Must-know sheet

Precalculus must-know sheet

The function families, rules, unit-circle values, identities and modeling tools you should know cold for AP Precalculus. The real exam gives you no formula sheet, so all of this has to be memorized; Unit 4 is included for classes that teach it, but it isn't on the AP exam.

Functions and rates of change

Unit 1

A function gives each input exactly one output: The domain is the set of allowed inputs and the range is the set of outputs. A zero is an input where f(x) = 0, which is an x-intercept of the graph.
Increasing and decreasing: f is increasing on an interval if bigger inputs always give bigger outputs there, and decreasing if bigger inputs give smaller outputs. A positive rate of change means increasing; a negative rate of change means decreasing.
Average rate of change on [a, b] = (f(b) − f(a)) / (b − a): It's the slope of the secant line through (a, f(a)) and (b, f(b)). Its units are output units per input unit, such as meters per second.
Rate of change at a point: Estimate it with average rates of change over smaller and smaller intervals around the point. It tells you how fast, and which way, the output is changing at that exact input.
Concave up: the rate of change is increasing. Concave down: the rate of change is decreasing: From a table with equal input steps, concave up means the average rates of change over consecutive intervals keep getting bigger; concave down means they keep getting smaller. A concave up graph bends upward like a cup.
Point of inflection: A point where the graph changes concavity, which is where the rate of change switches from increasing to decreasing or the reverse.
Relative (local) max and min: A relative max is where f changes from increasing to decreasing (the rate of change goes from positive to negative); a relative min is the reverse. An absolute (global) max or min is the greatest or least output on the whole domain.
Linear: constant rate of change: A linear function has the same average rate of change, its slope m, on every interval. Equal input steps give equal output differences.
Quadratic: the rate of change changes at a constant rate: Over consecutive equal-length intervals, the average rates of change of a quadratic change by the same amount each time, so second differences are constant. f(x) = ax² + bx + c is concave up everywhere if a > 0 and concave down everywhere if a < 0.

Polynomial functions

Unit 1

p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ with aₙ ≠ 0: The exponents are whole numbers. The degree is n, the leading term is aₙxⁿ, and the leading coefficient is aₙ.
Turning points and inflection points: A degree-n polynomial has at most n − 1 relative extrema and at most n − 2 points of inflection. Between any two distinct real zeros there is at least one relative max or min.
Absolute extrema by degree: An even-degree polynomial has an absolute max (if aₙ < 0) or an absolute min (if aₙ > 0). An odd-degree polynomial has neither, because its outputs go to both ∞ and −∞.
Zero ⇔ factor: a is a zero of p exactly when (x − a) is a factor of p(x). A real zero is an x-intercept. Every odd-degree polynomial with real coefficients has at least one real zero.
A degree-n polynomial has exactly n complex zeros, counting multiplicity: Some zeros may be non-real, like 2 + 3i. When the coefficients are real, non-real zeros come in conjugate pairs: if 2 + 3i is a zero, so is 2 − 3i.
Multiplicity: odd crosses, even touches: If (x − a) appears an odd number of times, the graph crosses the x-axis at a and the sign changes. If it appears an even number of times, the graph touches the axis and turns back, with no sign change. Multiplicity 2 or more also flattens the graph near a.
Even function: f(−x) = f(x). Odd function: f(−x) = −f(x): An even function's graph is symmetric over the y-axis; an odd function's graph is symmetric about the origin. A polynomial with only even powers (a constant counts as x⁰) is even; one with only odd powers is odd.
End behavior comes from the leading term: Even degree, aₙ > 0: up on both ends. Even degree, aₙ < 0: down on both ends. Odd degree, aₙ > 0: down on the left, up on the right. Odd degree, aₙ < 0: up on the left, down on the right.
Limit notation for end behavior: lim (x→∞) p(x) = ∞ means outputs grow without bound as x increases without bound. For example, for p(x) = −2x³ + x, lim (x→∞) p(x) = −∞ and lim (x→−∞) p(x) = ∞.
Sign chart: List the real zeros in order, then test one input in each interval between them to see where p(x) is positive or negative. The sign can only change at a zero of odd multiplicity.
Binomial theorem: (a + b)ⁿ uses row n of Pascal's triangle: The coefficients of (a + b)ⁿ are row n (counting the top 1 as row 0): 1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1. The powers of a count down from n while the powers of b count up from 0, so (x + 2)³ = x³ + 6x² + 12x + 8.
Polynomial long division: p(x) = d(x)·q(x) + r(x): The remainder r(x) has a smaller degree than the divisor d(x). Dividing by (x − a) leaves the remainder p(a), so a remainder of 0 means (x − a) is a factor.

