Pre-Calculus · Units 1–9 · Quick reference
Pre-Calculus
Functions and transformations, inverses, polynomial, rational, exponential and log functions, conics, matrices, parametric equations, series, counting, limits and rates of change.
1Functions & Their Featuresone input, one output, and how the graph behaves
- Function
- each input has exactly one output
- Domain / range
- allowed inputs / outputs produced
- Zero
- an input with f(x) = 0 (an x-intercept)
- y-intercept
- the point (0, f(0))
- Basics
- vertical line test, evaluating, f + g: Algebra §10
Interval notation
- [a, b] includes the ends; (a, b) leaves them out
- ∞ always takes a parenthesis; ∪ joins pieces
EXx ≤ −1 or x > 3 → (−∞, −1] ∪ (3, ∞)
EX−2 < x ≤ 5 → (−2, 5]
Domain of √(x + 3)x − 2 Worked
Increasing, decreasing, extrema
- Increasing on an interval: x₁ < x₂ gives f(x₁) < f(x₂).
- A local max (min) is a peak (valley) of nearby values.
- Name intervals by their x-values, with open ends.
Symmetry
- Even: f(−x) = f(x), mirror in the y-axis.
- Odd: f(−x) = −f(x), half-turn about the origin.
EXx³ − 3x: f(−x) = −x³ + 3x = −f(x) → odd
Difference quotient
The slope between x and x + h; it becomes the derivative in §13.
f(x + h) − f(x)h, h ≠ 0
f(x) = x² − 3x Worked
Piecewise Worked
Average rate of change Worked
DR. ZIPS
Describe any graph in six moves: Domain, Range, Zeros, Intervals (up or down), Peaks and valleys, Symmetry.
2Parent Functions & Transformationsa few shapes, moved and stretched
| Parent | Domain | Range | Symmetry |
|---|---|---|---|
| x, x³, ∛x | all reals | all reals | odd |
| x², |x| | all reals | [0, ∞) | even |
| √x | [0, ∞) | [0, ∞) | none |
| 1x | x ≠ 0 | y ≠ 0 | odd |
| bˣ | all reals | (0, ∞) | none |
| logb x | (0, ∞) | all reals | none |
| Change | Effect on the graph |
|---|---|
| f(x) + k | up k (down if k < 0) |
| f(x − h) | right h (left if h < 0) |
| −f(x) | reflect over the x-axis |
| f(−x) | reflect over the y-axis |
| a · f(x) | vertical stretch by |a| |
| f(bx) | horizontal shrink by 1b |
Inside Opposite, Outside Obvious
Changes inside f( ) act on x and run backwards: f(x − 3) moves right, f(2x) shrinks. Changes outside do just what they say.
Point mapping for y = a·f(b(x − h)) + k
Move a few key points of the parent; draw through them.
(x, y) → (xb + h, ay + k)
g(x) = −2√(x + 3) + 1 Worked
Absolute value Worked
Step function
Greatest integer ⟦x⟧: the largest integer ≤ x.
EX⟦2.7⟧ = 2, ⟦−1.2⟧ = −2
3Composition & Inversesfunctions in a chain, and undoing them
Composition
(f ∘ g)(x) = f(g(x)): apply g first. The domain keeps x in the domain of g and g(x) in the domain of f.
Basic examples: Algebra §10.
f(x) = √x, g(x) = x² − 4 Worked
Decompose h(x) = (3x + 1)⁵ Worked
f AFTER g
Read f ∘ g as “f after g”: the function written nearest x acts first.
Operations & domain Worked
Inverse functions
- f⁻¹ exists only if f is one-to-one (passes the horizontal line test).
- f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
- Domain and range trade places; the graphs mirror in y = x.
Inverse of f(x) = 2x + 1x − 3 Worked
One-to-one? Worked
Swap, Solve, Check
To invert: Swap x and y, Solve for y, Check that f(f⁻¹(x)) = x.
Restrict, then invert Worked
Inverse from a table Worked
4Polynomial Functionsend behavior, zeros, division (factoring: Algebra §4)
| Leading term | a > 0 | a < 0 |
|---|---|---|
| even degree | ↖ ↗ both up | ↙ ↘ both down |
| odd degree | ↙ ↗ down, up | ↖ ↘ up, down |
Even arms Agree
Even degree: both arms point the same way. Odd degree: the arms split. The sign of the leading coefficient sets the right arm.
