squares

x² - y² = (x + y)(x - y)

quadratic equation in standard form

y = ax² + bx + c C is the y-int

point slope form

y-y1=m(x-x1)

prep - exponent rule testing strategy

you can test out exponent rules with real numbers to figure out a rule

data inferences

​estimate = sample proportion ⋅ population (related to confidence level) range = estimate ± margin of error​ - lower confidence level = narrower range - higher confidence level = bigger range

exponential functions - asked to graph its transformation (ex. which graph is y=-g(x)? or y=-(1/2)g(x) or y=g(-x)?)

(use if it's is just a reflection across the x or y axis) sketch the parent function (note y-int, decrease or increase, vertical shifts) to help visualize the reflection, (use if the it's not just a reflection) make sure to also FIND POINTS FROM THE DAUGTHER FUNCTION by plugging in x values (or y-values depending on the reflection), such as -1, 0, 1, and 2, into the parent function/solve for y/then change point according to the daughter function

prep - circle geometry

- begin a hard geo question with an equation to outline how to solve (ex. radius - (1/2 of a square's side length) = answer, etc. etc.) OR by breaking down the shape in triangles - to match the answer choices: use ((central angle/360)*circumference or area FORMULA) to get the area of a sector or its length - a line segment from the center to the edge of a circle is the RADIUS - a square inscribed in a circle: the square's DIAGONAL = circle's DIAMETER - Lines that are TANGENT to a circle create a RIGHT ANGLE with the radius of the circle drawn to the point that the line touches. - sum of angles in a shape: (n-2) * 180 - central angle is two times the inscribed angle they both open up to the same sector - when asked to find the central angle use the formula: (central angle/360) = (area of sector/ area of triangle) - use PARAENTHESES in CALCULATOR when solving with PI - when asked which of the point don't lie in the interior of the circle: use the distance formula (center will be x1,y1 and answer choice will be x2,y2), SO a distance longer than the radius means that the point will lie outside of the circle - use completing the square to solve for the circle equation: factor out leading the coefficient, subtract constant term, sort out the x variable (use (b/2)^2), add constant to other side, replace the trinomials with the factored binomials), sort out y variable (repeat), finish simplifying

prep - ratios and proportions review

- conversions: we want unwanted units to cancel. - percent decrease: new value = original ·(1−(decrease/100)) - percent increase: new value = original ·(1+(increase/100)) - x and y are directly proportional if y=kx; they are inversely proportional if y=kx *

factoring reminders

- cross factor, divide solutions by ax if a>1

finding the center of circle from a graph

- draw a graph to visualize the points and the circle, guess and check - center is an equal length away from each point

prep - volume

- draw picture of solid objects - in volume to volume transfers, set the volumes equal - volume = area * height - finding line segments within objects (diagonal): - general rule that can be applied to unfamiliar shapes: identify the right triangles that the line segment belongs to, use the Pythagorean Theorem, and the info given - cylinders: d2=(2r)^2+h^2 - box (two right triangles - one with the base, one with the diagonal): d^2=h^2+w^2+l^2 (bc w^2+l^2 is a side length to the triangle with the diagonal - on the bottom of the box)

prep - exponential functions

- f(x)=a•b^x+c - word probs: find the INITIAL AMOUNT (a-value) and RATIO of change (b-value) - exponential tables: have a constant RATIO (NOT RATE) between consecutive outputs - linear tables: x and y both increase or decrease BY A SET VALUE - exponential graphs are curves with an increasing rate of change

