;
●​ Solve inequality to find range of values
●​ See which value not in range of values
●​ Note: best describes sol just means sol
●​ Simplify and open brackets to make linear form on both sides
●​ For NO SOL:
○​ co-eff of x same and same sign
○​ Diff constants
●​ For INFINITE SOl:
○​ Same equation on both sides
●​ For LINEAR MODELING:
○​ What question asks is dependant variable = Y
○​ What is changing / value changing is independent variable / “for each” = X
○​ Constant variable or something that is not changing in any case is intercept = C
●​ Thus: Y= 39X + 200
●​ Rate is always y over x thus in dist/time dist=y and time=x
●​ Initial Y value/distance at X/time=0 is Intercept C
●​ 1300 is constant amount thus intercept
●​ “In excess” is variable and thus 15% for every value of income: X-1300 where x is total
income
●​ For elimination 2 equations and their respective sides add
●​ To eliminate X or Y variables are made same through multiplying one equation
(in this case second eq is multiplied by 2)
●​ Simultaneous equations that do not have any answer (no intersect):
○​ Have same gradient
○​ Different y intercept
○​ Convert to standard y=mx+c form and compare
○​ Basically parallel lines
●​ For simultaneous eq word problems:
○​ Same variables X and Y representing 2 diff things
○​ 2 Diff answers or relationships with same variables and diff coefficients (amounts)
○​ Total is usually answer
●​ For inequality word problems:
○​ More than is x>5
○​ Less than is x<5
○​ Minimum (atleast) is x ->- 5
○​ Maximum (not more than) is x -<-5
●​ For (x-3) instead of just x:
○​ Constant value is not y intercept it is value of x=3
●​ Gradient is increase in Y for every x
●​ FOR F(3x) = x-6
○​ To find coefficient of x we equate 3x=x
■​ Thus x=x/3
■​ Thus f(6) = (6/3) -6
■​ Similarly f (x-3) = x +6
●​ Thus f(5)= (5+3) +6
■​ I.e operation on x in bracket f(x) is inverted on RHS
●​ FOR STATS:
●​ Part to whole ratios
○​ When in 1:5 ratio, 1 is a smaller part of 5 and can be represented by
fraction ⅕
○​ Remaining ingredients/parts are ⅘
○​ Parts are added i.e ⅕ butter and ⅖ sugar total make ⅗ parts, leaving
behind a ⅖ remainder milk
●​ In rate per minute/per hour/per cent i.e the divisor must be correct
●​ For square conversion, square the amount of normal conversion e.g:
○​ 1m=100cm
○​ 1m2 = (100)2
○​ For cubic conversion power or 3 is used
●​ For conversion of Rates e.g km/h to m/s
○​ When numerator value goes from smaller to larger unit e.g m to km; the
rate decreases and vice versa
○​ When denominator goes from smaller to larger unit e.g second to hour;
the rate increases and vice versa
●​ Complementary percentages:
○​ If a bag has blue and black marbles and 40% marbles are blue, then 60%
markers are black (100-40)
●​ IF SUM OF NUMBERS AND AMOUNT OF NUMBERS SAME THEN MEAN SAME
●​ For symmetrical datasets (same freq below and above the mid) the median and
mean is same i.e middle value
●​ If asks what percent
○​ Ans is 50 not 50%
●​ PERCENT DECREASE IS NOT SAME AS JUST DECREASE
●​ % change is difference/initial x 100
●​ Variance:
○​ Sum of (data point-mean)2/ no of data points
○​ Symbol is σ2
●​ Standard deviation:
○​ Symbol is σ
○​ It is square root of standard deviation
●​ Outlier is anomaly value
○​ which causes larger variation and standard deviation
○​ Affects mean more and median less
