Conquering AP Precalculus requires a solid grasp of numerous formulas and concepts. This comprehensive formula sheet serves as your ultimate resource, covering key areas to help you ace the exam. Remember, understanding the why behind each formula is as important as memorizing the formula itself. This sheet is designed to be a quick reference; consult your textbook and class notes for detailed explanations and examples.
Algebra Fundamentals
- Order of Operations (PEMDAS): Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
- Factoring Techniques:
- Greatest Common Factor (GCF): Find the largest factor common to all terms.
- Difference of Squares: a² - b² = (a + b)(a - b)
- Perfect Square Trinomial: a² + 2ab + b² = (a + b)² and a² - 2ab + b² = (a - b)²
- Sum/Difference of Cubes: a³ + b³ = (a + b)(a² - ab + b²) and a³ - b³ = (a - b)(a² + ab + b²)
- Quadratic Trinomial Factoring: Requires finding factors that multiply to the constant term and add to the coefficient of the linear term.
- Quadratic Formula: For ax² + bx + c = 0, x = [-b ± √(b² - 4ac)] / 2a
- Discriminant: b² - 4ac (determines the nature of the roots of a quadratic equation)
- Absolute Value: |x| = x if x ≥ 0, and |x| = -x if x < 0
Functions
- Function Notation: f(x) represents the output of the function f when the input is x.
- Domain: The set of all possible input values (x-values) for a function.
- Range: The set of all possible output values (y-values) for a function.
- Inverse Functions: f⁻¹(x) is the inverse of f(x) if f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. To find the inverse, switch x and y and solve for y.
- Composition of Functions: (f ∘ g)(x) = f(g(x))
- Transformations of Functions:
- Vertical Shift: f(x) + k (k > 0 shifts up, k < 0 shifts down)
- Horizontal Shift: f(x - h) (h > 0 shifts right, h < 0 shifts left)
- Vertical Stretch/Compression: af(x) (a > 1 stretches, 0 < a < 1 compresses)
- Horizontal Stretch/Compression: f(bx) (0 < b < 1 stretches, b > 1 compresses)
- Reflection across x-axis: -f(x)
- Reflection across y-axis: f(-x)
Exponents and Logarithms
- Exponent Properties:
- xᵃ * xᵇ = x⁽ᵃ⁺ᵇ⁾
- xᵃ / xᵇ = x⁽ᵃ⁻ᵇ⁾
- (xᵃ)ᵇ = x⁽ᵃᵇ⁾
- x⁻ᵃ = 1/xᵃ
- x⁰ = 1 (x ≠ 0)
- Logarithm Properties:
- logₐ(xy) = logₐx + logₐy
- logₐ(x/y) = logₐx - logₐy
- logₐ(xⁿ) = nlogₐx
- logₐx = y is equivalent to aʸ = x
- Change of Base Formula: logₐx = (logₓx / logₐx)
Trigonometry
- Unit Circle: Memorizing the unit circle is crucial for understanding trigonometric functions.
- Trigonometric Functions: sin θ, cos θ, tan θ, csc θ, sec θ, cot θ
- Trigonometric Identities: A large number of trigonometric identities exist; focus on the fundamental ones, such as Pythagorean identities (sin²θ + cos²θ = 1), reciprocal identities, and quotient identities.
- Inverse Trigonometric Functions: arcsin x, arccos x, arctan x
Sequences and Series
- Arithmetic Sequences: aₙ = a₁ + (n - 1)d, where aₙ is the nth term, a₁ is the first term, and d is the common difference.
- Geometric Sequences: aₙ = a₁ * r⁽ⁿ⁻¹⁾, where aₙ is the nth term, a₁ is the first term, and r is the common ratio.
- Arithmetic Series: Sₙ = n/2(a₁ + aₙ)
- Geometric Series: Sₙ = a₁(1 - rⁿ) / (1 - r) (r ≠ 1)
This formula sheet provides a solid foundation for your AP Precalculus studies. Remember to practice extensively and seek help when needed. Good luck!
