Overview
This lecture covers essential Algebra 2 concepts, including equations, functions, variations, exponentials, logarithms, matrices, and conic sections, with key formulas and definitions.
Inequalities and Intervals
- "Less than" and "greater than" use open intervals (e.g., 9 < x < 8).
- Solutions like x > 4 or x < -1 represent union of intervals.
Basic Algebraic Forms and Operations
- Sum/difference of cubes: a³ + b³ = (a+b)(a²–ab+b²), a³ – b³ = (a–b)(a²+ab+b²).
- Point-slope form: m = (y₂ – y₁)/(x₂ – x₁).
- Parallel lines have equal slopes; perpendicular lines have opposite reciprocal slopes.
Functions and Graphs
- Domain is all possible x-values, range is all possible y-values.
- Vertical line test determines whether a graph is a function.
- Inverse of a function switches x and y.
- One x cannot correspond to two different y-values.
Quadratic Equations and Parabolas
- Quadratic formula: x = [–b ± √(b² – 4ac)] / 2a.
- Discriminant (b²–4ac) determines root type: 0=one real root, negative=2 complex roots, positive=2 real roots.
- Maximum/minimum at vertex (h, k), with h=–b/2a.
Exponential and Logarithmic Functions
- Exponential growth shows rapid increase, never touches 0.
- Exponential decay, e.g., 3^–x, describes decreasing scenarios.
- Compound interest: A = P(1 + r/n)^(nt).
- Natural exponential base e ≈ 2.7.
- Exponential-logarithmic inverse relationship: y = b^x ⇔ log_b(y) = x.
- Log rules: log_b(mn) = log_b(m) + log_b(n), log_b(m/n) = log_b(m) – log_b(n), log_b(mⁿ) = n log_b(m).
- Logarithms undefined for negative or zero argument.
Polynomial and Rational Variation
- Direct variation: variable on top, f = kx.
- Inverse variation: variable in denominator, f = k/x.
- Joint variation includes two or more variables, f = kxy.
- Constant of variation denoted as k.
Complex Numbers
- i = √–1, i² = –1, i³ = –i, i⁴ = 1.
- Modulus represents distance of a complex number from the origin.
Matrices and Determinants
- Matrices equal if same size and entries.
- For multiplication, A's columns = B's rows.
- 2×2 determinant: |A| = ad–bc.
- Inverse matrix for 2×2: A⁻¹ = (1/ad–bc)[d –b/–c a].
Conic Sections
- Circle: (x–h)² + (y–k)² = r².
- Ellipse: x²/a² + y²/b² = 1 (horizontal), x²/b² + y²/a² = 1 (vertical).
- Hyperbola: (x–h)²/a² – (y–k)²/b² = 1 (sideways), (y–k)²/a² – (x–h)²/b² = 1 (up/down).
- Parabola: y = Ax² + Bx + C; general quadratic equation Ax² + By² + Cx + Dy + E = 0.
- Eccentricity for ellipse: e = (a²–b²)/a.
Other Important Concepts
- Asymptote: dashed line showing graph behavior at infinity.
- Projectile motion: s(t) = –gt² + v₀t + h₀, where g is gravity, v₀ is initial velocity, h₀ is initial height.
- Horizontal line test for inverse functions.
- Shading on graphs: shade above for > or ≥, below for < or ≤; solid line for ≥ or ≤, dotted for > or <.
Key Terms & Definitions
- Domain — All possible x-values of a function.
- Range — All possible y-values of a function.
- Discriminant — b² – 4ac; determines types of roots in quadratics.
- Exponential Function — Function of the form y = b^x.
- Logarithm — Inverse of exponential function.
- Matrix — Rectangular array of numbers.
- Determinant — Scalar value from a matrix, important for solving systems.
- Eccentricity — Measure of how much a conic section deviates from being circular.
Action Items / Next Steps
- Practice solving quadratic equations using the quadratic formula.
- Graph various conic sections and label their key features.
- Memorize and apply logarithm properties in exercises.
- Review function transformations and inverse testing.