prep - circles

- find the radius (area formula, circumference formula, (not given) distance formula) - arcs and sectors are proportions of circles - use equation of a circle (x-h)^2+(y-k)^2=r^2 - find angles using inverse trig - Any triangle formed by the center and two points on the circumference is isosceles (because all the radii must be the same length). - USE SHORTCUTS (trig ratios 3,4,5 and 5,12,13) (45 45 90, 30 60 90) - steps to complete the square to get the circle equation *Step 1: Divide all terms by the lead coefficient (the coefficient of the x^2 term)* Step 2: Subtract the constant term from both sides of the equation Step 3: Divide the middle coefficient by 2, then square it - *if a fraction is the middle term, then you are multiplying the denominator by 2 and then squaring !!* Step 4: Add the result from Step 3 to both sides of the equation (REPEAT FOR THE Y VARIABLE) Step 5: Factor the resultant trinomial *remember that a neg constant in the factored binomial is a pos center point (write out complete circle equation) *use parentheses when using pi in your calculator *use midpoint formula when given two endpoint of the diameter to find the center point *when given an arrow inscribed inside a circle - break up the arrow into two isosceles triangles - solve for the missing angles in both triangles - solve for y (x variable is from the isosceles triangles): 360-2x=y (ex. in doc) *SHORTCUTS - when given arc length and central angle, then asked to find another arc length in the same circle: use (degree measure of the angle)/(length of arc formed by angle) - In a circle, arc length is proportional to the central angle that forms it. (ex. in doc) - when given a graph with a circle and two points, then asked to find the center point: draw two triangles within the circle and solve (using special right triangle info) (ex. in doc)

prep mistake - solving for a variable in an exponent prob

- get rid of the root - simplify the denoms by factoring out the GCF - cross multiply - finish evaluating ex.) 3√(g/32) = g/2, what is g? g>0 g/32=g^3/2^3 g/4=g^3 4g^3=g 4g^2=1 g^2=1/4 g= + or - 1/2, but g>0 so g=1/2

exponential function on a graph

- has a constant ratio of change (difference between each input is the same, vice versa for outputs) - double check your equation with values from the table

key feature of graphs

- identify the trend - note (underline or highlight) the axis or intervals discussed - if asked to select a graph that represents a scenario, underline or highlight key phrases

coordinate geometry of nonlinear functions

- interpreting graphs is important: sign of the leading coefficient (even degree: neg=down, pos=up) (odd degree: neg=left side opens up, pos=right side opens up) - as leading coefficient get larger (absolute value), the graph gets *NARROWER* - *IMPORTANT* understand nonlinear equations by setting y=0 - graphs with leading coefficients that have a larger absolute value are narrower - different approaches: setting y=0 and analyzing the graph, plugging in values to find the solutions - finding the vertex: if given a quadratic in factored form, find roots and then x-value for the vertex (average of the roots) AND plug x-value for the vertex in to solve for the y-value *given a picture of the graph of y=-ax^2+k, then asked what is true about the parabola with the equation y=a(x−h)^2−k: - question is about *A NEW EQUATION AND GRAPH*, not ab the given picture of the graph *TWO DIFFERENT EQUATIONS* (with some of the same variables) - in your answer choice, base the vertex on the equation in question, and base whether or the parabola opens upward or downward on given graph picture (ex. the equation of the given graph picture has "a" and opens downward, so if the equation in question has "-a", then it opens upward) (ex. in goc) *when asked which function has a graph for which y is always greater than or equal to -2 (...does not have to be equal to -2) - graphing the function and using function transformations is MUCH FASTER - quickly sketch the parent and apply the transformations - correct answer will not be below -2 (CORRECT ANSWER DOES NOT NECESSARILY HAVE TO BE EQUAL TO -2, more important that the graph is above -2, but CHECK ALL ANSWER CHOICES) - translations: - *IF TRANSLATION IS INSIDE THE PARAENTHESES*: f(x+h) = positive = left f(x-h) = negative = right *opposite - *IF TRANSLATION ON THE OUTSIDE OF THE PARAENTHESES*: f(x)+k = positive = up f(x)-k = negative = down

khan - rewriting circle equations into standard form by completing the square

- make sure the coefficients of x^2 and y^2 are both 1 - use (b/2)^2 to create the binomials - substitute the quadratics into the circle equation - add the constants to the right side - cancel the constants from the left side - substitute in the binomials

when mean and median are or are not equal

- mean equals median: symm distribution - mean does not equal median: not symm distribution - there are outliers: mean will be pulled in their direction (can be smaller or larger), while median stays the same

standard deviation

- measure of the variability of a data set from its mean - a set with values grouped closely around the mean: lower SD - a set with values spread farther from the mean: higher SD

inequalities strategy - key phrases and symbol translation

- more than c: >c - less than c: <c - at least c: ≥c - at most c: ≤c - no less than c: ≥c - no more than c: ≤c - "minimum" value: smallest value that satisfies - "maximum" value: largest value that satisfies the inequality - "A possible" value: Any value that satisfies

dividing polynomial strategy

- question may ask what is equivalent but not directly ask you to divide - if you don't have to rationalize (no radicals in the denom), then start dividing - recognize that the answer format after dividing (ax^2+bx+c+(remainder/factor)

data table prob - finding and comparing the expected numbers from the survey results