●​ If same value e.g 10 is added to all values then mean and median increases by
10
●​ Use mean median mode Akif formulae even for graphs
●​ Median position change doesn't necessarily mean median value change
●​ Missing mean value= (mean*total freq)-other values
●​ Median:
■​ total/2=Y & total/2 + 1=X
■​ For even numbers median= (Y+X)/2
■​ For odd numbers median = X
■​ MEDIAN IS ALWAYS CALCULATED AFTER ARRANGING LOWEST
TO HIGHEST
●​ Cumulative freq will be sum of height of each bar
●​ Margin of error is upper and lower bound of average
●​ Greater sample size; small diff in sample and total population; better quality of
sample = small margin of error
●​ Clear line of best fit in scatter graph = linear relationship
●​ Y intercept of line of best fit represents initial value & Gradient represents rate e.g
questions solved (Y) per time (X) ; note: X value is always 1 unit
●​ -b/2a (when 1 x intercept)
●​ To identify equation of line of best fit / curve of best fit, look for C and Gradient
(shape)
●​ ALWAYS SUBTRACT Y INTERCEPT (C i.e THE INITIAL VALUE) FROM THE
MAXIMUM POINT
●​ Slope of line of best fit always means inc/decrease in Y for every
inc/decrease in 1 X always
●​ “BASED ON LINE OF BEST FIT”: ALWAYS TELL VALUE DERIVED FROM
LINE
●​ Linear relationship:
○​ X and Y have arithmetic progression
○​ Formula: y=mx+c
●​ Exponential (geometric) relationship:
■​ (also percentage increase/decrease & compound interest)
○​ X has arithmetic but Y has a geometric progression
○​ Formula: a(b)x wherein a= Y intercept (c) & b= factor/common ratio
○​ b= succeeding term divided by preceding term
○​ When a is negative, the graph’s shape is negative
○​ When b is decimal between 0 and 1 i.e 0<x<1, the graph’s shape is
horizontally inverted (decay/inversely proportional)
○​ IMP Note: Y intercept (a) is not 1st Y term, it is Y term at X=0
○​ C in a(b)x + C, is always asymptote (graph moves C places upwards)
●​ Quadratic relationship: X has arithmetic but Y has quadratic progression (2nd diff
same)
AT A CONSTANT RATE MEANS LINEAR EVEN IF PERCENTAGE DECREASE
●​
●​ Estimate= sampling proportion x size of entire population
●​ range= +/- margin of error
●​ Confidence interval:
○​ Is the percentage probability that the entire population lies within the moE range
○​ smaller range and = confidence interval decreases and vice versa
○​ i.e starting higher than lower bound and ending lower than higher bound
●​ Confidence interval DOES NOT MEAN 95% OF ALL POPULATION or 95% CHANCE
●​ Sampling errors (bias):
○​ Voluntary response
○​ Convenience (favorable sample)
○​ Undercoverage
●​ Studies:
○​ Observational
○​ Sampling
●​
●​
●​
●​
●​
●​
●​
●​
○​ Experiment
For SAT, causation (as opposed to correlation) is proved by Random assignment
For SAT, population generalizations are proven by random selection of sample
SAMPLE SHOULD BE RELATED TO CHANGING VARIABLE AND SHOULD HAVE
LESS VARIANCE = LESS MARGIN OF ERROR
For conversion and speed: large D unit and small t unit decrease value (viceversa)
“PER” VALUE IS IN DENOMINATOR E.G Y PER X = X IN DENOMINATOR
For ratio:
○​ convert ratio into fractions
○​ Square values follow inverse square laws
CONVERT PERCENTAGES INTO DECIMAL OR HUNDREDS
Keep in mind SUBTRACT THE REDUCED PERCENTAGE FROM 100
○​ E.g 40% less than x = (0.1-0.4) X = 0.6 X = 60% of X
