Math 30-1 / Recall practice

Flashcards

Read the term. Recall the rule, decide when you would use it, and write the formula before revealing the back.

Transformations and function operations

Domain

Rule: Keep every input for which the original expression exists.
Use when: You combine functions, take a root or simplify a denominator.
Formula or notation: D(f/g) = D(f) ∩ D(g), excluding g(x) = 0

Range

Rule: Collect the outputs actually attained. Check endpoints and excluded values.
Use when: You describe a graph or find the domain of its inverse.
Formula or notation: range(f) = {f(x) : x ∈ D(f)}

Parent function

Rule: Start from the simplest graph in the family and identify its anchor points.
Use when: You need to sketch a related curve without a table of many values.
Formula or notation: y = f(x)

Function operations

Rule: Combine outputs at the same allowed input.
Use when: A question asks for a sum, difference, product or quotient of graphs.
Formula or notation: (f ± g)(x) = f(x) ± g(x), (fg)(x) = f(x)g(x), (f/g)(x) = f(x)/g(x)

Composition

Rule: Apply the inner function first. Its output must be in the outer domain.
Use when: One process takes another process’s output as its input.
Formula or notation: (f ∘ g)(x) = f(g(x))

Translation

Rule: Inside subtraction moves a graph right. Outside addition moves it up.
Use when: A curve retains its shape but its anchor point changes position.
Formula or notation: y = f(x − h) + k

Vertical stretch or compression

Rule: Multiply each original height by the outside factor. A negative factor also reflects across the x-axis.
Use when: You know how far each point moves from the x-axis.
Formula or notation: y = a f(x)

Horizontal stretch or compression

Rule: Divide each original x-coordinate by the inside factor. A negative factor also reverses left and right.
Use when: The input is multiplied before the function acts.
Formula or notation: y = f(bx), b ≠ 0

Mapping rule

Rule: Transform an original point by undoing the inside operations and applying the outside operations.
Use when: Several transformations occur together or you must reconstruct a point.
Formula or notation: y = a f(b(x − h)) + k maps (u,v) → (u/b + h, av + k), b ≠ 0

Axis reflections

Rule: Reflect the output for the x-axis and the input for the y-axis.
Use when: You need to distinguish an outside minus sign from an inside one.
Formula or notation: y = −f(x) or y = f(−x)

Invariant point

Rule: Check which original points keep their coordinates after a transformation.
Use when: A graph and its image touch the line of reflection at a fixed point.
Formula or notation: Reflection across y = x fixes (t,t)

Inverse relation

Rule: Interchange input and output coordinates, then solve for the new output if possible.
Use when: You reverse a process or reflect a graph across y = x.
Formula or notation: (u,v) → (v,u)

One-to-one function

Rule: Each output comes from one input. Restrict the domain to one branch when needed.
Use when: You want the inverse relation to be a function.
Formula or notation: f⁻¹(f(x)) = x on the selected domain

Graphing window

Rule: Set horizontal and vertical limits and scale so the relevant features are visible.
Use when: An intercept or turning point is outside your calculator display.
Formula or notation: x: [min, max, scale], y: [min, max, scale]

Polynomial, radical and rational functions

Degree and leading coefficient

Rule: The largest exponent and its coefficient determine polynomial end behaviour.
Use when: You sketch the far left and far right of a polynomial.
Formula or notation: Leading term aₙxⁿ, aₙ ≠ 0

End behaviour

Rule: Even degree gives matching end directions. Odd degree gives opposite directions. The sign of the leading coefficient sets the right end.
Use when: You must choose among graphs with similar intercepts.
Formula or notation: As x → +∞ the sign of aₙxⁿ is the sign of aₙ

Zero and factor

Rule: A zero gives an x-intercept and a corresponding linear factor.
Use when: You translate between a polynomial equation, factors and a graph.
Formula or notation: P(r) = 0 ⇔ (x − r) is a factor

Multiplicity

Rule: A factor’s exponent determines whether the graph crosses or touches the axis at that zero.
Use when: A polynomial graph flattens at an intercept or turns back there.
Formula or notation: (x − r)ᵐ: odd m crosses, even m touches

