essential formulas for sat math

Linear Equations

Slope-InterceptForm:𝒚 = 𝒎𝒙 + 𝒃Slope=𝑚𝑦-intercept=𝑏

Point-SlopeForm:𝒚 − 𝒚𝟏 = 𝒎(𝒙 − 𝒙𝟏)

StandardForm:𝑨𝒙 + 𝑩𝒚 = 𝑪

Slope=−12

𝑦-intercept= 32

SlopeFormula:𝒎 = 𝒚𝟐5𝒚𝟏𝒙𝟐5𝒙𝟏

MidpointFormula:6𝒙𝟏7𝒙𝟐𝟐

, 𝒚𝟏7𝒚𝟐𝟐:

DistanceFormula:𝒅 = <(𝒙𝟐 − 𝒙𝟏)𝟐 + (𝒚𝟐 − 𝒚𝟏)𝟐

Systems of Equations

Intersectinglinesàonesolution

Parallellinesànosolutions

Samelineàinfinitesolutions

Quadratic Equations / Parabolas

Standard/QuadraticForm:𝒇(𝒙) = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄

𝑥-valueofvertex= − ABC

𝑦-valueofvertex= 𝑓 6− ABC:

Minimumwhen𝑎 > 0Maximumwhen𝑎 < 0

𝑦-intercept= 𝑐 →“constantorcoefficient”

VertexForm:𝒇(𝒙) = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌

Vertex:(ℎ, 𝑘)→“constantsorcoefficients”Minimum(when𝑎 > 0):𝑘Maximum(when𝑎 < 0):𝑘

FactoredForm:𝒇(𝒙) = 𝒂(𝒙 − 𝒔)(𝒙 − 𝒕)

𝑥-intercepts:𝑠and𝑡→“constantsorcoefficients”

𝑥-valueofvertex= S7TB

𝑦-valueofvertex= 𝑓 6S7TB:

Minimumwhen𝑎 > 0

Maximumwhen𝑎 < 0

ESSENTIAL FORMULAS FOR SAT MATH

Circles ArcLength= 6 𝒏

𝟑𝟔𝟎:𝟐𝝅𝒓SectorArea= 6 𝒏

𝟑𝟔𝟎:𝝅𝒓𝟐

𝑛 =centralangleofarc/sector

Center-RadiusEquation:(𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐Center:(ℎ, 𝑘)Radius=𝑟

Powers/Exponents/Roots 𝑥C × 𝑥A = 𝑥C7A 𝑥5C = ^

_`

_`

_a= 𝑥C5A 𝑥

`a = √𝑥Ca

(𝑥C)A = 𝑥CA (𝑥𝑦)C = 𝑥C𝑦C

<𝑥𝑦 = √𝑥 × <𝑦 𝑥c = 1

(−1)e = f 1, 𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛−1, 𝑖𝑓𝑛𝑖𝑠𝑜𝑑𝑑

Exponential Equations

GeneralForm: 𝒇(𝒙) = 𝒂𝒃𝒙 • If𝑏 > 1,exponentialgrowth• If0 < 𝑏 < 1,exponentialdecay

Growth/DecayFormula:𝑨(𝒕) = 𝑷6𝟏 + 𝒓𝟏𝟎𝟎:𝒕

• 𝑃 =Principle(initialamount)• 𝑟 =%increase/decrease• 𝑡 =timeinterval(inanyunit)

Trigonometry 𝑠𝑖𝑛𝑒 = noo

pqo 𝑐𝑜𝑠𝑖𝑛𝑒 = Crs

pqo 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 = noo

Crs

𝑖𝑓∠𝐴 + ∠𝐵 = 90°, 𝑡ℎ𝑒𝑛 z𝑠𝑖𝑛𝐴 = 𝑐𝑜𝑠𝐵𝑐𝑜𝑠𝐴 = 𝑠𝑖𝑛𝐵

Percentages % = oC|T

}pn~�× 100 %𝑐ℎ𝑎𝑛𝑔𝑒 = e�}5n~r

n~r× 100

Miscellaneous 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑅𝑎𝑡𝑒 × 𝑇𝑖𝑚𝑒

Quadratic Identities

(𝑥 + 𝑎)(𝑥 + 𝑏) = 𝑥B + (𝑏 + 𝑎)𝑥 + 𝑎𝑏 𝑎B − 𝑏B = (𝑎 + 𝑏)(𝑎 − 𝑏)(𝑎 + 𝑏)B = 𝑎B + 2𝑎𝑏 + 𝑏B(𝑎 − 𝑏)B = 𝑎B − 2𝑎𝑏 + 𝑏B

Eliminatebothvariables!

A mentor can change everything.

Components of Experiment Design

Population: Asetofitemsofinterestforsomequestionorexperiment.

RandomSample: Asubsetofthepopulationthatcanreasonablybestudiedinwhicheachitemhasanequalchanceofbeingselected.

Requiredinordertogeneralizesurveyresultstotheentirepopulation.

SampleBias: Whensomemembersofpopulationarelesslikelytobeincludedthanothers.

Randomsampling=nosamplebias

MarginofError: Howmanypercentagepointsasample’sresultswilldifferfromtherealpopulation’svalue.

ConfidenceInterval: A95%confidenceintervalwitha4%marginoferrormeansthatyourstatisticwillbewithin4pointsoftherealpopulationvalue95%ofthetime.

Statistical Measures

Mean: 𝑠𝑢𝑚𝑜𝑓𝑖𝑡𝑒𝑚𝑠#𝑜𝑓𝑖𝑡𝑒𝑚𝑠

Median: Themiddlenumberofanorderedsetofitems.

Median#term: e7^B,𝑛 =numberofitemsinset.

Range: Maximum–minimum

StandardDeviation: Measuresspreadofdataset

HighSD: dataspreadoutfrommeanLowSD: dataclosetomean

Outlier: Avaluethatissignificantlylargerorsmallerthantherestofthedata.

HighOutlier: Mean>Median

LowOutlier:Mean<Median

ESSENTIAL FORMULAS FOR SAT MATH

A mentor can change everything.

Box Plots

LowerQuartile: Lowest25%ofdata.

UpperQuartile: Highest25%ofdata

Extra Geometry Formulas

SurfaceAreaofRectangularPrism= 2(𝑤𝑙 + ℎ𝑙 + ℎ𝑤)

SurfaceAreaofCylinder= 2𝜋𝑟B + 2𝜋𝑟ℎ

AreaofEquilateralTriangle=√��𝑠B

Complex Numbers

𝑖^ = 𝑖 𝑖B = −1 𝑖� = −𝑖 𝑖� = 1

𝑖� = 𝑖 𝑖� = −1 𝑖� = −𝑖 𝑖� = 1

Direct/Inverse Variation DirectVariation:𝑦 = 𝑘𝑥

InverseVariation:𝑦 = �_

Complex Examples of Variation

𝑦 = 𝑘√𝑥 ∶ 9𝑥 → 3𝑦 𝑦 = 𝑘𝑥B ∶ 3𝑥 → 9𝑦

𝑦 = �√_ ∶ 9𝑥 → ^

�𝑦 𝑦 = �

_� ∶ 3𝑥 → ^

�𝑦

Arithmetic Sequences

𝑎e = 𝑎^ + (𝑛 − 1)𝑑 𝑆e =eB(𝑎^ + 𝑎e)

𝑘 = constantofvariation