Rational functions

Unit 1

r(x) = p(x) / q(x), with p and q polynomials: The domain is every real number except the zeros of q. Factor both parts first: factored form shows zeros, vertical asymptotes and holes at a glance.
End behavior: compare the leading terms: Far to the left and right, r(x) acts like (leading term of p) ÷ (leading term of q). Check what that quotient does as x → ±∞.
Degree of top < degree of bottom: horizontal asymptote y = 0: The outputs shrink toward 0 as x → ±∞.
Degrees equal: horizontal asymptote y = (leading coefficient of p) / (leading coefficient of q): For example, r(x) = (6x² − 1)/(3x² + x) has horizontal asymptote y = 2.
Degree of top is exactly one more: slant asymptote: Divide p by q with long division; the line y = quotient (ignore the remainder) is the slant asymptote. If the top's degree is two or more higher, there is no horizontal or slant asymptote, and the graph acts like the quotient of the leading terms.
A graph can cross a horizontal or slant asymptote: Those asymptotes only describe what happens far to the left and right. A graph never crosses a vertical asymptote, because the function is undefined there.
Zeros of r: zeros of the numerator that are in the domain: r(x) = 0 where p(x) = 0 and q(x) ≠ 0. A value that makes both zero is a hole or an asymptote, not a zero.
Vertical asymptote at x = a: Happens when a is a zero of q and (x − a) appears more times in q than in p, so it doesn't fully cancel. Near it the outputs grow without bound, written with one-sided limits such as lim (x→a⁺) r(x) = −∞.
Which way it goes near a vertical asymptote: Test the sign of r(x) just left and just right of a. If the leftover (x − a) factor in the bottom has odd multiplicity, the two sides go opposite ways; if even, both go the same way.
Hole at x = c: Happens when (x − c) appears at least as many times in p as in q, so it cancels completely from the bottom. Cancel, plug c into the simplified function to get L, and the hole is at (c, L); lim (x→c) r(x) = L even though r(c) is undefined.
Sign chart for a rational function: Mark the zeros of the numerator and of the denominator on a number line, then test each interval. The sign can change at either kind of point.

Transformations

Units 1, 2, 3

g(x) = f(x) + k: vertical shift: Moves the graph up k units (down if k < 0). This is an additive change to the output, so the range shifts by k.
g(x) = f(x + h): horizontal shift: Moves the graph LEFT h units if h > 0, right if h < 0. So f(x − 3) moves f right 3. The domain shifts too.
g(x) = a·f(x): vertical dilation: Stretches or shrinks the graph vertically by a factor of |a|. If a < 0 it also reflects over the x-axis.
g(x) = f(bx): horizontal dilation: Stretches or shrinks the graph horizontally by a factor of 1/|b|, so b = 2 squeezes it to half as wide. If b < 0 it also reflects over the y-axis.
g(x) = a·f(b(x − h)) + k moves the point (x₀, y₀) to (x₀/b + h, a·y₀ + k): Factor out b before reading the shift: f(2x + 6) = f(2(x + 3)) is a shift left 3, not 6. Use the point rule to find the new domain, range, zeros, asymptotes and extrema.
Parent functions to know: x, x², x³, √x, 1/x, |x|, bˣ, log_b x, sin x, cos x and tan x. Know each one's shape, domain, range, key points and asymptotes so you can transform it.

