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Formulas & shortcuts

SAT · 36

Every formula in this course and the easiest ways to remember them, on one page.

1. Getting to know the exam2. The Reading and Writing section3. Math: Algebra and Advanced Math4. Math: data, geometry and Desmos5. Score strategies

1Getting to know the examBeginner

The Digital SAT: format, adaptive modules and scoring

Open lesson
Total score = Reading and Writing (200–800) + Math (200–800) = 400–1600

Scores come in 10-point steps: for example, 650 + 700 = 1350.

Registration, test day and a study plan

Open lesson

2The Reading and Writing sectionIntermediate

Craft and Structure: words, purpose and paired texts

Open lesson

Information and Ideas: central ideas, evidence and inferences

Open lesson

Standard English Conventions: punctuation and grammar

Open lesson

Expression of Ideas: transitions and rhetorical synthesis

Open lesson

3Math: Algebra and Advanced MathIntermediate

SAT Algebra: linear equations, functions and systems

Open lesson
ax + b = cx + d
where:
  • a ≠ cone solution: x = (d − b)/(a − c)
  • a = c, b ≠ dno solution
  • a = c, b = dinfinitely many solutions

After expanding brackets and collecting like terms, compare the coefficients of x and the constants.

m = (y₂ − y₁) / (x₂ − x₁) y = mx + bm = (y₂ − y₁) / (x₂ − x₁) y = mx + b
where:
  • mslope — the change in y when x increases by 1: a rate or a price per unit
  • by-intercept — the value of y when x = 0: the starting value
  • (x₁, y₁), (x₂, y₂)two points on the line

Parallel lines have equal slopes; for perpendicular lines m₁ · m₂ = −1.

a₁x + b₁y = c₁, a₂x + b₂y = c₂
where:
  • a₁/a₂ ≠ b₁/b₂a₁/a₂ ≠ b₁/b₂one solution — the lines intersect
  • a₁/a₂ = b₁/b₂ ≠ c₁/c₂a₁/a₂ = b₁/b₂ ≠ c₁/c₂no solution — the lines are parallel
  • a₁/a₂ = b₁/b₂ = c₁/c₂a₁/a₂ = b₁/b₂ = c₁/c₂infinitely many solutions — the same line

Advanced Math I: quadratic and exponential functions

Open lesson
y = ax² + bx + c y = a(x − r₁)(x − r₂) y = a(x − h)² + k
where:
  • cin standard form, the y-intercept: (0, c)
  • r₁, r₂in factored form, the roots (x-intercepts)
  • (h, k)in vertex form, the vertex; k is the minimum (a > 0) or maximum (a < 0) value
  • aa > 0 opens up, a < 0 opens down
x = (−b ± √(b² − 4ac)) / (2a) x₁ + x₂ = −b/a x₁ · x₂ = c/a h = −b/(2a)x = (−b ± √(b² − 4ac)) / (2a) x₁ + x₂ = −b/a x₁ · x₂ = c/a h = −b/(2a)
where:
  • b² − 4acdiscriminant: > 0 two real solutions, = 0 one solution, < 0 no real solutions
  • hthe x-coordinate of the vertex

The quadratic formula is not on the SAT reference sheet — you must know it by heart.

f(t) = a · bᵗ a(1 + r)ᵗ a(1 − r)ᵗ
where:
  • athe starting value (t = 0)
  • bthe factor per period: b > 1 growth, 0 < b < 1 decay
  • a(1 + r)ᵗ / a(1 − r)ᵗa(1 + r)ᵗ / a(1 − r)ᵗgrowth by r per period / decay by r per period
  • rthe percent as a decimal: 3% → 0.03

A linear function adds the same amount at each step; an exponential function multiplies by the same factor. On the SAT: growth and decay.

Advanced Math II: functions, polynomials, radicals and rational expressions

Open lesson
y = a · f(x − h) + k
where:
  • hthe graph shifts h units right (left if h < 0)
  • kthe graph shifts k units up (down if k < 0)
  • avertical stretch; if a < 0 the graph is reflected over the x-axis

f(−x) reflects the graph over the y-axis.

p(a) = 0 ⇔ (x − a) is a factor of p(x) p(x) ÷ (x − a) ⇒ remainder = p(a)
where:
  • p(x)a polynomial
  • aa zero of the polynomial: the graph crosses the x-axis at x = a

Remainder theorem: when p(x) is divided by (x − a), the remainder equals p(a).

x^(m/n) = ⁿ√(xᵐ) = (ⁿ√x)ᵐ x⁻ⁿ = 1/xⁿx^(m/n) = ⁿ√(xᵐ) = (ⁿ√x)ᵐ x⁻ⁿ = 1/xⁿ
where:
  • nthe index of the root (the denominator)
  • mthe power (the numerator)

For example: 8^(2/3) = (∛8)² = 2² = 4.

4Math: data, geometry and DesmosAdvanced

Problem-Solving and Data Analysis: percentages, statistics and probability

Open lesson
percent change = (new − old) / old × 100% successive: (1 + r₁)(1 + r₂)percent change = (new − old) / old × 100% successive: (1 + r₁)(1 + r₂)
where:
  • r₁, r₂the changes as decimals: +25% → 0.25, −20% → −0.20
P(A) = n(A) / n(total) P(A | B) = n(A and B) / n(B)P(A) = n(A) / n(total) P(A | B) = n(A and B) / n(B)
where:
  • n(…)the number of outcomes (people) in that group
  • P(A | B)the probability of A given B — only group B goes in the denominator

Geometry and Trigonometry: shapes, triangles and circles

Open lesson
sin θ = a/c cos θ = b/c tan θ = a/b sin θ = cos(90° − θ)sin θ = a/c cos θ = b/c tan θ = a/b sin θ = cos(90° − θ)
where:
  • athe leg opposite angle θ
  • bthe leg adjacent to angle θ
  • cthe hypotenuse

Memory aid SOH-CAH-TOA: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

radians = degrees × π/180 s = rθ S = ½r²θradians = degrees × π/180 s = rθ S = ½r²θ
where:
  • sarc length (θ in radians)
  • Ssector area (θ in radians)

In degrees: s = (angle/360°) · 2πr. For example, r = 9 and angle 120° = 2π/3: s = 9 · 2π/3 = 6π.

(x − h)² + (y − k)² = r²
where:
  • (h, k)the center of the circle
  • rthe radius (the square root of the right side)

Desmos strategies: faster with the calculator

Open lesson

5Score strategiesAdvanced

Pacing, guessing and process of elimination

Open lesson
average time per question = module time / number of questionsaverage time per question = module time / number of questions

Reading and Writing: 32 min / 27 questions ≈ 71 seconds. Math: 35 min / 22 questions ≈ 95 seconds.

expected correct answers = number of questions × 1 / (options left)expected correct answers = number of questions × 1 / (options left)

Since wrong answers are not penalized, a guess can never lower your score.

Avoiding traps and the final-week plan

Open lesson