- random selection = ratio from survey can be applied to the larger population - "expected total number" = asking you to apply the percentage in the data to the larger population (multiplication)

Quadratic strategy

- recognize the format of a solution from the quadratic formula, helps solve for a missing value question x = -b ± √(b² - 4ac)/2a

radical and rational expressions strategy

- remember the domain restrictions given in the question, helps rule out some solutions ex.) If y is a solution to the equation above and y>0, is greater than, 0, what is the value of y?

solving system of linear equations strategy - what is the constant if there are no solutions

- rewrite equations into slope intercept *no solution in system when the equations represent parallel lines *(a,b) = (x,y) - if constant is in the slope, set equations equal to each other and solve for the constant

prep mistake - finding equivalent statement in an exponent prob

- rewrite the exponent to match the answer choices - simplify the exponent - finish evaluating ex.) what is equivalent to 4^(4/3) 4^(4/3) = (2^2)^(3/4) = 2^(6/4) = 2^(3/2)

congruent triangles

- same angles measures and side lengths

given a formula and told about two different situations, then asked to find the relationship between them

- set the equations equal to each other according to the word problem - cancel out like terms - answer the question

similar triangles and proportions

- similar triangles: same shape and angle measures, not same size - hypotenuse/leg - if the triangles are similar you can write proportions of the side height/side length=toy height/toy side length - the ratio of their areas = (ratio of their side lengths)^2

prep - lines and slope

- the steeper a line, the greater the slope's magnitude - ALWAYS USE THE WHAT YOUR GIVEN to QUICKLY create an equation for the line - if you have y-int and slope use y=mx+b - if you have slope and a point use y−y0=m(x−x0) - use info given to find relationship (convert what you can, recognize patterns or relationships, apply that knowledge to solve) - (ex. if asked about two lines and given two points on a line, find slope...) - make sure you are plugging in the correct points when using the slope formula (DOUBLE CHECK - through careful examination OR by creating an equation with point-slope or slope-intercept then plugging in given points)(eliminate careless errors) - when creating a linear equation from a data table, if x/in-put ALREADY represents the number of years after a certain date, THERE IS NO NEED TO SUBTRACT THAT DATE FROM THE X/IN PUT in the equation (ex. which functions models the number of penguins t years after 1900? N(t)=934−1.5t - YES, N(t)=934−1.5(t−1,900) - NO) (if we were looking at the year 1901, that's 1 year after 1900. The value of t would therefore be 1.) *when given a graph with a line and two points AND given two point from another line that is parallel (ex. (7,7) and (4,k), then asked to find the value of k? (ex. in doc) - get the slope of the lines (parallel so same slope) - plug the second line's two points into the slope formula (that include K) - set the slope you found and the slope equation equal to each other *when given three points and asked to find a variable (ex. - "...a line passes through the points (0,0), (2,p), and (p,18). Which of the following is a possible value of p?" - create a y=mx+b equation with (0,0): y=mx - plug in the points into line's equation: p=2m and 18=mp - still have two unknowns (m and p) but need to solve for "p" so cancel out m with substitution: p/2=m --> 18=(p/2)p - solve to get answer: p= -6, +6

prep - absolute value MEMORIZE TWO ABSOLUTE VALUE RULES

Absolute values are never negative. never true statements: *|any expression|+ any positive number=0 *|x|=any negative number *|any expression|=any negative number two rules to MEMORIZE: 1) If |x|< a, then x<a and x>−a. Similarly, if |x|≤ a, then x≤a and x≥−a. (on a number line, you can think of it as x being sandwiched between negative and positive a.) 2) If |x|> a, then x<−a or x>a. Similarly, if |x|≥ a, then x≤−a or x≥a. (on a number line, you can think of it as x being outside of the space between negative and positive a.) strategy: - check if a statement appears in all of the answer choices, do not test it and assume that it is true/do not even worry about that statement and thinking through it logically - use real numbers to replace the variables, - think through each answer choice logically, - follow general thinking guidelines * general thinking: - (ex. bc h<k, in |h−k| you would get a neg inside the absolute value and a pos value overall, which is the same as k-h just as is) - when given a absolute value statement with the variables multiplied by something, try to rewrite each answer choice into the given statement - if you can find even one set of numbers that makes the answer choice and given statements true, then that answer choice is correct (wrong answers usually deal with the extremes such as zero) - a positive cannot be less than a negative - a number can't be in between zero