●​ For probability:
○​ ALWAYS FIRST IDENTIFY SUPERSET AND SUBSET
○​ First add numbers, then find probability (for either or cases)
○​ FOCUS ON
■​ FROM STATEMENT (XYZ group selected from/ ABC group)
■​ BY STATEMENT (BY XYZ)
■​ GIVEN THAT (must be) STATEMENT (DENOMINATOR)
○​ Focus on words like at least (XYZ and above) or maximum (XYZ and below)
○​ Usually the first value is numerator and second is denominator in
statement
○​ XYZ is numerator while ABC is denominator
○​ In the case of 2 characteristics, DON'T USE 2 SEPERATE PROBABILITY AND
MULTIPLY; INSTEAD FIND COMMON GROUP
○​ TO FIND A MISSING VALUE IN TABLE U NEED
■​ 1) TOTAL
■​ 2) ALL OTHER VALUES
○​ For percentages : they cant be added across categories (rows/columns)
○​ Percentage ratio :
■​ E.g 2 data columns wherein
●​ X percent of variable a = 2X percent of variable B
○​ 2 things:
■​ 1. Conditional term which is variable (numerator)
■​ 2. Permanent term which is fixed and is must (denominator)
●​ ADVANCED MATHS:
○​ In x2+bx+c = (x+d) (x+g)
■​ Sum of roots= -b/a
■​ Product of roots= c/a
■​ “Maximum” in quadratics means vertex
■​ When in completing the square/vertex form, even if constant is 0,
BRACKET IS ALWAYS WHOLE SQUARED
■​ CONSTANT IN COMPLETING THE SQUARE REPRESENTS Y VALUE
OF THE VERTEX, NOT THE Y INTERCEPT
■​ For x2-bx+c:
●​ x2-bx+(b/2)2 becomes (x-b/2)2 and -(b/2)2 makes constant
always
■​ WHEN GRAPH GIVEN AND VALUES ASKED, PUT POINTS ON
GRAPH IN VALUES TO VERIFY
■​ The factor (x) = (x+0) i.e root = 0
■​ To verify an equation w graph, use either substitution of roots to see
if they are factors, OR use desmos
■​ If one X intercept is 0; half of the 2nd intercept is midpoint/ vertex X
value
●​ VERTEX GIVEN=COMPLETING SQUARE FORM
●​ ROOTS GIVEN=ROOT FORM
○​ THEN 1) USE FOR -B/A & C/A
○​ OR 2) USE BY EXPANDING ROOTS TO Ax2+Bx+C
■​ COMPARE CO EFFs
■​ PLUGGING IN POINTS
■​ To find points of intersection:
●​ 1. Merge 2 equations and find roots
●​ Plug in roots into separate equations not merged one to find y
●​ To find unknown, merge and use discriminant
■​ Calc equation mode
■​ VERTEX FORM COEFFICIENTS AFTER EXPANSION CAN BE
COMPARED WITH ax2+bx+c form WHENEVER VERTEX GIVEN
■​ WHENEVER OPTIONS IN ROOT: USE QUADRATIC FORMULA
Q) WHAT IS MIN MAX X WHEN SOMETHING Y
ans) 0 since any number squared is
not negative
○​ In Synthetic division:
■​ Convert X+2 into -2
■​ Use division to find resultant equation
■​ The remainder i.e the polynomial constant (d) after addition is remainder
and it is written as (remainder) / x+2 (factor)
■​ If constant doesn’t exist, we still use it in Synthetic division as 0
■​ For long division ONLY if R is unknown, use remainder theorem to
find
●​ For above polynomial: EXPAND AND COMPARE COEFFICIENTS
●​ Same thing can be applied in PARTIAL FRACTIONS FORMAT
○​ ABSOLUTE VALUE:
○​ A modulus equation with ans= negative, has NO solutions
○​ When Modulus…
■​ Has same module (expression in the modulus) on both sides and
ans is a constant then use method: solving for positive and negative
constant
■​ Has different module on both sides e.g |2x+1|=|3x+2| use method:
square both sides
○​ MODULUS range WORD PROBLEMS WHEREIN IF 16<W<18 THEN |W-17|<1
■​ Wherein w-17 is distance of w value from 17
●​ ZOOM IN FOR TANGENTS ON DESMOS BRUH
●​ QUADRATIC AND EXPONENTIAL WORD PROBLEMS
