Claude math hiset 1 and 2 sets combined

Slope formula

m = (y₂ − y₁) / (x₂ − x₁) | "Rise over run." Subtract y's on top, x's on bottom. | Use when: finding steepness or given two points. | Watch out: keep the order the same top and bottom.

Slope-intercept form

y = mx + b | m = slope, b = y-intercept (where line crosses y-axis) | Use when: graphing a line or writing its equation. | Trick: "b is where the line begins."

Point-slope form

y − y₁ = m(x − x₁) | Use when: you know the slope and ONE point but not the y-intercept. | Plug in: m = slope, (x₁, y₁) = the known point.

Quadratic formula

x = (−b ± √(b² − 4ac)) / 2a | Use when: solving ax² + bx + c = 0 and can't factor easily. | The ± means you get TWO answers — that's normal. | Watch out: don't forget the negative in front of b.

Distance formula

d = √((x₂ − x₁)² + (y₂ − y₁)²) | Use when: finding straight-line distance between two points on a graph. | Trick: it's the Pythagorean theorem in disguise — distance is the hypotenuse. | Watch out: square BEFORE adding, then take the root.

Midpoint formula

M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2) | Use when: finding the exact middle point between two coordinates. | Trick: "Average the x's, average the y's."

Pythagorean theorem

a² + b² = c² | a and b = legs (shorter sides), c = hypotenuse (longest side, opposite the right angle) | Use when: right triangle, know TWO sides, need the third. | Watch out: c is ALWAYS the hypotenuse, never a leg.

45-45-90 triangle — side ratios

Both legs = x, Hypotenuse = x√2 | Use when: right triangle with two equal legs, or a square cut diagonally. | Example: legs = 5 → hypotenuse = 5√2 ≈ 7.07 | Trick: two equal sides = two equal angles (both 45°).

30-60-90 triangle — side ratios

Short leg (opposite 30°) = x, Long leg (opposite 60°) = x√3, Hypotenuse (opposite 90°) = 2x | Use when: problem shows 30° or 60°, or an equilateral triangle cut in half. | Example: short leg = 5 → long leg = 5√3, hypotenuse = 10 | Trick: hypotenuse is always DOUBLE the short leg.

Surface area of a cylinder

SA = 2πr² + 2πrh | 2πr² = two circular ends (top + bottom), 2πrh = curved side | r = radius, h = height | Use when: asked how much material covers the outside of a cylinder.

Surface area of a rectangular prism

SA = 2(lw + lh + wh) | l = length, w = width, h = height | Think: find the area of each pair of opposite faces, add them, multiply by 2. | Use when: wrapping a box, painting walls, covering a rectangular solid.

Mean (average)

Mean = sum of all values ÷ number of values | Use when: finding the "average" of a data set. | Example: 70, 80, 90 → (70+80+90) ÷ 3 = 80 | Don't confuse with: median (middle value) or mode (most frequent).

Probability

P = favorable outcomes ÷ total possible outcomes | Answer is always between 0 (impossible) and 1 (certain). | Example: rolling a 3 on a die = 1/6 | Use when: asked "what is the probability that..."

Percent change

% change = (new − old) / old × 100 | Positive = increase, Negative = decrease. | Example: $50 to $60 → (60−50)/50 × 100 = 20% increase | Trick: "New minus old, over old."

Simple interest

I = P × r × t | I = interest, P = principal (starting amount), r = rate as decimal, t = time in years | Example: $1,000 at 5% for 3 years → 1000 × 0.05 × 3 = $150

Percent proportion

part / whole = percent / 100 | Use when: finding what percent one number is of another, or finding a missing part/whole. | Example: "What is 30% of 80?" → x/80 = 30/100 → x = 24

Proportions / cross-multiplication

If a/b = c/d, then a × d = b × c | Use when: two ratios are equal and you need a missing value. | Example: 3/4 = x/12 → 3×12 = 4×x → x = 9 | Use for: unit rates, scale problems, similar triangles.

Complementary angles

Two angles that add up to 90°. | Example: 30° and 60° are complementary. | Signal: problem mentions a right angle split into two parts. | Trick: "C" is for Corner (90°).