prep - experimental interpretation

Experimental interpretation questions are about interpretation, not numbers. Questions about flaws in the study are about identifying a bias that means the sample doesn't represent the population. - check if the data was collected randomly out of a population Questions about drawing conclusions are about making statements that don't go too far. - must include terms like likely/unlikely (nothing can be too certain) *if an experiment is conducted correctly, then what is true of a sample is PROBABLY true of the population - if you get a certain percentage from a sample, you MAY GET THE EXACT SAME PERCENTAGE FOR THE POPULATION

prep - lines and angles

Figure out the important relationship between angles. Are they equal? Supplementary? Do they add up to a known amount, like 180° or 360°? supplementary = add up to 180 complementary = add up to 90 When two parallel lines are cut by a transversal (a line going through both of them): - Opposite angles are equal - Adjacent angles are supplementary (they add up to 180°). - Corresponding angles are equal *if the sum of angles is "a+2b=180", 2b must be an even value, so in "a=180-2b," the variable "a" must also be even bc even-even=180 *when trying to find equivalent angles, remember: - supplementary angle rules (add up to 180) - congruent angle rules (angles on **opposite sides of transversal** with PARALLEL line and angles **directly opposite** to each other/reflections of each other are congruent) *when trying to chose which statements are true, test each answer by thinking logically and use these general guidelines - identify which angles are congruent (note them with the symbols) AND remember congruency rules - use the given equation and substitute in the equivalent angles as needed, then cancel out variables to find relationships *when trying to test an answer statement but you cannot substitute values into the given equations, try to make the answer statement untrue (ex. if a statement is c<m then try to make c=m) plug in values for all the variable in the given equation (using congruent angle and supplementary angle knowledge)... see if the question is true if so then the answer statement incorrect *when told that six angles are formed by three lines that intersect at a common point, and ASKED WHAT IS THE LARGEST POSSIBLE VALUE OF THREE ADJACENT ANGLES - should have DREW OUT THE Q AND CIRCLED YOU WERE TOLD TO FIND - but remember that any three adjacent angles will meet back at the same line and equal 180 (ex. in doc)

prep - data and probability

Tables give us numbers that we can use in these problems. The probability of something happening is the (number of desired outcomes/the number of possible outcomes). Similarly, the proportion of (things in a set is the number of things/the number of things in the entire set). Read the questions carefully to find out which values (or combinations of values!) - even if the question say "selected randomly" but then says "given", the given tells you to restrict your data to a specific category SO the denom in your probability will not be the overall total but just the total in a specific category - if asked to find a probability from data table concerning pairs, just add together the values of whenever the data applies for the numerator AND the denominator will usually be the overall total (add together value from each category) *check sat doc for an example - QUESTION DETERMINES THE DENOM AND NUMERATOR, "(denom) ARE (numerator)"? (ex. what proportion of the movies ARE comedy movies with a PG-13?" SO 4/50, what propportion of movies with a PG-13 rating that ARE comedy movies? SO 11/25)

Volume Formula of Cone

V = (1/3) π r² h (right angle from tip to base) (height is from tip to base)

Volume Formula of Sphere

V = (4/3) π r^3

volume formula of a rectangular prism

V = lwh

Volume Formula of Cylinder

V = π r² h

Pythagorean Theorem

a²+b²=c²

transformations: y=g(-x)

changing all the x values/a reflection across the y-axis

transformations: y=-g(x)

changing all the y values/a reflection across the x-axis

transformations (can apply to exponential and quadratic equations)

changing y values f(x)+d = up and f(x)-d=down changing x values f(x+c) = left and f(x-c) = right af(x) = vertical stretch IaI>1 compression 0<IaI<1 f(bx) = horizontal compression IbI>1 stretch 0<IbI<1

If (ax+2)(bx+7)=15x^2+cx+14 for all values of x, and a+b=8, what are the two possible values for c?