○​ Quadratic:
■​ Ball height & time
■​ Length and width
■​ Area quadratic equation: length and width are in terms of x
■​ C is always initial Y value whether Y be height or length etc
■​ Turning point is max/min quantity depending on shape
■​ Height of object against time: h(t)= -at2+bt+c
■​ X intercept= the X when Y is 0; Y intercept= the Y when X is 0
○​ Exponential:
■​ Population & time
■​ Interest
■​ Increasing at 30%= multiplied by 1.3
■​ Decreasing by 30% = multiplied by 0.7
■​ NATURAL Y INTERCEPT OF EXPONENTIAL IS 1 FOR a(0.1)x
(horizontally inverted) & a(0.1)x
●​ -1 FOR -a(0.1)x
■​ Constant i.e Y intercept is the value on T=0 NOT T=1
●​ In the above eq, N is decreasing, for decrease: rate is in denominator
○​ 2 TYPES OF EXPONENTIAL MODELS:
■​ “EVERY MODEL” = TIME IN DENOMINATOR
■​ “UNTIL MODEL” E.G INTEREST: GAIN OR DECREASE UNTIL/FOR A
SPAN OF TIME = TIME IN NUMERATOR
■​ MOREOVER, for y=c(a)x + b; b= y intercept -1 or y intercept-initial y
intercept
●​ It takes 5.5 seconds to half N
●​ IF exponent is 5.5t then t / k = t / (1/5.5) = t x 5.5/1
●​ ALWAYS CONVERT T AND K TO SAME UNIT
●​ V V IMP:
○​ Gaining 4/5th or X fraction means 1+X fraction
○​ Similarly gaining X percent means 100+X percent or 1 + ​
X decimal
○​ Similarly losing 2% = 100-2 percent or losing X fraction = 1-X fraction
○​ The rate is only applied when its exponent is equal to 0
●​ IN DETERMINING WHETHER EXPO OR LINEAR, FOCUS ON
○​ Y VALUE
○​ TYPE OF SLOPE/ CHANGE
●​ inc/dec by Percent of original is linear
●​ When equation not solving by algebra, plug in options to see which one is right
○​ FUNCTIONS:
○​ 2 to1 function exists while 1 to 2 does not exist
○​ When no middle term exists, solve quadratics by taking square root on both sides
(+/-)
○​ Radical equations = have variables in roots
○​ Rational equations = have fractions with variables in denominator
○​ Domain of radical and rational equations excludes undefined values
○​ ^^(x-1)=g(2)
○​ IF F(x+1)=7(x) THEN F(4)=7(3)
■​ Since x+1=4 === x=3
○​ No of solutions of f(x)=k means number of times the line y=k intercepts f(x)
○​ EXTRANEOUS SOLUTIONS to radical and rational equations
○​ ALWAYS identify extraneous solutions for radical and rational equations by
putting answers in equations
○​ When +/- is not written next to square root then answer is always positive
○​ REMEMBER WHEN MORE THAN 1 UNKNOWN VARIABLE, CANCEL
OTHERS BY:
■​ TAKING COMMON (INCLUDING VARIABLES IN EXPONENT)
■​ DIVIDING EQ BY IRRELEVANT VARIABLE
○​ 3x+2= 3x x 3 x 3
○​ 4* square root 3x = (42)*3x
■​ Similarly 4* cube root 3x = (43)*3x
●​ In quadratic equation ax2+bx+c magnitude of a also defines how narrow or wide a
parabola is: greater magnitude means narrow and vice versa
●​ C value shifts up and down
●​ (x-h)2 shifts to right or (x+h)2 shifts to left
●​ Polynomials is = 0 when the factors in (x+a)(x+b)(x+c) = 0 e.g (x+a) = 0 and thus
x=-a
●​ Y intercept in polynomials is D (constant)
●​ In the polynomial term a(xn):
○​ N is even = ends point in same direction (U shape)
○​ N is odd = ends point in opposite direction (N shape)
●​ In rational functions,
○​ Vertical asymptote i.e Y = infinity/undefined/asymptote is at x value found when
denominator expression=0 (undefined)
○​ E.g In below case: 2x+1=0 means asymptote at x=-0.5
○​ Y = 0 is at x value found when numerator expression=0
