SAT Math Formula Sheet.

Example

Points (1, 3) and (4, 9): m = (9−3)/(4−1) = 6/3 = 2

SAT tip

Slope questions appear in nearly every SAT. Identify two points or the rate of change in context.

Example

y = 2x + 5 has slope 2 and y-intercept 5.

SAT tip

Most common line form on the SAT. Rearrange standard form into this to read slope/intercept instantly.

Example

Through (3, 4) with slope 2: y − 4 = 2(x − 3)

SAT tip

Fastest way to write an equation when given a point and slope, or two points.

Example

2x + 3y = 12 → slope = −A/B = −2/3

SAT tip

SAT often gives lines in standard form. Convert to slope-intercept or use x/y intercepts.

Example

Slope 3/4 → perpendicular slope is −4/3

SAT tip

Classic SAT trap: "which line is perpendicular to y = 2x + 5?" Answer: slope −1/2.

Example

$200 with 15% increase: 200 × 1.15 = $230

SAT tip

Used in word problems involving discounts, taxes, tips, and salary changes.

Example

From 80 to 100: (100−80)/80 × 100 = 25% increase

SAT tip

SAT data interpretation questions frequently ask for percent change.

Example

If y = 12 when x = 4, then k = 3 and y = 3x.

SAT tip

Look for "directly proportional" language. The ratio y/x is always constant.

Example

If y = 6 when x = 4, then k = 24 and y = 24/x.

SAT tip

Look for "inversely proportional." As x doubles, y halves.

Example

x + y = 10, x − y = 4 → add → 2x = 14 → x = 7, y = 3

SAT tip

SAT tests both algebraic and "no solution / infinite solutions" cases.

Example

y = 2x + 3 and y = 2x + 7 → no solution (parallel)

SAT tip

SAT asks "for what value of k does the system have no solution?" Look for matching slope.

Example

|2x − 3| = 7 → 2x−3 = 7 (x=5) or 2x−3 = −7 (x=−2)

SAT tip

Always check both solutions in the original equation if there are restrictions.

Example

x² + 5x + 6 = 0 → x = (−5 ± √1)/2 → x = −2 or −3

SAT tip

Use when factoring is not obvious. Always check if a = 1 first for easy factoring.

Example

y = 2(x − 3)² + 1 → vertex (3, 1), opens up.

SAT tip

SAT frequently asks for the vertex or minimum/maximum directly — read it from this form.

Example

y = x² − 6x + 8: h = 6/2 = 3, k = 9 − 18 + 8 = −1 → vertex (3, −1)

SAT tip

Axis of symmetry = h. Use this to find max/min without completing the square.

Example

x² + 2x + 5: D = 4 − 20 = −16 → no real roots.

SAT tip

"How many x-intercepts?" = "what is the sign of the discriminant?"

Example

x² − 9 = (x+3)(x−3)

SAT tip

SAT uses this to factor quickly or to rationalize. Recognize the pattern on sight.

Example

(x + 5)² = x² + 10x + 25

SAT tip

Used in completing the square and recognizing factorable trinomials.

Example

x² − 5x + 6 = 0: sum = 5, product = 6 → roots 2 and 3.

SAT tip

SAT uses this to find a missing coefficient without fully solving the quadratic.

Example

$1000 at 5%/yr for 3 yr: 1000 × (1.05)³ ≈ $1157.63

SAT tip

"Increases by r% per period" → b = 1 + r. "Decreases by r%" → b = 1 − r.

Example

x³ · x⁴ = x⁷; (x²)³ = x⁶; x⁻² = 1/x²

SAT tip

Exponent manipulation is tested heavily in the advanced module. Know all 5 rules cold.

Example

8^(2/3) = (∛8)² = 2² = 4

SAT tip

Convert between radical and exponent form to simplify expressions.

Example

f(x) = x² + 3x − 4 divided by (x−2): f(2) = 4+6−4 = 6, remainder is 6.

SAT tip

SAT asks "what is the remainder when p(x) is divided by (x−a)?" Just plug in a.

Example

y = x² and y = x + 2 → x² = x + 2 → x² − x − 2 = 0 → x = 2 or −1

SAT tip

The number of solutions = number of intersection points; use discriminant to check.

Example

f(x) = 2x + 1 → f(3) = 2(3) + 1 = 7

SAT tip

SAT tests f(a+b), f(f(x)), and table-based function reading.

Example

f(x) = x² on [1, 3]: ARC = (9−1)/(3−1) = 8/2 = 4

SAT tip

Equivalent to slope formula but applied to functions over an interval.

Example

f(x) = x²; g(x) = (x−2)² + 3 shifts right 2, up 3.

SAT tip

Inside the function = horizontal (counter-intuitive direction). Outside = vertical.

Example

g(x) = −2x² reflects over x-axis and stretches by 2.

SAT tip

SAT may show a graph and ask which transformation produces another graph.

Example

f(x) = x+1, g(x) = 2x → f(g(3)) = f(6) = 7

SAT tip

Evaluate inside-out. The SAT tests this both with formulas and function tables.