Sequences and exponential functions

Unit 2

Arithmetic sequence: aₙ = a₀ + d·n, or aₙ = aₖ + d(n − k): Add the common difference d each step. From two terms, d = (aₙ − aₖ)/(n − k). An arithmetic sequence is a linear function with whole-number inputs.
Geometric sequence: gₙ = g₀·rⁿ, or gₙ = gₖ·r^(n − k): Multiply by the common ratio r each step; r = (next term)/(term). A geometric sequence is an exponential function with whole-number inputs.
Linear vs exponential: Over equal-length input steps, a linear function adds the same amount (equal differences) and an exponential function multiplies by the same factor (equal ratios, or proportional change).
Line through two points: f(x) = y₁ + m(x − x₁), with m = (y₂ − y₁)/(x₂ − x₁): This is point-slope form. Slope-intercept form is f(x) = b + mx.
Exponential through two points: f(x) = y₁·r^(x − x₁), with r = (y₂/y₁)^(1/(x₂ − x₁)): Or set up a·b^(x₁) = y₁ and a·b^(x₂) = y₂ and divide the equations to get b. Example: f(1) = 150 and f(4) = 1200 give b³ = 8, so b = 2 and a = 75.
f(x) = a·bˣ with a ≠ 0, b > 0, b ≠ 1: a is the initial value f(0) and b is the growth factor per 1-unit step. Domain: all real numbers. With a > 0 the range is y > 0, the function grows if b > 1 and decays if 0 < b < 1, and it is always concave up.
Exponential graphs: no extrema, no inflection points, horizontal asymptote y = 0: For b > 1, lim (x→−∞) a·bˣ = 0; for 0 < b < 1, lim (x→∞) a·bˣ = 0. Adding k moves the asymptote to y = k, so a·bˣ + k approaches k.
Percent change: b = 1 + r for growth, b = 1 − r for decay: r is the percent as a decimal. Growing 3.5% per year gives b = 1.035; losing 12% per hour gives b = 0.88. A bank rate r compounded n times a year gives P(1 + r/n)^(nt) after t years.
Half-life h: A(t) = A₀·(1/2)^(t/h). Doubling time d: A(t) = A₀·2^(t/d): The amount halves (or doubles) every h (or d) units of time.
Natural base e ≈ 2.718: A(t) = A₀·e^(kt): Used for continuous growth (k > 0) or decay (k < 0). Since e^(kt) = (eᵏ)ᵗ, it's an ordinary exponential with b = eᵏ.
Changing the time unit: A factor of b per year is a factor of b^(1/12) per month and b¹⁰ per decade; you change the exponent, not the percent. So 1.035 per year is about 1.002871 per month, not 1 + 0.035/12.

Exponent and log rules

Unit 2

bᵐ·bⁿ = b^(m+n) bᵐ/bⁿ = b^(m−n) (bᵐ)ⁿ = b^(mn): Same base: add exponents to multiply, subtract to divide, multiply exponents for a power of a power. Also (ab)ⁿ = aⁿbⁿ.
b⁰ = 1 b^(−n) = 1/bⁿ b^(1/k) = ᵏ√b b^(m/k) = (ᵏ√b)ᵐ: A negative exponent means a reciprocal; a fractional exponent means a root. These hold for b > 0.
b^(x+k) = bᵏ·bˣ: A horizontal shift of an exponential is the same as a vertical dilation: 2^(x+3) = 8·2ˣ.
b^(cx) = (bᶜ)ˣ: A horizontal dilation of an exponential is the same as changing the base: 4^(x/2) = 2ˣ.
log_b c = a means bᵃ = c: A log is an exponent: it answers "b to what power gives c?" The base must satisfy b > 0 and b ≠ 1, and you can only take the log of a positive number. log x means base 10 (common log) and ln x means base e (natural log).
log_b 1 = 0 log_b b = 1 log_b(bˣ) = x b^(log_b x) = x: The last two say log_b and bˣ undo each other: ln(eˣ) = x and e^(ln x) = x for x > 0.
Product: log_b(xy) = log_b x + log_b y: For x, y > 0. On a graph, log_b(kx) = log_b k + log_b x, so a horizontal dilation of a log function is the same as a vertical shift.
Quotient: log_b(x/y) = log_b x − log_b y: For x, y > 0.
Power: log_b(xⁿ) = n·log_b x: For x > 0. On a graph, raising the input to a power is the same as a vertical dilation of the log function.
Change of base: log_b x = ln x / ln b = log x / log b: Use it to evaluate any log on a calculator. It also shows that every log function is a vertical dilation of every other.
Rules that do NOT exist: log(x + y) ≠ log x + log y, log x / log y ≠ log(x/y), (log x)ⁿ ≠ n·log x, and (a + b)ⁿ ≠ aⁿ + bⁿ.