○​ Horizontal asymptote i.e x=infinity/undefined is at y value when inverse of
function’s denominator expression= 0
●​ Distinct 0 = x
intercept
●​ Curve as
tangent to x
axis = 2 x
intercepts
2
repeating = still x (still quadratic equation)
●​ Sign of Coefficient of a in highest power term of x determines
shape i.e a(xn)
●​ FOR TRIGONOMETRY:
●​ Check which sides are corresponding in similar figures by
checking on which sides the angles are the same
●​ RULE: FOR TRIANGLE WITH SIDES XYZ:
■​ X-Y< Z <X+Y
●​ Sin(x) = Cos(90-x) and vice versa
●​ Trigonometric ratios are same for similar figures
●​ Degrees to radian: (Degrees X pi) / 180
●​ Unit circle: angle in radians and radius is 1
●​ CIRCLE ARC RATIOS: S1/ANGLE OF ARC 1 x ANGLE OF ARC 2 = S2
●​ NOTE: CIRCLE ANGLES CAN BE IN DEGREE WHILE OPTION CAN BE IN
RADIAN IF Pi SYMBOL SHOWNTHUS DEGREE TO RAD CONVERSION IS
FREQUENT
●​ Give answers of trig ratios as fractions
●​ R x pi = S (circumference)
●​ S x R = area
●​ Circle equation which is basically distance formula from center
○​ center = inverse sign of constants in brackets of x and y
○​ Distance = root of answer
○​ Answer of eq is always SQUARE of radius/dist from center
○​ Conversion to completing square form by addition of (X/2)2 and (Y/2)2
●​
●​
●​
●​
●​
●​
●​ Take constant to other side of equation
●​ Convert X2 - 10X + 25 = (X-5)2
○​ Wherein 25 = (10/2=5)2
○​ Note how we take absolute value of -10x
○​ No negative square as in just +52 is added to x2 -10x
2
■​ If (x+2) + y2 (no constant w y), it means center is y=0
○​ https://www.khanacademy.org/test-prep/v2-sat-math/x0fcc98a58ba3bea7:
geometry-and-trigonometry-easier/x0fcc98a58ba3bea7:circle-equations-e
asier/a/v2-sat-lesson-circle-equations​
Midpoint of diameter is center.
Look for:
○​ 30 and 90
○​ 60 and 90
○​ 45 and 90
○​ 90 and isosceles (45-45-90)
○​ 2 angles have same measure (isosceles)
○​ A RIGHT ANGLE TRIANGLE W/ HYP DOUBLE OF ONE OF THE SIDES
it will be 30-60-90 triangle
Sim figs: Ratio gives total length not desired length & perimeter also proportional
Look for 2 same angles for similar figures
THERE ARE ALOT OF ISOSCELES
A DIAGONAL IN SQUARE MAKES 2 45-45-90 TRIANGLES
●​ x/4 = 2/x (parallel lines cut by 2 non parallel lines)
●​
●​ Ans is A since from center to tangential point is the radius which is perpendicular to
tangent: thus apply gradient formula
●​ For questions regarding points lying on diameter of circle AND circumference, use
equation of diameter/radius line method\
●​ OR apply midpoint method to find opposite point on circumference
○​ (x1-x2)/2 = x of center
●​ In this Q, when they say SOME
VALUE IS BETWEEN SUCH AND SUCH
●​ Find upper and lower limits i.e range
by using both such and such
●​ Find required answer from range
●​ For 2) When the value of a trigonometric ratio is given and another ratio is asked
for the SAME angle, THEN:
○​ CONVERT VALUE INTO ANY FRACTION AND SOLVE FOR MISSING
TRIANGLE SIDE
○​ THEN USE VALUES TO SOLVE FOR TRIGONOMETRIC RATIO
○​ Note: tanx = Z also means tanx= Z/1 where Z=opp & 1=adj
○​ Sin of 1 angle is cos of the other
○​ Trig ratios are same for similar figs
○​ If HYP = 1 then cos is just adjacent and sin is just opposite
●​ In 16: quadrants question: focus on ASTC concept of + and ○​ Ans is C where both quadrants positive/negative