Supplementary angles

Two angles that add up to 180°. | Example: 110° and 70° are supplementary. | Signal: two angles on a straight line always add to 180°. | Trick: "S" is for Straight line (180°).

Vertical angles

The angles directly across from each other when two lines intersect. They are always equal. | Example: two lines cross making angles of 120°, 60°, 120°, 60° — the two 120° angles are vertical, the two 60° angles are vertical. | Signal: any time two lines cross (an X shape).

Adjacent angles

Angles that share a side and a vertex (corner point). They sit next to each other. | On a straight line, adjacent angles are supplementary (add to 180°). | In a right angle, adjacent angles are complementary (add to 90°).

Triangle angle rule

All three angles inside any triangle always add up to 180°. | Example: if two angles are 45° and 75°, the third = 180 − 45 − 75 = 60°. | Use when: given two angles of a triangle, find the third.

Parallel lines cut by a transversal

When a line crosses two parallel lines, it creates 8 angles. | Corresponding angles (same position) are equal. | Alternate interior angles (inside, opposite sides) are equal. | Co-interior angles (inside, same side) add to 180°. | Signal: two parallel lines with a diagonal line cutting through them.

Exponent rule — multiplying same base

x^a × x^b = x^(a+b) | Add the exponents when multiplying with the same base. | Example: x^3 × x^4 = x^7 | Watch out: only works when the BASE is the same. 2^3 × 3^4 cannot be combined.

Exponent rule — dividing same base

x^a ÷ x^b = x^(a−b) | Subtract the exponents when dividing with the same base. | Example: x^7 ÷ x^3 = x^4

Exponent rule — power to a power

(x^a)^b = x^(a×b) | Multiply the exponents when raising a power to another power. | Example: (x^3)^4 = x^12 | Trick: "power to a power = multiply."

Exponent rule — zero exponent

x^0 = 1 (for any x except 0) | Anything raised to the zero power equals 1. | Example: 5^0 = 1 | Exception: 0^0 is undefined — won't appear on the HiSET.

Exponent rule — negative exponent

x^(−a) = 1 / x^a | A negative exponent means "flip it to the denominator." | Example: x^(−3) = 1/x^3 | Example: 2^(−2) = 1/4 | Watch out: negative exponent ≠ negative number.

Exponent rule — fractional exponent

x^(1/n) = the nth root of x | A fraction in the exponent means a root. | Example: x^(1/2) = √x | Example: 27^(1/3) = ∛27 = 3 | Trick: the denominator tells you which root.

Similar triangles — what it means

Two triangles are similar if they have the same angles but different sizes. | Their sides are proportional — the ratios of matching sides are equal. | Signal words: "similar," scale drawings, shadow problems, indirect measurement.

Similar triangles — how to solve

Set up a proportion matching corresponding sides: (small side) / (big side) = (other small side) / (other big side) | Cross-multiply and solve for x. | Example: small triangle sides 3 and 5, big triangle has matching side 9 → 3/9 = 5/x → x = 15

Similar triangles — AA rule

If two triangles share two equal angles, the third must also match (angles always add to 180°). | Two matching angles is enough to prove triangles are similar. | This is called the Angle-Angle (AA) similarity rule.

Function notation — what f(x) means

f(x) means "the output of function f when the input is x." | It is NOT f times x — the parentheses mean INPUT, not multiplication. | Example: if f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11

Function notation — evaluating

Plug the input number in everywhere you see x, then simplify. | Example: g(x) = x^2 − 5. Find g(3): g(3) = 3^2 − 5 = 9 − 5 = 4 | Watch out: if input is negative, use parentheses: g(−2) = (−2)^2 − 5 = 4 − 5 = −1

Function notation — domain and range

Domain = all valid INPUTS (x values). Range = all possible OUTPUTS (y values). | Example: f(x) = √x → domain is x ≥ 0 (can't square root a negative). | Example: f(x) = 1/x → domain is x ≠ 0 (can't divide by zero). | On HiSET: they may give a table or graph and ask which values are in the domain or range.