Composition, inverses and log functions

Unit 2

f(g(x)): apply g first, then f: The output of g becomes the input of f, so x must be in g's domain and g(x) must be in f's domain. Order matters: f(g(x)) and g(f(x)) are usually different.
The identity function f(x) = x: Composing with it changes nothing: if i(x) = x, then f(i(x)) = f(x) and i(f(x)) = f(x). If f(g(x)) = x and g(f(x)) = x on their domains, f and g are inverses.
Inverse function f⁻¹: f⁻¹ undoes f: if f(a) = b then f⁻¹(b) = a. Its graph is f's graph reflected over y = x, its domain is f's range and its range is f's domain. To find it, swap x and y and solve for y.
Only one-to-one functions have inverses: Each output must come from exactly one input (the horizontal line test). If f isn't one-to-one, restrict its domain to a piece where it is, such as x ≥ 0 for x². Note f⁻¹(x) is not 1/f(x).
bˣ and log_b x are inverses: Input-output pairs swap: (0, 1) and (1, b) on bˣ become (1, 0) and (b, 1) on log_b x. The exponential's horizontal asymptote y = 0 becomes the log's vertical asymptote x = 0.
f(x) = a·log_b x: domain x > 0, range all real numbers: Vertical asymptote x = 0, no horizontal asymptote, no extrema, no inflection point. With a > 0 and b > 1 it increases without bound, very slowly, and is concave down.
Logs turn equal ratios into equal differences: When inputs are multiplied by the same factor k, the outputs of log_b x change by the same amount, log_b k. That's the opposite of an exponential, where equal input differences give equal output ratios.
Inverse of f(x) = a·b^(x − h) + k is f⁻¹(x) = log_b((x − k)/a) + h: Undo the steps in reverse order: subtract k, divide by a, take log_b, add h. The domain of the inverse is the range of f.

Solving exponential and log equations

Unit 2

Exponential equation: isolate the power, then take a log of both sides: 3·2ˣ = 24 gives 2ˣ = 8, so x = 3. For 5ˣ = 12, x = log₅ 12 = ln 12 / ln 5 ≈ 1.544. If both sides can be written with the same base, set the exponents equal instead.
Log equation: combine into one log, then rewrite in exponential form: log₂ x + log₂(x − 2) = 3 becomes log₂(x(x − 2)) = 3, so x² − 2x = 8 and x = 4 or x = −2.
Check for extraneous solutions: Every log's argument must be positive in the ORIGINAL equation. In the example above, x = −2 makes log₂ x undefined, so the only solution is x = 4.
Exponential and log inequalities: Solve the matching equation first, then use the graph or a test value. Applying bˣ or log_b with b > 1 keeps the inequality direction; with 0 < b < 1 it flips. Remember the log's domain restriction.
Give exact answers when asked: On the two no-calculator free-response questions (Part B), leave answers like ln 7 / ln 3 or log₃ 7 unless told to approximate.

Choosing, building and checking models

Units 1, 2, 3

Equal input steps, equal output differences: linear: For equally spaced inputs: constant first differences point to linear, constant second differences to quadratic, constant third differences to cubic.
Equal input steps, equal output ratios: exponential: Successive outputs are proportional. Sometimes you must subtract a constant first, such as room temperature for a cooling drink.
Inputs multiply by a constant, outputs add a constant: logarithmic: Logarithmic models fit quantities that grow fast at first and then much more slowly.
Outputs repeat: sinusoidal (periodic): Use a sine or cosine model for tides, seasons, daylight hours, Ferris wheels and anything else that cycles.
Inversely proportional: rational: y = k/x when y is inversely proportional to x, and y = k/x² for an inverse-square law.
Context clues: Area contexts often give quadratics and volume contexts cubics. Situations with distinct phases call for a piecewise-defined function. n + 1 points with different inputs determine a polynomial of degree at most n.
Calculator regressions to know: Linear, quadratic, cubic, quartic, exponential (y = a·bˣ), logarithmic (y = a + b·ln x) and sinusoidal. Use radian mode, store the full equation and use it unrounded, and report constants to three decimal places.
Residual = actual value − predicted value: A positive residual means the model underestimated; a negative residual means it overestimated.
A good model's residual plot shows no pattern: Residuals scattered randomly around 0 support the model. A curved or other clear pattern means a different function type would fit better.
State assumptions and restrictions: Say what the model assumes stays constant (like a steady percent rate), and restrict the domain and range to what makes sense: no negative time, whole numbers for counts, and caution outside the data's input range.
Semi-log plot: exponential data look linear: With the y-axis on a log scale, y = a·bˣ becomes log y = log a + x·log b: a line with slope log b and intercept log a. If a line fits (x, log_n y) with log_n y = mx + c, then y = nᶜ·(nᵐ)ˣ.
Answer in context with units: Say what an average rate of change means: "the population grew by an average of 1.2 thousand people per year from 2010 to 2020."