Remainder theorem

Rule: Evaluate the polynomial at the zero of a monic linear divisor.
Use when: The question asks for a remainder or an unknown coefficient.
Formula or notation: P(x) = (x − r)Q(x) + P(r)

Polynomial division

Rule: Divide out a known factor to reduce the degree, then factor or solve the quotient.
Use when: A cubic, quartic or quintic has a known root.
Formula or notation: P(x) = D(x)Q(x) + R(x), degree R < degree D for nonzero R

Factor by grouping

Rule: Rearrange terms into groups with the same factor, then factor again.
Use when: Several terms repeat a binomial or a quadratic pattern.
Formula or notation: x³ + 2x² − x − 2 = (x + 2)(x − 1)(x + 1)

Turning points

Rule: The degree limits the number of local turns. A graph may have fewer.
Use when: You infer a minimum possible degree from a sketch.
Formula or notation: At most n − 1 turning points for degree n ≥ 1

Constructing a polynomial

Rule: Build a factor for each given root and use a known nonzero point to solve for the scale.
Use when: Intercepts, multiplicities and another point are provided.
Formula or notation: P(x) = a(x − r₁)ᵐ¹⋯(x − rₖ)ᵐᵏ when these account for the degree

Quadratic formula

Rule: Write the equation in standard form before substituting the signed coefficients.
Use when: A quadratic factor or trig substitution does not factor conveniently.
Formula or notation: x = (−b ± √(b² − 4ac))/(2a), a ≠ 0

Radical domain

Rule: A real square root requires a nonnegative radicand. The principal square root is nonnegative.
Use when: You find the starting point or allowed intervals of a radical graph.
Formula or notation: y = √f(x) requires f(x) ≥ 0

Square-root graph

Rule: Keep original nonnegative heights and take their square roots. Heights zero and one stay fixed.
Use when: You are given a graph of f and asked to sketch its square root.
Formula or notation: (x,y) → (x,√y) for y ≥ 0

Extraneous root

Rule: Check every candidate in the original equation after squaring or changing domains.
Use when: An algebraic step can introduce extra solutions.
Formula or notation: √u = v requires u ≥ 0 and v ≥ 0 before squaring

Rational restriction

Rule: Record excluded denominator zeros before cancelling any factors.
Use when: You simplify a quotient or identify breaks in its graph.
Formula or notation: f(x) = P(x)/Q(x), Q(x) ≠ 0

Removable discontinuity

Rule: After cancelling, evaluate the reduced expression at the excluded input if it is finite. Plot an open point.
Use when: A common factor disappears completely from the denominator.
Formula or notation: Hole at (r, reduced value at r)

Vertical asymptote

Rule: A denominator zero left after cancellation gives a vertical asymptote for a rational function.
Use when: Values grow in magnitude near an excluded input.
Formula or notation: x = r

Horizontal asymptote

Rule: Compare degrees. A smaller numerator degree gives zero. Equal degrees give the ratio of leading coefficients.
Use when: You describe the far ends of a rational graph. The curve can cross the horizontal asymptote at a finite input.
Formula or notation: y = 0 or y = aₙ/bₙ in these cases

Exponential and logarithmic functions

Exponential function

Rule: Equal input increments multiply the output by a constant factor for an unshifted exponential.
Use when: A table has a constant ratio at equally spaced inputs.
Formula or notation: y = Abˣ, A ≠ 0, b > 0, b ≠ 1

Growth or decay factor

Rule: Convert the fractional rate per interval to a multiplier.
Use when: A quantity increases or decreases by a fixed percentage each interval.
Formula or notation: Growth b = 1 + r, decay b = 1 − r

Repeated growth or decay

Rule: Divide elapsed time by the length of one growth interval.
Use when: The time unit and the stated doubling or decay period differ.
Formula or notation: y = A b^(t/p), p > 0

Half-life and doubling time

Rule: The multiplier is one half or two per stated period.
Use when: You know the period and need the amount or elapsed time.
Formula or notation: y = A(1/2)^(t/H) or y = A·2^(t/D)

