Master advanced functions, trigonometric identities, conic sections, sequences, vectors, and limits — the complete bridge to college-level calculus
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Parts I–IV with free-response questions and scoring rubrics
30-question diagnostic quiz testing precalculus prerequisites
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Master Precalculus concepts with our advanced worksheet collection
This is the most important course before calculus. You'll master advanced functions, trigonometric identities, conic sections, sequences and series, vectors, parametric equations, and an introduction to limits. Every lesson includes detailed concept explanations, worked examples, and 30 practice questions with complete step-by-step solutions.
— Mr. Augustine, Mathematics Department
Master function notation, composition, inverses, and transformations at the precalculus level
A function assigns exactly one output to each input. Precalculus deepens your understanding of functions as the foundation for calculus. **Function Notation:** f(x) reads "f of x" — x is the input, f(x) is the output. f(a + h) means substitute (a + h) everywhere you see x. **Composition of Functions:** (f ∘ g)(x) = f(g(x)) — apply g first, then f. Domain of f ∘ g: all x in domain of g where g(x) is in domain of f. **Inverse Functions:** f⁻¹ is the inverse of f if f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. To find f⁻¹: swap x and y, then solve for y. A function has an inverse only if it is one-to-one (passes horizontal line test). **Transformations Summary:** • f(x) + k → shift up k • f(x) - k → shift down k • f(x + h) → shift left h • f(x - h) → shift right h • -f(x) → reflect over x-axis • f(-x) → reflect over y-axis • af(x) → vertical stretch/compression by factor a • f(bx) → horizontal compression/stretch by factor 1/b **Piecewise Functions:** Defined by different formulas on different intervals. Always check which piece applies for a given x-value. **Difference Quotient:** [f(x + h) - f(x)] / h — the foundation of the derivative in calculus.
If f(x) = 2x² - 3x + 1 and g(x) = x + 4, find (f ∘ g)(x) and (g ∘ f)(x)
Find the inverse of f(x) = (3x - 2)/(x + 1)
Compute the difference quotient for f(x) = x² + 3x
The difference quotient [f(x+h) - f(x)]/h is the single most important formula in precalculus — it becomes the derivative in calculus! Practice it with every function type. For inverses, always verify by checking that f(f⁻¹(x)) = x. And remember: composition is NOT commutative — f∘g ≠ g∘f in general!
Analyze complex polynomials, rational functions, partial fractions, and real-world modeling
**Rational Root Theorem:** If f(x) = aₙxⁿ + ... + a₀ has integer coefficients, then any rational zero p/q satisfies: • p divides the constant term a₀ • q divides the leading coefficient aₙ **Descartes' Rule of Signs:** • Number of positive real zeros = number of sign changes in f(x), or less by an even number • Number of negative real zeros = number of sign changes in f(-x), or less by an even number **Fundamental Theorem of Algebra:** Every polynomial of degree n ≥ 1 has exactly n zeros (counting multiplicity) in the complex number system. **Complex Conjugate Pairs:** If a + bi is a zero of a polynomial with real coefficients, then a - bi is also a zero. **Partial Fraction Decomposition:** Used to break rational expressions into simpler fractions for integration. For distinct linear factors: A/(x-a) + B/(x-b) For repeated factors: A/(x-a) + B/(x-a)² For irreducible quadratics: (Ax+B)/(x²+bx+c) **Upper and Lower Bounds:** Synthetic division with positive c: if all values in bottom row are non-negative → c is upper bound. Synthetic division with negative c: if values alternate in sign → c is lower bound.
Find all rational zeros of f(x) = 2x³ - 3x² - 11x + 6
Decompose into partial fractions: (3x + 5)/[(x+1)(x-2)]
Use Descartes' Rule to determine possible positive/negative zeros of f(x) = x⁴ - 3x³ + x² + 2x - 5
The Rational Root Theorem gives you a finite list to test — don't try random numbers! Once you find one zero, use synthetic division to reduce the degree, then factor or use the quadratic formula on the quotient. For partial fractions, the 'cover-up' method is fastest for distinct linear factors: cover the factor in the denominator and substitute its zero into the rest.