Radians and the unit circle

Unit 3

Radian measure: θ = arc length / radius, so s = rθ: One radian cuts off an arc as long as the radius. A full turn is 2π radians = 360°; π radians = 180°. Multiply degrees by π/180 to get radians.
Standard position and coterminal angles: The vertex is at the origin and the initial ray is on the positive x-axis. Positive angles turn counterclockwise, negative angles clockwise. θ and θ + 2πk (k an integer) end on the same ray.
On a circle of radius r: sin θ = y/r, cos θ = x/r, tan θ = y/x: On the unit circle, the point at angle θ is (cos θ, sin θ), and tan θ = sin θ / cos θ is the slope of the terminal ray (undefined when x = 0). On a circle of radius r the point is (r cos θ, r sin θ).
Special right triangles: 30°-60°-90° sides are in the ratio 1 : √3 : 2. 45°-45°-90° sides are in the ratio 1 : 1 : √2.
0 (0°): cos = 1, sin = 0, tan = 0: The point (1, 0).
π/6 (30°): cos = √3/2, sin = 1/2, tan = √3/3: The point (√3/2, 1/2). Note √3/3 = 1/√3.
π/4 (45°): cos = √2/2, sin = √2/2, tan = 1: The point (√2/2, √2/2).
π/3 (60°): cos = 1/2, sin = √3/2, tan = √3: The point (1/2, √3/2).
π/2, π, 3π/2: the points (0, 1), (−1, 0), (0, −1): tan π/2 and tan 3π/2 are undefined; tan π = 0. At 2π you are back to (1, 0).
Quadrant II: 2π/3, 3π/4, 5π/6: (−1/2, √3/2), (−√2/2, √2/2), (−√3/2, 1/2). Only sine is positive here.
Quadrant III: 7π/6, 5π/4, 4π/3: (−√3/2, −1/2), (−√2/2, −√2/2), (−1/2, −√3/2). Only tangent is positive here.
Quadrant IV: 5π/3, 7π/4, 11π/6: (1/2, −√3/2), (√2/2, −√2/2), (√3/2, −1/2). Only cosine is positive here.
Reference angles: The acute angle back to the x-axis: π − θ in quadrant II, θ − π in quadrant III, 2π − θ in quadrant IV. The trig values match the reference angle's in size; the quadrant sets the sign.
Symmetry rules: sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ, sin(π − θ) = sin θ and cos(π − θ) = −cos θ.

Trig graphs and sinusoidal functions

Unit 3

Periodic: f(x + k) = f(x) for all x: The period is the smallest such positive k. Measure it from max to the next max (not max to min, which is half a period). Every feature repeats each period.
y = sin θ: Domain all reals, range [−1, 1], period 2π. Zeros at θ = kπ, max 1 at π/2 + 2πk, min −1 at 3π/2 + 2πk. It is an odd function.
y = cos θ: Domain all reals, range [−1, 1], period 2π. Zeros at θ = π/2 + kπ, max 1 at 2πk, min −1 at π + 2πk. It is an even function, and cos θ = sin(θ + π/2): the cosine graph is the sine graph shifted left π/2.
f(θ) = a·sin(b(θ + c)) + d (same for cos): Amplitude |a|, period 2π/|b|, frequency |b|/(2π), phase shift c units left (right if c < 0), midline y = d. Max is d + |a| and min is d − |a|. If a < 0, the graph is reflected over the midline.
Building a sinusoid from data or a context: Amplitude = (max − min)/2, midline d = (max + min)/2, b = 2π/period. Use cos if the cycle starts at a max, −cos if it starts at a min, and sin if it starts on the midline going up.
Concavity and rate of change of a sinusoid: It is concave down above the midline and concave up below it, with inflection points on the midline. It changes fastest at the midline crossings and slowest (rate 0) at maxes and mins.
y = tan θ: Period π, range all reals, zeros at θ = kπ, vertical asymptotes at θ = π/2 + kπ. It increases between each pair of asymptotes, concave down left of each zero and concave up right of it. f(θ) = a·tan(b(θ + c)) + d has period π/|b|.
csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ = cos θ/sin θ: Each has vertical asymptotes where its partner is 0: csc and cot at θ = kπ, sec at θ = π/2 + kπ. csc and sec have period 2π and range y ≤ −1 or y ≥ 1; cot has period π and range all reals.