Logarithm

Rule: A logarithm gives the exponent producing the argument.
Use when: The unknown is an exponent or you need to switch representations.
Formula or notation: log_b M = x ⇔ bˣ = M, M > 0, b > 0, b ≠ 1

Common logarithm

Rule: Use base ten to bring an exponent down with the power law.
Use when: You solve an exponential equation with unlike bases.
Formula or notation: log x = log₁₀ x

Product law

Rule: A logarithm of a product splits into a sum for positive factors.
Use when: Two logged factors with the same base appear in a sum.
Formula or notation: log_b(MN) = log_b M + log_b N

Quotient law

Rule: A logarithm of a quotient splits into a difference for positive arguments.
Use when: You condense a difference of same-base logarithms.
Formula or notation: log_b(M/N) = log_b M − log_b N

Power law

Rule: A factor multiplying a logarithm can become an exponent of its positive argument.
Use when: An exponent must come down or several log terms must be combined.
Formula or notation: log_b(Mᵖ) = p log_b M, M > 0

Change of base

Rule: Divide logs taken with the same new base.
Use when: Your calculator has no direct key for the requested base.
Formula or notation: log_b M = log₁₀ M / log₁₀ b

Logarithmic equation

Rule: Record each original argument’s positivity condition, condense terms, solve and test candidates.
Use when: A solution might make an original logarithm undefined.
Formula or notation: log_b U = log_b V ⇒ U = V, U,V > 0

Logarithmic scale

Rule: Read the sign and multiplier in the supplied model. A rise of 10 dB gives ten times the sound intensity. A rise of one pH unit divides hydrogen ion concentration by ten in the concentration model.
Use when: You translate a difference of readings into a ratio.
Formula or notation: ΔL = 10 log₁₀(I₂/I₁), ΔpH = −log₁₀([H⁺]₂/[H⁺]₁)

Inverse graphs

Rule: Exponential and logarithmic graphs with the same base reflect across y = x.
Use when: You match equations to asymptotes, domains and ranges.
Formula or notation: y = bˣ has asymptote y = 0, y = log_b x has asymptote x = 0

Angles, ratios and equations

Standard position

Rule: Put the vertex at the origin and the initial arm on the positive x-axis. Counterclockwise angles are positive.
Use when: You must locate the terminal arm before choosing a sign.
Formula or notation: One complete turn = 360° = 2π radians

Radians and degrees

Rule: Scale the measure by the ratio between a half-turn in the two units.
Use when: A question supplies degrees but the formula or interval uses radians.
Formula or notation: radians = degrees × π/180, degrees = radians × 180/π

Arc length

Rule: Multiply the radius by the angle in radians for a nonnegative central angle.
Use when: A wheel or circular path sweeps out an arc.
Formula or notation: s = rθ, θ = s/r

Coterminal angle

Rule: Add or subtract whole turns to keep the same terminal arm.
Use when: You need a representative inside a stated interval.
Formula or notation: θ + 2πk or θ + 360°k, k integer

Reference angle

Rule: For a non-quadrantal angle, use the acute angle between its terminal arm and the x-axis.
Use when: You know an acute-angle value but need the sign in a different quadrant.
Formula or notation: In quadrant II, α = π − θ for θ in [0,2π)

Unit circle

Rule: A point at angle θ has horizontal coordinate cosine and vertical coordinate sine.
Use when: You recover a missing coordinate using the quadrant.
Formula or notation: x² + y² = 1, (x,y) = (cos θ,sin θ)

Point on a terminal arm

Rule: Normalize the horizontal and vertical coordinates by the distance to the origin.
Use when: The given point lies on a circle with radius other than one.
Formula or notation: r = √(x²+y²), sin θ = y/r, cos θ = x/r, tan θ = y/x where x ≠ 0

Special angles

Rule: Read exact sine and cosine from the special triangles, then apply quadrant signs.
Use when: An exact answer is required for a multiple of 30° or 45°.
Formula or notation: sin(π/6)=1/2, sin(π/4)=√2/2, sin(π/3)=√3/2; cosine uses the reversed list

