Math

Given

Arithmetic and Algebra

• Linear - Slope-Intercept Form of a Line: y=mx+b

  • m is the slope

  • b is the y-intercept

  • same slope, different y-intercept = no solutions (lines are parallel)

  • different slope, y-intercept is irrelevant = 1 solution (lines intersect)

  • same slope, same y-intercept = infinitely many solutions (same line)

• Slope: m=y2-y2/x2-x1

• Midpoint: (x₁ + x₂/2, y₁ + y₂/2)

• Vertex Form of a Parabola/ Quadratic: y=a(x-h)²+k

• Standard Form: y=ax²+bx+c

  • a>0 opens upwards

  • a<0 opens downwards

  • (h,k) is the vertex

  • c is the y-intercept

• Distance: d=√(x2 - x1)²+(y2 - y1)²

• Difference of Squares: a²-b²=(a+b)(a-b)

• Quadratic Equation: x = -b ± √b^2 - 4ac / 2a

  • Discriminant is b²-4ac

    • b²-4ac>0 = 2 real solutions

    • b²-4ac = 1 = 1 real solution

    • b²-4ac<0 = no real solutions

• Sum of Solutions: -b/2a

  • also the x-value at the vertex

  • also the axis of symmetry

• Product of Solutions: c/a

• Exponents

  • Multiplying: (a^n)(a^m)=a^n+m

  • Dividing: a^n/a^m=a^n-m

  • To a Power: (a^n)^m=a^n*m

• Binomial Product

  • Difference of Squares: (x-y)(x+y)=x²-y²

  • Perfect Squares Trinomial

    • Positive: (x+y)²=x²+2xy+y²

    • Negative: (x-y)²=x²-2xy-y²

Ratios, Percentages, and Statistics

• Simple Interest: A=Prt

  • P is the principle amount

  • r is the interest rate

  • t is time (usually in years)

• Average/ Mean: average = sum of terms/ number of terms

• Median: middle number

• Range: max/min

• Standard Deviation: the measure of spread in the data set

  • higher standard deviation = greater spread

  • lower standard deviation = smaller spread

Geometry and Trigonometry

• Equation of a Circle: (x-h)²+(y-k)²=r²

  • (h,k) is the center of the circle

  • r is the radius

• Radians:

  • radians=degrees*(n/180)

  • degrees=radians*(180/n)

• Arc Length: s=r/theta

  • theta is the radians

• Degrees in a Polygon: (n-2)*180

• Regular Polygon Interior Angle: (n-2)180/n

• SOHCAHTOA:

  • sin=opp/hyp

  • cos=adj/hyp

  • tan=opp/adj

• Sine/Cosine Relationship:

  • sin(x)=cos(90-x)

  • cos(x)=sin(90-x)

• Pythagorean Theorem: a²+b²=c²

  • c is the hypotenuse

Statistics & Science

• Percent: part/100

• Percentage Change: (new-old)/old

• Probability: desired possibilities/total possibilities

• Speed: S=d/T

• Density: mass/volume

English

• Period = semicolon

  • Used to separate 2 complete sentences

• Comma + it, this, s/he, they (sometimes) = comma splice = WRONG

• Colon = single dash = explanation/ list

• 2 Commas = 2 dashes = 2 parentheses = non-essential clause

• Semicolon = period = comma + FANBOYS

• Coordinating Conjunctions: FANBOYS - joins 2 independent clauses

  • For, and, nor, but, or, yet, so

  • Comma + coordinating conjunctions

• Subordinating Conjunctions: joins an independent and a dependent clause

  • After, although, before, because, since, until, when…

• Continuers: words that indicate an idea is continuing where it began

  • ex. Also, and, likewise, moreover…

• Cause-and Effect:

  • ex. Because, consequently, thus, colons, dashes…

• Contradictors: words that indicate a sentence is shifting directions/ contrasting

  • ex. But, in contrast, despite, however…

• Comma splices are ALWAYS INCORRECT: caused by 2 constructions

  • pronoun as subject

  • adverb at the start of a clause