Inverse trig and trig equations

Unit 3

arcsin x: inputs [−1, 1], outputs [−π/2, π/2]: Returns the angle in quadrant I or IV (or on the axis) whose sine is x. Also written sin⁻¹ x, which is NOT 1/sin x.
arccos x: inputs [−1, 1], outputs [0, π]: Returns the angle in quadrant I or II (or on the axis) whose cosine is x.
arctan x: inputs all reals, outputs (−π/2, π/2): Returns the angle in quadrant I or IV whose tangent is x. Its graph has horizontal asymptotes y = −π/2 and y = π/2.
Inverse compositions: sin(arcsin x) = x for x in [−1, 1], but arcsin(sin θ) = θ only when θ is in [−π/2, π/2]. Likewise arccos(cos θ) = θ only for θ in [0, π].
Solving: isolate, find solutions in one period, then add the period: sin θ = k has θ = arcsin k and π − arcsin k in each cycle; cos θ = k has θ = ±arccos k; tan θ = k has θ = arctan k. Add 2πn (n any integer) to sine and cosine solutions (πn for tangent) to get all of them.
Inside angle bθ: solve for bθ first, over a longer interval: For sin(2θ) = 1/2 on [0, 2π), 2θ ranges over [0, 4π): 2θ = π/6, 5π/6, 13π/6, 17π/6, so θ = π/12, 5π/12, 13π/12, 17π/12.
Factor; don't divide by a trig function: For sin θ cos θ = sin θ, write sin θ(cos θ − 1) = 0 to keep the solutions where sin θ = 0. Treat 2sin²θ − sin θ − 1 = 0 like a quadratic in sin θ.
Trig inequalities: Solve the matching equation, then use the graph or unit circle to pick the intervals where the inequality holds.

Trig identities

Unit 3

sin²θ + cos²θ = 1: Comes from x² + y² = 1 on the unit circle. Use it to find one value from another: cos θ = ±√(1 − sin²θ), with the sign set by the quadrant.
1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ: Divide sin²θ + cos²θ = 1 by cos²θ or by sin²θ.
sin(α + β) = sin α cos β + cos α sin β: Difference: sin(α − β) = sin α cos β − cos α sin β.
cos(α + β) = cos α cos β − sin α sin β: Difference: cos(α − β) = cos α cos β + sin α sin β. The sign in the middle is the opposite of the sign inside.
sin 2θ = 2 sin θ cos θ: Set α = β = θ in the sine sum identity.
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ: Pick the form that leaves only one trig function in your equation.
Verifying an identity: Start with the more complicated side and change it into the other side step by step: rewrite everything in sine and cosine, use a Pythagorean identity, factor or combine fractions. Half-angle and tangent sum identities aren't in the course.