General sine solutions

Rule: Use the reference branch and its supplement, then add full turns. Remove duplicate angles.
Use when: You solve sin x = c with −1 ≤ c ≤ 1.
Formula or notation: x = α + 2πk or π − α + 2πk, α = arcsin c

General cosine solutions

Rule: Use both signs of the principal angle, then add full turns.
Use when: You solve cos x = c with −1 ≤ c ≤ 1.
Formula or notation: x = ±arccos c + 2πk

Tangent graph and solutions

Rule: Tangent repeats each half-turn, has every real output, and is undefined where cosine is zero.
Use when: You sketch y = tan x or solve a tangent equation.
Formula or notation: x = arctan c + πk for tan x = c; asymptotes x = π/2 + πk

Double-angle equation

Rule: Replace the doubled angle with an identity in the original angle. Factor or solve the resulting equation and check the interval.
Use when: A double angle appears alongside a ratio of the original angle.
Formula or notation: cos 2x = cos x becomes (2cos x + 1)(cos x − 1) = 0

Quadratic trig equation

Rule: Substitute for the trig ratio, solve the quadratic and recover every allowed angle.
Use when: The same ratio appears both squared and to the first power.
Formula or notation: Let u = sin x or cos x and keep −1 ≤ u ≤ 1

Sinusoidal graphs and models

Sinusoidal model

Rule: Identify the vertical scale, horizontal scale, starting position and midline from the situation.
Use when: A quantity oscillates regularly between a minimum and maximum.
Formula or notation: y = a sin(b(x − c)) + d or a cos(b(x − c)) + d

Amplitude

Rule: Take half the vertical distance between maximum and minimum.
Use when: Height, depth or temperature oscillates around an average.
Formula or notation: A = (maximum − minimum)/2 = absolute value of a

Midline

Rule: Average the highest and lowest values.
Use when: A graph is centred above or below the horizontal axis.
Formula or notation: d = (maximum + minimum)/2

Period

Rule: Measure the distance between matching points travelling in the same direction.
Use when: You know a cycle length or need the input coefficient.
Formula or notation: P = 2π/abs(b) in radians or 360°/abs(b) in degrees

Frequency

Rule: Count complete cycles per unit of input.
Use when: The model gives a cycle length and asks for cycles per second or hour.
Formula or notation: frequency = 1/P

Phase shift

Rule: Read the horizontal displacement after factoring the input coefficient.
Use when: Use the time of a known maximum, minimum or directed midline crossing to position the model.
Formula or notation: bx − q = b(x − q/b), so c = q/b

Range of a wave

Rule: Move one amplitude above and below the midline.
Use when: You check if a requested height is reachable.
Formula or notation: d − A ≤ y ≤ d + A for the unrestricted periodic model

First time or next time

Rule: Find all candidate times in a cycle, then apply the physical interval and direction of motion.
Use when: A rider reaches the same height twice during one cycle.
Formula or notation: Successive cycles repeat at t + kP

Trigonometric identities

Identity and domain

Rule: A proof gives equal expressions at every input where both original sides are defined.
Use when: You must justify a statement for a variable angle.
Formula or notation: LHS = RHS on their common domain

Numerical verification

Rule: Substitute a permissible value into each side. A counterexample disproves a proposed identity.
Use when: You test a conjecture before deciding whether to prove it.
Formula or notation: A unequal to B at one allowed x rules out an identity

Reciprocal identities

Rule: Invert the matching ratio and retain every restriction.
Use when: Secant, cosecant or cotangent is stopping a cancellation.
Formula or notation: sec x = 1/cos x, csc x = 1/sin x, cot x = 1/tan x where both sides exist

Quotient identities

Rule: Rewrite tangent or cotangent as a ratio of the two coordinates.
Use when: A proof mixes tangent with sine and cosine.
Formula or notation: tan x = sin x/cos x, cot x = cos x/sin x on their domains

Pythagorean identities

Rule: Recast a sum or difference of squared ratios.
Use when: A squared term appears beside one or beside another squared ratio.
Formula or notation: sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = csc²x

Conjugate

Rule: Multiply numerator and denominator by the same nonzero conjugate to expose a difference of squares.
Use when: A denominator contains one plus or minus a trig ratio.
Formula or notation: (u + v)(u − v) = u² − v²