Zeros & multiplicity
- Factor (x − c)ᵐ: odd m crosses the x-axis, even m touches and turns.
- Degree n: at most n real zeros and at most n − 1 turning points.
Long division Worked
Remainder & factor theorems
- P(c) is the remainder of P(x) ÷ (x − c).
- P(c) = 0 exactly when (x − c) is a factor.
EXx⁴ − 3x² + 2x − 5 at 2: 16 − 12 + 4 − 5 = 3
Polynomial inequality Worked
Counting zeros
- Fundamental Theorem of Algebra: degree n has exactly n complex zeros, counting multiplicity.
- Conjugate pairs: with real coefficients, a + bi brings a − bi.
- Descartes' rule: sign changes in P(x) give the most positive zeros (or fewer by 2s); use P(−x) for negatives.
All zeros of x³ − x² + 4x − 4 Worked
Build from zeros Worked
Intermediate Value Theorem
If P is continuous and P(a), P(b) have opposite signs, P has a zero between a and b.
EXx³ − x − 1: P(1) = −1, P(2) = 5 → a zero in (1, 2), ≈ 1.3247
5Rational Functionsa polynomial over a polynomial
Happy Vets Help Injured Sparrows
Graph in order: Holes (canceled factors), Vertical asymptotes, Horizontal or slant asymptote, Intercepts, Sign chart.
Analyze f(x) = x² − x − 2x² − 4 Worked
Rational equation Worked
Horizontal asymptote (degrees n over d)
- n < d: y = 0 · n = d: ratio of leading coefficients · n > d: none (BOBO: Algebra §10)
- A graph may cross its HA; it never crosses a VA.
Slant asymptote
When n = d + 1, divide: the quotient line is the asymptote.
EXx² + 3x + 1x − 1 = x + 4 + 5x − 1 → y = x + 4
Near a vertical asymptote
- Odd multiplicity: the two sides go opposite ways (±∞).
- Even multiplicity: both sides go the same way.
Partial fractions
Split a proper fraction over its factored denominator: one constant per linear factor.
EX5x − 4x² − x − 2 = Ax − 2 + Bx + 1: x = 2 → A = 2; x = −1 → B = 3
Rational inequality x + 3x − 1 ≤ 0 Worked
6Exponential & Logarithmic Functionsgrowth, decay and the exponents that measure them
| y = bˣ | y = logb x | |
|---|---|---|
| domain | all reals | (0, ∞) |
| range | (0, ∞) | all reals |
| key point | (0, 1) | (1, 0) |
| asymptote | y = 0 | x = 0 |
Laws (b, M, N > 0)
- logb MN = logb M + logb N; logb MN = logb M − logb N
- logb Mᵖ = p logb M; logb x = ln xln b
- blogb x = x and logb bˣ = x
More examples: Algebra §10.
Logs step down
A log lowers each operation one step: × becomes +, ÷ becomes −, a power becomes a multiplier in front.
Exponential equations Worked
Log equation Worked
Quadratic in eˣ Worked
I Love Solving Carefully
Isolate the power or log, Log (or exponentiate) both sides, Solve, Check every answer in the original.
Growth & decay
k > 0 growth, k < 0 decay. Doubling time ln 2k; half-life h: A = A₀(½)t/h.
A = A₀ekt
Find k Worked
Half-life Worked
The number e
e = the limit of (1 + 1n)ⁿ ≈ 2.71828. Continuous growth: A = Pert.
EX$2,000 at 5% for 10 years: 2000e0.5 ≈ $3,297.44
Log scales
Each step of 1 is a factor of 10: pH = −log[H⁺], decibels, earthquake magnitude.