Master advanced exponential models, logarithmic equations, natural log, and real-world applications
**The Number e:** e ≈ 2.71828... is the base of natural logarithms. e = lim(n→∞) (1 + 1/n)ⁿ Continuous compounding: A = Pe^(rt) **Natural Logarithm:** ln(x) = log_e(x) ln(e) = 1, ln(1) = 0, ln(e^x) = x, e^(ln x) = x **Solving Exponential Equations:** 1. Same base: set exponents equal 2. Different bases: take ln or log of both sides 3. Use power rule: ln(a^x) = x·ln(a) **Solving Logarithmic Equations:** 1. Condense to single log 2. Convert to exponential form 3. Solve and CHECK (domain: argument must be positive) **Exponential Growth/Decay Models:** • Growth: A(t) = A₀e^(kt), k > 0 • Decay: A(t) = A₀e^(-kt), k > 0 • Half-life: t₁/₂ = ln(2)/k • Doubling time: t_d = ln(2)/k **Logistic Growth:** P(t) = L / (1 + Ce^(-kt)) where L = carrying capacity **Real-World Logarithm Applications:** • pH = -log[H⁺] • Richter scale: M = log(I/I₀) • Decibels: dB = 10·log(I/I₀) • Compound interest: A = P(1 + r/n)^(nt)
Solve: 3^(2x-1) = 27^(x+2)
Solve: ln(x + 3) + ln(x - 1) = ln(5)
A radioactive substance has a half-life of 10 years. How long until only 25% remains?
Always CHECK logarithmic equation solutions — you can only take the log of positive numbers, so extraneous solutions are common! For exponential equations, if you can't match bases, take ln of both sides and use the power rule. Remember: ln and e are inverses, so they cancel each other. For real-world problems, identify whether it's growth (k > 0) or decay (k < 0) first.
Master sum/difference formulas, double/half-angle identities, and solving trig equations
**Pythagorean Identities:** sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = csc²θ **Sum & Difference Formulas:** sin(A ± B) = sinA cosB ± cosA sinB cos(A ± B) = cosA cosB ∓ sinA sinB tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB) **Double-Angle Formulas:** sin(2θ) = 2sinθ cosθ cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ tan(2θ) = 2tanθ / (1 - tan²θ) **Half-Angle Formulas:** sin(θ/2) = ±√[(1 - cosθ)/2] cos(θ/2) = ±√[(1 + cosθ)/2] tan(θ/2) = sinθ/(1 + cosθ) = (1 - cosθ)/sinθ **Solving Trig Equations:** 1. Isolate the trig function 2. Find the reference angle 3. Determine all solutions in [0, 2π) 4. Add 2πk (or πk for tan/cot) for general solution **Cofunction Identities:** sin(π/2 - θ) = cosθ cos(π/2 - θ) = sinθ tan(π/2 - θ) = cotθ
Find the exact value of sin(75°)
If sinθ = 3/5 and θ is in QI, find sin(2θ) and cos(2θ)
Solve: 2sin²x - sinx - 1 = 0 on [0, 2π)
Memorize the sum formulas — everything else derives from them! For double-angle, just use sum formula with A = B = θ. When solving trig equations, always find ALL solutions in [0, 2π) first, then add the period for the general solution. Factor trig equations just like algebraic ones — substitute u = sinx or u = cosx to see the structure clearly.