Example

f(x) = 2x + 3 → x = 2y + 3 → y = (x−3)/2 → f⁻¹(x) = (x−3)/2

SAT tip

"Find f⁻¹(5)" means "what input gives f(input) = 5?"

Example

2, 4, 6, 8 → differences = 2 (linear). 2, 4, 8, 16 → ratios = 2 (exponential).

SAT tip

SAT uses tables; check if consecutive differences are equal (linear) or ratios are equal (exponential).

Example

f(x) = √(x−3): domain is x ≥ 3; range is y ≥ 0.

SAT tip

For rational functions, set denominator ≠ 0. For even roots, set radicand ≥ 0.

Example

Base 8, height 5: A = ½ × 8 × 5 = 20

SAT tip

The height must be perpendicular to the base. For oblique triangles, drop an altitude.

Example

Legs 3 and 4: c² = 9 + 16 = 25 → c = 5

SAT tip

Memorize Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, and multiples.

Example

30-60-90 with short side 3: sides are 3, 3√3, 6.

SAT tip

The SAT reference sheet gives these — but recognizing them saves time.

Example

r = 7: C = 14π ≈ 43.98

SAT tip

Given diameter, halve it to get radius first. Diameter = 2r.

Example

r = 5: A = 25π

SAT tip

Always check whether the problem gives radius or diameter.

Example

90° arc, r = 4: L = (90/360) × 8π = 2π

SAT tip

Arc length is a fraction of the full circumference; the fraction is θ/360.

Example

120° sector, r = 6: A = (120/360) × 36π = 12π

SAT tip

Same fraction (θ/360) as arc length, but multiply by area not circumference.

Example

r = 3, h = 10: V = 90π

SAT tip

Think: base area × height. This pattern works for any prism.

Example

r = 3, h = 9: V = (1/3) × π × 9 × 9 = 27π

SAT tip

The SAT reference sheet gives this formula — but know when to use it.

Example

r = 3: V = (4/3)π(27) = 36π

SAT tip

Provided on the SAT reference sheet. Cube the radius, not the diameter.

Example

l = 8, w = 5: A = 40

SAT tip

For a square, A = s². Perimeter = 2(l + w).

Example

Bases 6 and 10, height 4: A = ½ × 16 × 4 = 32

SAT tip

Identify the two parallel sides (bases). Height is perpendicular distance between them.

Example

(1,2) to (4,6): d = √[(3)²+(4)²] = √25 = 5

SAT tip

Recognize Pythagorean triples to avoid unnecessary calculation.

Example

Midpoint of (2,4) and (8,10): M = (5, 7)

SAT tip

If given the midpoint and one endpoint, work backwards to find the other endpoint.

Example

Center (3, −1), r = 4: (x−3)² + (y+1)² = 16

SAT tip

SAT may give an equation in general form; complete the square to find center/radius.

Example

Opposite = 4, hypotenuse = 5: sin θ = 4/5 = 0.8

SAT tip

"SOHCAHTOA" — memorize this acronym. The SAT uses all three ratios.

Example

sin 30° = cos 60° = 0.5

SAT tip

The SAT tests: "sin(x°) = cos(y°) means x + y = 90." Use this to solve for unknowns.

Example

Set {4, 7, 9, 12}: mean = 32/4 = 8

SAT tip

You can work backwards: if mean = 8 and n = 4, the sum must be 32.

Example

{2, 5, 7, 9, 11}: median = 7. {2, 5, 7, 9}: median = (5+7)/2 = 6

SAT tip

SAT often uses median vs. mean to test understanding of outlier effects.

Example

{3, 4, 4, 5, 6, 6, 6}: mode = 6

SAT tip

SAT rarely tests mode alone — it appears in "which measure is greatest/least?" comparisons.

Example

{3, 7, 12, 19}: range = 19 − 3 = 16

SAT tip

A simple measure of spread. SAT also tests interquartile range (IQR = Q3 − Q1).

Example

20 students avg 80, 30 students avg 90: (20×80 + 30×90)/50 = 86

SAT tip

Appears in SAT word problems with groups of different sizes.

Example

18 out of 24: (18/24) × 100 = 75%

SAT tip

"What percent of 40 is 14?" → 14/40 × 100 = 35%

Example

Drawing a red card: 26/52 = 0.5

SAT tip

SAT tests conditional probability: P(A | B) = P(A and B) / P(B).

Example

P(heads twice) = 0.5 × 0.5 = 0.25

SAT tip

"At least one" problems: use P(at least one) = 1 − P(none).

Example

$1000 at 6%/yr monthly for 2 yr: A = 1000(1.005)²⁴ ≈ $1127.16

SAT tip

SAT uses this formula; plug in carefully. If compounded annually, n = 1.

Example

$500 at 4%/yr for 3 yr: I = 500 × 0.04 × 3 = $60; A = $560

SAT tip

Simple interest is linear (constant dollar growth). Compound interest is exponential.

Example

{10,10,10}: σ = 0. {0,10,20}: σ > 0.

SAT tip

SAT asks: "which set has a larger standard deviation?" Compare the spread visually.

Example

y = 3x + 10: for each 1-unit increase in x, y increases by 3.

SAT tip

SAT scatter plot questions ask what the slope or intercept means in context.