EX[H⁺] = 3.2 × 10⁻⁴ → pH ≈ 3.49
7Conic Sectionsfoci, eccentricity and general form (basic forms: Algebra §9)
| Conic (center or vertex at origin) | Focus relation | Eccentricity e | Defining distance |
|---|---|---|---|
| Parabola x² = 4py | focus (0, p), directrix y = −p | e = 1 | to focus = to directrix |
| Ellipse x²a² + y²b² = 1, a > b | c² = a² − b², foci (±c, 0) | e = ca < 1 | sum to foci = 2a |
| Hyperbola x²a² − y²b² = 1 | c² = a² + b², foci (±c, 0) | e = ca > 1 | difference = 2a |
| Circle | both foci at the center | e = 0 | to center = r |
Ellipse from general form Worked
Parabola y² = 12x Worked
Hyperbola x²9 − y²16 = 1 Worked
H is for Hypotenuse
In a hyperbola, c is the longest, like a hypotenuse: c² = a² + b². In an ellipse, a is the longest: c² = a² − b².
Classify Ax² + Bxy + Cy² + Dx + Ey + F = 0
- B = 0: AC = 0 parabola; AC > 0 ellipse (circle if A = C); AC < 0 hyperbola.
- Any B: B² − 4AC < 0 ellipse, = 0 parabola, > 0 hyperbola.
Earth’s orbit Worked
8Systems & Matricessolve many equations at once (2×2 basics: Algebra §6)
Nonlinear system Worked
Matrix arithmetic
- Size m × n = rows × columns. Add same-size matrices entry by entry.
- AB needs columns of A = rows of B; the result is rows(A) × cols(B).
- Row of A times column of B, summed. In general AB ≠ BA.
EX1234201−1 = 4−210−4
2 × 2 determinant & inverse
For A = abcd, det A = ad − bc. A⁻¹ exists only if det A ≠ 0.
A⁻¹ = 1ad − bcd−b−ca
Swap, Negate, Divide
For a 2 × 2 inverse: Swap the main diagonal, Negate the other diagonal, Divide everything by the determinant.
Solve with A⁻¹ Worked
Cramer's rule
x = DxD, y = DyD: replace that variable’s column with the constants.
EXsame system: D = 1, Dx = 10 − 8 = 2, Dy = 16 − 15 = 1 → (2, 1)
3 × 3 determinant (row 1)
a(ei − fh) − b(di − fg) + c(dh − eg)
EXthe coefficient matrix of the elimination system at right: 1(1 − 2) − 1(−2 − 1) + 1(4 + 1) = 7
Row operations
- Swap two rows · multiply a row by a nonzero number · add a multiple of one row to another.
- Aim for a staircase (row-echelon form), then back-substitute.
3 × 3 by elimination Worked
No unique answer
- A row 0 0 0 | c with c ≠ 0: no solution.
- A row of all zeros: infinitely many (dependent).
- det = 0 means one of these two.
9Parametric Equations & the Trig Toolkitx and y driven by a third variable
Parametric curves
x = f(t), y = g(t): each t gives a point, and increasing t gives the curve a direction.
- Line from (x₁, y₁) to (x₂, y₂): x = x₁ + (x₂ − x₁)t, y = y₁ + (y₂ − y₁)t
- Circle: x = h + r cos t, y = k + r sin t
Eliminate the parameter Worked
Line segment Worked
Projectile motion
g ≈ 9.8 m/s² (32 ft/s²); air resistance ignored.
x = (v₀ cos θ)t
y = h₀ + (v₀ sin θ)t − ½gt²
That launch Worked
Trig in Pre-Calculus (see the Trigonometry guide)
- Radians and the unit circle: Trig §1, §3
- Sinusoids y = A sin(B(x − C)) + D, period 2πB: Trig §6
- Inverse trig and identities: Trig §7–9
- Vectors, u · v = |u||v| cos θ: Trig §11
- Polar graphs; De Moivre (r cis θ)ⁿ = rⁿ cis nθ: Trig §12–13
The puppeteer t
One hand (t) works two strings: x and y move separately, and the curve is where the puppet goes.
Direction of travel
x = cos t, y = sin t runs counterclockwise from (1, 0); x = sin t, y = cos t runs clockwise from (0, 1). Same circle, different motion.