Explore parabolas, ellipses, hyperbolas, and circles with equations, graphs, and applications
**Circles:** Standard form: (x - h)² + (y - k)² = r² Center: (h, k), Radius: r **Parabolas:** Vertical: (x - h)² = 4p(y - k) Horizontal: (y - k)² = 4p(x - h) • Vertex: (h, k) • Focus: p units from vertex along axis • Directrix: p units from vertex, opposite focus • If p > 0: opens up/right; if p < 0: opens down/left **Ellipses:** Horizontal major axis: (x-h)²/a² + (y-k)²/b² = 1, a > b Vertical major axis: (x-h)²/b² + (y-k)²/a² = 1, a > b • Center: (h, k) • a = semi-major axis, b = semi-minor axis • c² = a² - b² (c = distance from center to focus) • Eccentricity: e = c/a (0 < e < 1) **Hyperbolas:** Horizontal: (x-h)²/a² - (y-k)²/b² = 1 Vertical: (y-k)²/a² - (x-h)²/b² = 1 • c² = a² + b² • Asymptotes: y - k = ±(b/a)(x - h) [horizontal] • Eccentricity: e = c/a (e > 1) **Identifying Conics from General Form Ax² + Bxy + Cy² + Dx + Ey + F = 0:** • Circle: A = C (and B = 0) • Parabola: A = 0 or C = 0 (but not both) • Ellipse: A and C have same sign, A ≠ C • Hyperbola: A and C have opposite signs
Find the equation of the ellipse with vertices (±5, 0) and co-vertices (0, ±3)
Find the vertex, focus, and directrix of x² = 12y
Find the asymptotes of the hyperbola x²/9 - y²/16 = 1
The key to conic sections is recognizing which form you have. Always complete the square to convert to standard form — it reveals the center, vertices, and foci immediately. Remember: for ellipses c² = a² - b² (c is smaller than a), but for hyperbolas c² = a² + b² (c is larger than a). The eccentricity tells you the shape: e = 0 is a circle, 0 < e < 1 is an ellipse, e = 1 is a parabola, e > 1 is a hyperbola.
Master arithmetic and geometric sequences, series, sigma notation, and proof by induction
**Arithmetic Sequences:** aₙ = a₁ + (n-1)d, where d = common difference Sum of n terms: Sₙ = n/2 · (a₁ + aₙ) = n/2 · [2a₁ + (n-1)d] **Geometric Sequences:** aₙ = a₁ · r^(n-1), where r = common ratio Sum of n terms: Sₙ = a₁(1 - rⁿ)/(1 - r), r ≠ 1 **Infinite Geometric Series:** S = a₁/(1 - r), only if |r| < 1 (converges) If |r| ≥ 1, the series diverges. **Sigma Notation:** Σᵢ₌₁ⁿ aᵢ = a₁ + a₂ + ... + aₙ **Useful Summation Formulas:** Σᵢ₌₁ⁿ 1 = n Σᵢ₌₁ⁿ i = n(n+1)/2 Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6 Σᵢ₌₁ⁿ i³ = [n(n+1)/2]² **Mathematical Induction:** To prove P(n) for all n ≥ 1: 1. Base case: Verify P(1) is true 2. Inductive step: Assume P(k) is true, prove P(k+1) is true **Binomial Theorem:** (a + b)ⁿ = Σₖ₌₀ⁿ C(n,k) · aⁿ⁻ᵏ · bᵏ where C(n,k) = n! / [k!(n-k)!]
Find the 20th term and sum of the first 20 terms of the arithmetic sequence 3, 7, 11, 15, ...
Find the sum of the infinite geometric series: 8 + 4 + 2 + 1 + ...
Expand (x + 2)⁴ using the Binomial Theorem
For arithmetic sequences, the common difference d is constant — check by subtracting consecutive terms. For geometric sequences, the common ratio r is constant — check by dividing consecutive terms. The infinite geometric series formula S = a₁/(1-r) only works when |r| < 1! For the Binomial Theorem, Pascal's Triangle gives you the coefficients C(n,k) quickly for small n.
