Pre-Calculus (US High School)
A comprehensive pre-calculus deck for US high school students (grades 11-12) bridging Algebra 2 and Calculus. Covers functions, polynomial and rational functions, exponential and logarithmic functions, trigonometry, analytic trigonometry, polar coordinates, vectors, parametric equations, conic sections, sequences and series, introduction to limits, matrices, probability, combinatorics, and complex numbers.
Ämne: Matematik · Nivå: Gymnasium (16–19) · 488 kort
Innehåll
- A function f is a rule that assigns to each input x in the domain exactly one output f(x) in the range.
- The domain of a function is the set of all permissible input values, and the range is the set of all output values produced.
- The vertical line test: a graph in the xy-plane represents y as a function of x if and only if no vertical line intersects the graph more than once.
- The horizontal line test: a function is one-to-one (and therefore has an inverse) if and only if no horizontal line intersects its graph more than once.
- An even function satisfies f(−x) = f(x), and its graph is symmetric about the y-axis. Example: f(x) = x².
- An odd function satisfies f(−x) = −f(x), and its graph is symmetric about the origin. Example: f(x) = x³.
- Composition of functions: (f ∘ g)(x) = f(g(x)). The inner function g is evaluated first, then f is applied to the result.
- If f and f⁻¹ are inverse functions, then f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in their respective domains.
- The graph of f⁻¹ is the reflection of the graph of f across the line y = x.
- Transformations of y = f(x): y = f(x) + k shifts up by k; y = f(x − h) shifts right by h; y = af(x) stretches vertically by factor |a|; y = f(bx) compresses horizontally by factor |b|.
- A piecewise function is defined by different expressions on different intervals of its domain, such as the absolute value function f(x) = x for x ≥ 0 and f(x) = −x for x < 0.
- A polynomial function of degree n has the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ··· + a₁x + a₀, where aₙ ≠ 0 and the exponents are non-negative integers.
- End behavior of a polynomial f(x) = aₙxⁿ + ···: if n is even and aₙ > 0, both ends rise; if n is even and aₙ < 0, both ends fall; if n is odd and aₙ > 0, left falls and right rises; if n is odd and aₙ < 0, left rises and right falls.
- The Fundamental Theorem of Algebra: every non-constant polynomial with complex coefficients has at least one complex root. A polynomial of degree n has exactly n complex roots, counting multiplicities.
- The Remainder Theorem: when a polynomial p(x) is divided by (x − c), the remainder equals p(c).
- The Factor Theorem: (x − c) is a factor of polynomial p(x) if and only if p(c) = 0.
- Rational Root Theorem: if a polynomial with integer coefficients has a rational root p/q (in lowest terms), then p divides the constant term and q divides the leading coefficient.
- Synthetic division is a shortcut method for dividing a polynomial by a linear factor (x − c), using only the coefficients arranged in a row.
- Descartes' Rule of Signs: the number of positive real roots of p(x) equals the number of sign changes in p(x), or fewer by an even number. For negative roots, count sign changes in p(−x).
- Complex roots of polynomials with real coefficients always occur in conjugate pairs: if a + bi is a root, then a − bi is also a root.
- A rational function is a function of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) is not the zero polynomial. The domain excludes any x for which q(x) = 0.
- A vertical asymptote of a rational function occurs at x = c if q(c) = 0 but p(c) ≠ 0 (after canceling common factors).
- Horizontal asymptote rules for rational function p(x)/q(x) where deg p = n and deg q = m: if n < m, y = 0; if n = m, y = ratio of leading coefficients; if n > m, no horizontal asymptote.
- A slant (oblique) asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. Find it using polynomial long division — the asymptote is the quotient ignoring the remainder.
- A hole (removable discontinuity) in a rational function's graph occurs at x = c when (x − c) is a common factor of both numerator and denominator that cancels.
- An exponential function has the form f(x) = a·bˣ where b > 0, b ≠ 1, and a ≠ 0. If b > 1, the function grows; if 0 < b < 1, the function decays.
- The natural exponential function is f(x) = eˣ, where e ≈ 2.71828 is Euler's number, defined as the limit of (1 + 1/n)ⁿ as n → ∞.
- Logarithm definition: logₘ(x) = y if and only if mʸ = x, where m > 0, m ≠ 1, and x > 0. The logarithm is the inverse of the exponential function.
- Common logarithm: log(x) = log₁₀(x) (base 10). Natural logarithm: ln(x) = logₑ(x) (base e). Both are widely used in scientific notation and continuous growth.
- Product rule for logarithms: logₘ(xy) = logₘ(x) + logₘ(y), provided x > 0 and y > 0.
- Quotient rule for logarithms: logₘ(x/y) = logₘ(x) − logₘ(y), provided x > 0 and y > 0.
- Power rule for logarithms: logₘ(xⁿ) = n·logₘ(x), provided x > 0 and n is a real number.
- Change of base formula: logₘ(x) = logₖ(x) / logₖ(m), useful for computing logs in any base using a calculator that has only log₁₀ or ln.
- Continuous compound interest formula: A = Peʳᵗ, where A is the final amount, P is the principal, r is the annual rate (as a decimal), and t is the time in years.
- Compound interest formula (n times per year): A = P(1 + r/n)ⁿᵗ. As n → ∞, this approaches the continuous formula A = Peʳᵗ.
- Exponential growth/decay model: A(t) = A₀e^(kt). If k > 0, the quantity grows; if k < 0, it decays. Half-life is the time required for A to reduce to A₀/2.
- Radians and degrees: 180° = π radians. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.
- Arc length formula: s = rθ, where r is the radius and θ is the central angle measured in radians.
- Sector area formula: A = (1/2)r²θ, where r is the radius and θ is the central angle in radians.
- Unit circle: a circle of radius 1 centered at the origin. For an angle θ measured from the positive x-axis, the terminal point on the unit circle is (cos θ, sin θ).
- Special angle values: sin(0) = 0, sin(π/6) = 1/2, sin(π/4) = √2/2, sin(π/3) = √3/2, sin(π/2) = 1.
- Special angle values: cos(0) = 1, cos(π/6) = √3/2, cos(π/4) = √2/2, cos(π/3) = 1/2, cos(π/2) = 0.
- Special angle values: tan(0) = 0, tan(π/6) = √3/3, tan(π/4) = 1, tan(π/3) = √3, tan(π/2) is undefined.
- Reciprocal trig functions: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ = cos θ / sin θ.
- Pythagorean identity: sin²θ + cos²θ = 1. This identity holds for all real θ and follows directly from the unit circle definition.
- Pythagorean identity variants: 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. Both derived by dividing sin²θ + cos²θ = 1 by cos²θ or sin²θ.
- Period of sine and cosine is 2π; period of tangent and cotangent is π; period of secant and cosecant is 2π.
- Amplitude of y = A sin(Bx) is |A|; period is 2π/|B|. For tangent y = A tan(Bx), the period is π/|B| (no amplitude since tan is unbounded).
- Phase shift of y = A sin(B(x − C)) + D is C (horizontal), and D is the vertical shift (midline of oscillation).
- Inverse sine arcsin(x) (or sin⁻¹(x)) has domain [−1, 1] and range [−π/2, π/2]. It returns the angle whose sine is x.