expand the polynomial, find equivalent values for each corresponding term, ax^2+7ax+2bx+14=ax^2+(7a+2b)x+14=15x^2+cx+14 SO ab=15 and 7a+2b=c find possible values for the "a" and "b" variables, ab=15 and a+b=8 SO a=3 or 5 and b=5 or 3 plug in the possible variables value to solve for c 7(3)+2(5)=31 7(5)+2(3)=41

exponential form

f(x) = AB^x + C - A = initial amount (not always the y-int, depending on vertical shift) - identifying the a-value in an exponential function: b-value will always be positive SO assume a= 1 or -1 (ex. - in g(x)=-3^x+2, a=-1 and b=3) - B = ratio of change (less than 1 = decrease, greater than 1 = increase) - C = vertical shift

Qs dealing w/ bar graphs and mean/median

given a bar graph that displays the percentages from a data set: - you can determine the mean without calculations - if a multiple percentages are similar to one another for a majority of the data set, then the mean should be around those values ex. in doc

exponent prob strategy

given: 18*16^(.0125t) asked: "After how many days did the total number of people viewing the video double from the original number of people?" 18*16^(.0125t)=18*(2^4)^(.0125t)=18*2^(.05t) *exponent must equal 1 so that the initial value can double .05t=1 t=20

related to the remainder theorem

if something is a factor of a polynomial, that means the polynomial is divisible by that thing

reminder ab slope in slope intercept equation

in slope intercept form the slope (always next to the x variable) relates x and y (or cups of tea to tea bags), slope of 1 = 1/1 OR each cup of tea requires 1 new tea bag

given an quadratic and its factored form with variables, asked what variable k equals (a variable in the factored equations) - (1/3)x^2-2 can be rewritten as (1/3)(x-k)(x+k), what is the value of k?

make sure the given equations are have the same value factored set the same term from both equations equal to each other solve for k

prep - mean, median, mode, standard deviation

mean: - average= sum of n numbers/n - average×n=sum of n numbers *if given 2 averages in a Q, write an equation for each and substitute as needed to solve *if asked to find an average hat would not be possible: plug in the extreme values, find a range, anything outside of that range is not possible *weighted average Qs dealing with speed: remember that trip there and back may have different times, so find each time from the given rates. Then, divide the total distance/total time to find the average speed for the entire trip. median - make sure numbers are in order stdev - no clear center in a data set/multiple numbers appear multiple times = larger stdev compared to a data with a clear center range range=[largest value]−[smallest value] *when the median of 8 numbers is negative and asked how many numbers are positive at max: - remember that the median of an even set will be (#+#)/2 = median - rule out each possible scenario when numbers in the set are placed from least to greatest - the median can be a large neg - small pos / 2 = neg median (so at most 4 pos numbers in the set) *when given a description of a set of # and asked which can be the mean - create the set of number according to the description - determine the lowest possible mean by plugging in values that don't conflict with the description - rule out values answer statements *when given a scenario and the mean for the first 5 games, and then asked to find the points for the 6th game, so that the average for all 10 games is a certain number: - use the formula ([1stgame]+[2ndgame]+...+[9thgame]+[10thgame])/10 = average (use inequalities if needed according to the Q) - plug in values for the first 5 games from the given info - then ask what do we do about the [7th game] through [10th game]? are you trying to make the 6th game as big as possible/find the greatest possible points for this game? (then make 7-10 the smallest possible values) are you trying to find least possible value for the 6th game? (then make 7-10 the max possible value) - solve for game 6 (ex in google doc) *CARELESS ERRORS: - when calculating median MAKE SURE VALUES ARE IN ORDER (pay attention to this when using precents) - when putting mean, median, and mode is increasing order WRITE DOWN WHICH IS BIGGEST AND SMALLEST - make sure that stdev and range you calculated MATCH THE ANSWER CHOICE (don't rush and make sure to check both stdev and range) *given the mean and median (different value) and asked what could explain the different between the mean and median values: REMEMBER THAT WHEN THE MEAN AND MEDIAN ARE DIFFERENT, THE MEAN WILL BE PULLED IN THE DIRECTION OF THE OUTLIERS!!! (mean smaller than the mean = outliers are less than mean (a few values are much less than the rest)