10Sequences, Series, Induction & Binomialspatterns, sums, proofs and (a + b)ⁿ
- Sequence
- a list a₁, a₂, a₃, …
- Series
- the sum of its terms
- Recursive
- each term from the one before
- Explicit
- aₙ straight from n
Recursive to explicit Worked
Arithmetic & geometric (Algebra §11)
- aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ)2
- aₙ = a₁rⁿ⁻¹; Sₙ = a₁(1 − rⁿ)1 − r
- Infinite geometric: S = a₁1 − r if |r| < 1, else no sum
Power sums
Sums run k = 1 to n. Constants and sums split: Σ(cak + bk) = cΣak + Σbk; Σc = cn.
Σk = n(n + 1)2
Σk² = n(n + 1)(2n + 1)6
Σk³ = n²(n + 1)²4
Use the formulas Worked
Repeating decimal Worked
Bouncing ball Worked
Mathematical induction
- Base: show the statement for n = 1.
- Assume it holds for n = k.
- Show it then holds for n = k + 1.
Prove 1 + 3 + ⋯ + (2n − 1) = n² Worked
BAS: the dominoes
Base knocks over the first domino; Assume one falls; Show it knocks over the next. Then they all fall.
Binomial theorem
Term k + 1 is ₙCₖ an−kbᵏ.
(a + b)ⁿ = Σk=0n ₙCₖ an−kbᵏ
Expand (2x − 1)⁴ Worked
One term Worked
Down, up, total n
Powers of a count down from n, powers of b count up from 0, and each term’s powers add to n.
11Counting & Probabilityhow many ways, and how likely
Counting principle
m ways, then n ways: m · n ways in all.
EX3 shirts, 4 pants, 2 pairs of shoes: 24 outfits
Arrangements & selections
n! = n(n − 1)⋯1, 0! = 1. P counts ordered lists; C counts groups.
ₙPᵣ = n!(n − r)!
ₙCᵣ = n!r!(n − r)!
| Order matters? | Example (from 10 people) | Count |
|---|---|---|
| yes: P | president, vice president, secretary | ₁₀P₃ = 720 |
| no: C | a committee of 3 | ₁₀C₃ = 120 |
P for Position, C for Committee
If swapping two people makes a new result (positions), use P. If not (a committee), use C.
Repeated letters Worked
Probability rules
- P(A) = favorable outcomesall equally likely outcomes; P(not A) = 1 − P(A)
- P(A or B) = P(A) + P(B) − P(A and B)
- Independent: P(A and B) = P(A) · P(B)
EXtwo dice, sum 7: 636 = 16; at least one 6: 1 − (56)² = 1136
Probability by counting Worked
12Limits & Continuitywhere a function is heading
What a limit means
limx→c f(x) = L: f(x) gets as close to L as we like for x close enough to c, from both sides. The value f(c) does not matter.
| x | 2.9 | 2.99 | 2.999 | 3.001 | 3.01 | 3.1 |
|---|---|---|---|---|---|---|
| f(x) | 5.9 | 5.99 | 5.999 | 6.001 | 6.01 | 6.1 |
One-sided limits
The limit exists only if the left and right limits agree.
EXx + 1 for x < 2, x² for x ≥ 2: left 3, right 4 → no limit (a jump)
Please Fix Really Carefully
Plug in first. If you get 00: Factor, Rationalize, or Combine fractions, then plug in again.
Factor Worked
Rationalize Worked
Combine fractions Worked
At infinity; a special limit
- Divide by the highest power: limx→∞ 3x² − x2x² + 5 = 32 (the HA).
- limx→0 sin xx = 1 (x in radians)
Continuity at x = c
f(c) is defined, limx→c f(x) exists, and the two are equal.
Breaks: removable (hole), jump, infinite (vertical asymptote).
D-E-E
Continuous means Defined, Exists, Equal: three checks, in that order.
13From Average to Instantaneous Ratethe first step into calculus
The derivative
Secant slope (average rate, §1) → tangent slope (instantaneous rate) as h → 0.
f′(x) = limh→0 f(x + h) − f(x)h
Power rule preview
ddx xⁿ = nxⁿ⁻¹, term by term (proved in Calculus).
EXddx(3x⁴ − 5x + 2) = 12x³ − 5
Zoom until straight
Zoom in on a smooth curve until it looks like a line. That line’s slope is the derivative.