Polar coordinates and polar graphs

Unit 3

(r, θ): signed distance r along the ray at angle θ: If r < 0, go |r| units in the opposite direction, so (−r, θ) is the same point as (r, θ + π). Every point has infinitely many polar names.
Polar to rectangular: x = r cos θ, y = r sin θ: Works for negative r too.
Rectangular to polar: r = √(x² + y²), tan θ = y/x: arctan(y/x) only gives quadrants I and IV, so if x < 0 use θ = arctan(y/x) + π. For (−1, 1): r = √2 and θ = 3π/4.
Complex number a + bi is the point (a, b): Real part on the horizontal axis, imaginary part on the vertical. In polar form, a + bi = r cos θ + i·r sin θ = r(cos θ + i sin θ), with r = √(a² + b²).
Graph r = f(θ) on regular (θ, r) axes first: It shows where r is positive, negative, zero and largest. r = 0 means the curve passes through the origin; negative r draws on the opposite side of the origin; the largest |r| is the farthest point.
Circles: r = a, r = a cos θ, r = a sin θ: r = a is a circle of radius |a| centered at the origin. r = a cos θ and r = a sin θ are circles of diameter |a| through the origin, centered at (a/2, 0) and (0, a/2). θ = constant is a line through the origin.
Rose: r = a cos(nθ) or r = a sin(nθ), n a whole number ≥ 2: Petal length |a|. n odd gives n petals; n even gives 2n petals.
Limaçon: r = a + b cos θ or r = a + b sin θ: With a and b nonzero: |a| < |b| gives an inner loop. |a| = |b|: cardioid (touches the origin). |a| > |b|: no loop. The farthest point is |a| + |b| from the origin. The cos versions are symmetric over the x-axis, the sin versions over the y-axis.
Moving toward or away from the origin: The distance from the origin is |r|. If r > 0 and increasing, or r < 0 and decreasing, the point moves away. If r > 0 and decreasing, or r < 0 and increasing, it moves closer.
Relative extrema of r: Where r switches between increasing and decreasing, the point is locally farthest from or closest to the origin. Careful when r < 0: a relative min of r is then a locally farthest point.
Average rate of change of r on [θ₁, θ₂] = (r(θ₂) − r(θ₁))/(θ₂ − θ₁): Units are radius units per radian. Use it to estimate r between known values, the same way you would for any function.

Unit 4 (taught in some classes, not on the AP exam)

Unit 4

Parametric function: (x(t), y(t)): Plot points in order of increasing t to trace the path and its direction. x increasing means moving right; y increasing means moving up. Slope of the secant between two times = Δy/Δx = (average rate of y)/(average rate of x).
Parametric circle and segment: x = h + r cos t, y = k + r sin t traces the circle of radius r centered at (h, k) counterclockwise. x = x₁ + (x₂ − x₁)t, y = y₁ + (y₂ − y₁)t for 0 ≤ t ≤ 1 traces the segment from (x₁, y₁) to (x₂, y₂).
Conic sections: Parabola: y − k = a(x − h)² or x − h = a(y − k)². Ellipse: (x − h)²/a² + (y − k)²/b² = 1. Hyperbola: (x − h)²/a² − (y − k)²/b² = 1 opens left and right; (y − k)²/b² − (x − h)²/a² = 1 opens up and down.
Parametrizing ellipses and hyperbolas: Ellipse: x = h + a cos t, y = k + b sin t. Hyperbola opening left and right: x = h + a sec t, y = k + b tan t.
Vector ⟨a, b⟩: magnitude √(a² + b²): Add component by component and multiply each component by a scalar. A unit vector in the direction of v is v/|v|. A vector of length m at angle θ is ⟨m cos θ, m sin θ⟩.
Dot product: u·v = u₁v₁ + u₂v₂ = |u||v| cos θ: Use it to find the angle θ between two vectors. A dot product of 0 means the vectors are perpendicular.
Matrix multiplication: An m × n matrix times an n × p matrix gives an m × p matrix; each entry is a row of the first times a column of the second. Order matters: AB is usually not BA.
2 × 2 determinant and inverse: For [a b; c d], det = ad − bc and |det| is the area of the parallelogram formed by the two row (or column) vectors. The inverse exists only if det ≠ 0 and equals (1/(ad − bc))·[d −b; −c a].
Linear transformations: Every linear transformation of the plane is multiplication by a 2 × 2 matrix, and it keeps the origin fixed. The matrix's columns are where ⟨1, 0⟩ and ⟨0, 1⟩ land. Rotation by θ counterclockwise: [cos θ −sin θ; sin θ cos θ]. |det| is the area scale factor.
Composing and undoing transformations: Doing B then A is the matrix product AB. The inverse matrix undoes the transformation.
Transition matrices: Multiplying a state vector by the transition matrix gives the next state; repeated multiplication predicts later states, and the inverse matrix estimates earlier ones.