Prep - single variable equations

six steps to solving single variable equations: - If there are fractions or decimals, multiply both sides by the fractions' *LCD* (or if there are binomials and variables in the denominator, COMBINE THE TERMS INTO ONE FRACTION ON EACH SIDE then use CROSS MULTIPLICATION) - deal with parentheses - combine like terms - ALWAYS TRY TO MAKE NUMBERS SMALLER - use for FRACTIONS, CROSS MULTIPLICATION, FACTORING (smaller numbers = prevent errors) - finish solving *3(4x−(3/2)x+7)=−x−2 - to get rid of the fraction just first multiply on inside the parentheses with 2 (on the left) Solve word problems by translating them from English to algebra - when you have to create TWO EQUATIONS FROM A WORD PROBLEM and USE SUBSTITUTION TO SOLVE, refrain from substituting into the denominator of a fraction (MAKE IT EASIER FOR YOURSELF = saves time and prevent error) Get variables out of denominators by multiplying by a common denominator or taking a reciprocal. Look for shortcuts to solve for expressions. (Stated in sat doc!!) Solve inequalities just like equations, being sure to flip the inequality sign if you multiply by a negative number or take a reciprocal (from both sides). - if given two equations set equal to each other and that has no solution, then asked to find the value of c: ASK YOURSELF WHAT VALUE OF C WOULD MAKE THE TERM WITH THE VARIABLE EQUAL, so you would be left with the constants (ex. if 10cy+4=20y−44 AND c=2, then 4=-44 (no solution)) - use shortcuts, save time (ex. If 8z+14=102, what is the value of 8z−14?) 8z+14=102 8z+14−14=102−14 8z=88 8z−14=88−14=74 *when given an equation for last month and this month, but asked for the value of next month (you must use the formula twice)

prep - solving systems of linear equations

solve a system by combining the equations (with addition or subtraction) OR using substitution If the two equations are multiples = same equation and there will be an infinite number of solutions REMEMBER ax+by=c AND mx+ny=k RATIO TRICK *ex in doc when given two linear equations, write a system of equation and solve **cancel a third variable when combining the equations in a question with a lot of words SKIP TO THE QUESTION BEFORE READING ALL THE TEXT (Q provides a lot of unnecessary info), from the question FIND THE TOTAL and write the first equation (going into the text when needed) - use this structure for the first equation (usually): total = (variable*rate) + (variable*rate) - second equation structure (usually): another total (USES THE SAME UNIT AS THE VARIABLE) = variable + variable *asked about which "inequalities is satisfied by the y-coordinates of all the points that are solutions" (ex. in doc) - use the boundary of the x inequality and plug in to solve for the y inequality (just bc boundary for the x inequalities (ex. x<4) is not a solution does not mean you cannot plug it in to solve for y) *when asked "Which of the following ordered pairs (x,y) satisfies the system of inequalities above?" - write down and check each answer OR just plug into your calculator (if working on the ok calc section) Review questions and examples in doc! SUPER IMPORTANT (height and edge formula, substituting given points into given equations, given two inequalities and a point) Q may give you the boundaries in your inequality (check doc for example) "Prachi needs between 240 and 260 total feet of fence panels.....which of the following combinations of fence panels could she buy?": 240 <= 6x+8y <= 260