Master vector operations, dot products, parametric curves, and applications in physics and geometry
**Vectors:** A vector has both magnitude and direction. Component form: v = ⟨a, b⟩ Magnitude: |v| = √(a² + b²) Direction angle: θ = arctan(b/a) **Vector Operations:** • Addition: ⟨a,b⟩ + ⟨c,d⟩ = ⟨a+c, b+d⟩ • Subtraction: ⟨a,b⟩ - ⟨c,d⟩ = ⟨a-c, b-d⟩ • Scalar multiplication: k⟨a,b⟩ = ⟨ka, kb⟩ • Unit vector: û = v/|v| **Dot Product:** u · v = a₁a₂ + b₁b₂ |u||v|cosθ = u · v • If u · v = 0, vectors are perpendicular • If u · v > 0, angle is acute • If u · v < 0, angle is obtuse **Angle Between Vectors:** cosθ = (u · v) / (|u| · |v|) **Parametric Equations:** Express x and y as functions of a parameter t: x = f(t), y = g(t) To eliminate the parameter: solve one equation for t, substitute into the other. **Common Parametric Curves:** • Line: x = x₀ + at, y = y₀ + bt • Circle: x = r·cos(t), y = r·sin(t) • Ellipse: x = a·cos(t), y = b·sin(t) **Projectile Motion:** x = v₀cos(θ)·t y = v₀sin(θ)·t - (1/2)gt²
Find the magnitude and direction angle of v = ⟨3, -4⟩
Find the angle between u = ⟨2, 1⟩ and v = ⟨1, 3⟩
Eliminate the parameter: x = t + 1, y = t² - 2
Vectors are everywhere in physics and engineering! The dot product is your tool for finding angles and checking perpendicularity. Remember: two vectors are perpendicular if and only if their dot product is zero. For parametric equations, think of t as time — the curve traces out a path as t increases. When eliminating the parameter, always check if there are domain restrictions on t that limit the curve.
Understand limits intuitively and formally, continuity, and the bridge to differential calculus
**The Concept of a Limit:** lim(x→c) f(x) = L means f(x) gets arbitrarily close to L as x approaches c. The limit does NOT depend on f(c) — only on values near c. **One-Sided Limits:** • Left-hand limit: lim(x→c⁻) f(x) • Right-hand limit: lim(x→c⁺) f(x) • Two-sided limit exists ⟺ both one-sided limits exist and are equal **Limit Laws:** • lim[f(x) ± g(x)] = lim f(x) ± lim g(x) • lim[f(x)·g(x)] = lim f(x) · lim g(x) • lim[f(x)/g(x)] = lim f(x) / lim g(x), if lim g(x) ≠ 0 • lim[f(x)]ⁿ = [lim f(x)]ⁿ **Continuity:** f is continuous at x = c if: 1. f(c) is defined 2. lim(x→c) f(x) exists 3. lim(x→c) f(x) = f(c) **Types of Discontinuity:** • Removable: hole in graph (limit exists, but f(c) ≠ limit) • Jump: left and right limits exist but are unequal • Infinite: vertical asymptote **Indeterminate Forms:** 0/0, ∞/∞ — need algebraic manipulation or L'Hôpital's Rule **The Derivative (Preview):** f'(x) = lim(h→0) [f(x+h) - f(x)] / h This is the instantaneous rate of change — the slope of the tangent line. **Special Limits:** lim(x→0) sinx/x = 1 lim(x→0) (1-cosx)/x = 0 lim(x→∞) (1 + 1/x)^x = e
Evaluate: lim(x→3) (x² - 9)/(x - 3)
Determine if f(x) = {x² if x < 2; 4 if x = 2; x + 3 if x > 2} is continuous at x = 2
Use the limit definition to find f'(x) for f(x) = x²
Limits are the foundation of all calculus! When you get 0/0, don't panic — it just means you need to do more algebra (factor, rationalize, or use trig identities). The limit exists only when BOTH one-sided limits agree. For continuity, think of it as 'you can draw the graph without lifting your pencil.' The derivative definition lim(h→0)[f(x+h)-f(x)]/h is the most important formula in calculus — understand it deeply!