Sine sum and difference

Rule: Combine the cross-products, keeping the sign of the angle operation.
Use when: An angle splits into a sum or difference of known angles.
Formula or notation: sin(A ± B) = sin A cos B ± cos A sin B

Cosine sum and difference

Rule: Combine the like-products with the opposite sign between them.
Use when: You see two cosine factors and two sine factors.
Formula or notation: cos(A ± B) = cos A cos B ∓ sin A sin B

Tangent sum and difference

Rule: Build a quotient and check that all parts are defined.
Use when: The given information is in tangent ratios for two angles.
Formula or notation: tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)

Sine double angle

Rule: Replace a sine-cosine product or a doubled angle.
Use when: The angle in one term is twice the angle in another.
Formula or notation: sin(2x) = 2 sin x cos x

Cosine double angle

Rule: Choose the form containing the ratio already present.
Use when: You want to eliminate sine squared or cosine squared.
Formula or notation: cos(2x) = cos²x − sin²x = 2cos²x − 1 = 1 − 2sin²x

Tangent double angle

Rule: Use the tangent addition rule with equal angles, where both expressions exist.
Use when: You need tan(2x) from tan x.
Formula or notation: tan(2x) = 2tan x/(1 − tan²x)

Counting and the binomial theorem

Fundamental counting principle

Rule: Multiply the available choices at successive stages. Account for any earlier choice that changes later options.
Use when: A complete result requires choosing one item at every stage.
Formula or notation: total = n₁ × n₂ × ⋯ × nₖ for fixed stage counts

Disjoint cases

Rule: Split into non-overlapping cases and add their counts.
Use when: A restriction changes how many choices remain in different branches.
Formula or notation: total = count(case A) + count(case B) for disjoint cases

Factorial

Rule: Multiply the descending positive integers to count orders of distinct objects.
Use when: All n objects are arranged in a row.
Formula or notation: n! = n(n−1)⋯1 and 0! = 1

Permutation

Rule: Count ordered selections without replacement.
Use when: Jobs, positions or prizes differ and changing the order changes the outcome.
Formula or notation: nPr = n!/(n−r)!, 0 ≤ r ≤ n

Repeated objects

Rule: Divide by the permutations of each group of indistinguishable objects.
Use when: Rearranging identical letters would otherwise count the same string more than once.
Formula or notation: n!/(a!b!⋯)

Together or apart

Rule: Treat required neighbours as a block and count internal orders. For an exclusion, subtract the forbidden cases from the total.
Use when: Two named people must sit together or apart in a row.
Formula or notation: allowed = total − forbidden

Combination

Rule: Count unordered selections of distinct objects without replacement.
Use when: A group’s membership matters and its order does not.
Formula or notation: nCr = n!/[r!(n−r)!], 0 ≤ r ≤ n

At least or at most

Rule: Subtract all unwanted cases from the unrestricted total.
Use when: A crew needs at least one new member.
Formula or notation: count(at least one) = count(all crews) − count(crews with none)

Choosing from groups

Rule: Count the required selections inside each group and multiply. Add across permitted group compositions.
Use when: A committee must include specified numbers from two groups.
Formula or notation: C(m,i) × C(n,j) for disjoint groups

Pascal’s triangle

Rule: Add two adjacent coefficients to obtain the coefficient beneath them.
Use when: You want a row of binomial coefficients for a small exponent.
Formula or notation: C(n,k) = C(n−1,k−1) + C(n−1,k) for 0 < k < n

Binomial theorem

Rule: In each successive term, lower the first exponent and raise the second. Carry signs and coefficients with their terms.
Use when: You expand a positive integer power of a two-term expression.
Formula or notation: (u+v)ⁿ = Σ C(n,k)u^(n−k)vᵏ for k = 0,…,n

General binomial term

Rule: Set the exponent of the variable to the requested power, then find the matching index.
Use when: You need one coefficient or a constant term without expanding everything.
Formula or notation: Tₖ₊₁ = C(n,k)u^(n−k)vᵏ