prep - triangles and polygons

solve polygons by dividing them into triangles and quadrilaterals use special right triangle knowledge (to get side lengths info) - 45:x 45:x 90:x√2, 30:x 60:x√3 90:2x, 3-4-5, 5-12-13 (n-2)*180=sum of the angles regular polygons have equal side lengths and equal interior angles if told the ratio of angles in a triangle (such as 3:2:1), then use 3x+2x+x=180 to solve for the angles. when trying to find the side length of hexagon, DIVIDE THE HEXAGON INTO SIX EQUAL TRIANGLES then use 6*(area of one triangle) = area of the hexagon **"shorter sides OF LENGTH T" means that the side of triangle = the side of the hexagon when a Q asks about the relationship of interior angles to the number of sides , WRITE AN INEQUALITY FROM THE QUESTION, solve substitute into the given formula, solve for the variable you want **ex in doc when asked to ind the fraction of the triangle shaded/given info ab midpoints on the triangle AND an image of a small triangle inside a big triangle: - identify that they are similar because of the same top angle and midpoint info tells us that the smaller triangle's side are exactly half of the bigger triangle = ratio of 1:2 between the triangles - write an area formula for the big and small triangles bigger tri area = bh/2, smaller tri area = 1/2*(1/2b)*(1/2h) = 1/4 (bh/2) - subtract small area to big area to find the fraction of the shaded area (1)(1/2bh)−(1/4)(1/2bh)=(3/4)(1/2bh) *USE (1/2bh) AS A VARIABLE* *example in doc *the area formula for an isosceles triangle or 45-45-90 is area=(1/2)x^2 (b/c base and height are the same) *diagonals divide squares into 45-45-90 right triangles - correct the orientation of triangles to see this relationship (ex. in doc) *when given that a side segment is a *fraction of another side segment* AND asked to find the *value of the side segment* - write an equation that includes the side segment (ex. in doc) *when given that a side segment is a *fraction of another side segment* AND asked to find the *value of a side length* - label the triangle with the variables - find the ratio between the large triangle and the small triangle - use the ratio to find the side length in question (ex. in doc) errors from 9/26 prep assignment: - use (1/2bh) as a variable when finding the fraction of the triangle shaded - slow down and clearly write down #s - careless error - note what the Q actually wants you to find! (ex. what you solved for length, but actually asked to find width) - when using pythag, the hypotenuse goes with c^2 (correct orientation of the right triangle to reveal relationships)

line of best fit prep adjustments

when asked to FIND VALUES on a graph relative to the line of best fit strategy: - find the given value on an axis (draw a vertical or horizontal line from that value) - then find the first whole number where our line is above or below the line of best fit (depending on the question) (ex. If the gym's membership size is MORE THAN that predicted by the line of best fit, what is the least number of comparable gyms this gym could have?) - if a POINT on the line of best fit does not make sense (ex. (16,-50) = when there are 16 comparable gyms near a gym, the gym will have −50 members), THEN the CORRECT ANSWER is THE LINE OF BEST FIT WILL NOT MODEL THE NUMBER WELL - when asked to choose the correct STATEMENT COMAPRING AVERAGE RATES ON A GRAPH, remember GREAT MAGNITUDE = MORE STEEP LINE - when asked ab what GRAPH REPRESENTS A STUDENT POURING LIQUID into a TAPERED GLASS (FLASK), remember the height of the liquid in the flask will increase EXPONENTIALLY bc the glass in TAPERED at the top - when asked to CALCULATE VALUES use EXACT VALUES FROM A DATA TABLE (don't estimate values from points on a graph)

exponent rule - x^(-n) (negative exponents)

whole number with negative exponent: - take reciprocal - add positive version of the exponent to the denom fraction with negative exponent: - flip num and denom - exponent stays where it is but turns positive

when asked ab slope use...

y2-y1/x2-x1

vertex form

y=a(x-h)^2+k a will tell us if the parabola opens up or down

when x is inversely proportional

y=k/x

when x is directly proportional

y=kx

prep - define and interpret linear functions

y=mx+b, where m is the slope or rate of change and b is the y-intercept or starting point. Interpret values and select models for linear functions by looking for starting points and rates of change - "d in terms of s" = function should have d as the output and s as the input. - our slope is defined as: when the input goes up by 1, the output changes by m. - in y=mx+b: if x increases by 1, then y increases by m. BUT if y increases by 1, then x increases by 1/m (IMPORTANT!) Plug in numbers if there are letters in your answer options. *Take note of the wording in a Q it can change your equation or inequality (ex. "her plan was to teach a total of no less than 22 weeklong lecture courses. She DID NOT achieve this goal.." SO instead of m+n>22, it is actually m+n<22 *IMPORTANT: The POINTS on the scatterplot are ACTUAL data that is collected. The LINE OF BEST FIT is a PREDICTED line based on the points on the scatterplot. *an equation for "Lily received a 25-piece jewelry collection. She purchased 12 new pieces of jewelry each month and gave away 8 pieces of jewelry each month" = total pieces of jewelry = 25+12t−8t = 25 + 4t (WORD PROBLEM GAVE 2 SLOPES) *when asked about objects and containers (questions relates to slope) (check ex. in doc!!!!! - https://docs.google.com/document/d/10hUNKAWHNEII5Qdhrjvj_gu9tiVvz3CO1foV_J5xBf4/edit) - write an equation (steps stated in google doc)