Overview
This lecture reviews essential Algebra 2 (Common Core) concepts and formulas that are critical for success on the Regents Exam, covering core topics including polynomials, functions, systems, quadratics, trigonometry, statistics, probability, and applications.
Number Systems, Polynomials, & Algebra
- Division Algorithm: Dividend = (Divisor × Quotient) + Remainder.
- Synthetic division simplifies dividing polynomials when divisor is in the form (x - k).
- Remainder Theorem: The remainder of f(x) ÷ (x - k) is f(k).
- Factor Theorem: If f(k) = 0, then (x - k) is a factor of f(x).
- Factoring order: GCF, DOTS, Trinomials, AC method, Factor by grouping.
- Quadratic Formula: x = [ -b ± √(b² - 4ac) ] / 2a.
- Rational expressions require common denominators for addition/subtraction; flip second fraction when dividing.
Exponentials, Logarithms, & Complex Numbers
- Exponential and logarithmic functions are inverses.
- Key logarithm properties: log_b(MN) = log_b M + log_b N; log_b(M/N) = log_b M - log_b N.
- i is the imaginary unit with i² = -1.
- Rationalize denominators using conjugates for complex or radical denominators.
Functions & Their Properties
- A function pairs each x-value to one unique y-value.
- Domain: All possible input values; restrictions occur for denominators ≠ 0 and radicals ≥ 0.
- Range: All possible output values.
- Inverse functions reflect over y = x; switch x and y, then solve.
- One-to-one functions pass horizontal and vertical line tests.
- End behavior depends on leading term’s degree and coefficient.
Linear Systems & Equations
- Slope formula: m = (y₂ - y₁) / (x₂ - x₁).
- Slope-intercept form: y = mx + b.
- Average rate of change is the same as slope.
- Solve 3-variable systems by elimination and substitution.
Quadratics & Conics
- Standard quadratic: ax² + bx + c = 0.
- Sum of roots: -b/a; product of roots: c/a.
- Axis of symmetry: x = -b/(2a); vertex uses this x-value.
- Vertex form: y = a(x - h)² + k.
- Completing the square rewrites quadratics and circles.
- Focus and directrix relate to the vertex via distance.
Trigonometry & Trig Functions
- Convert degrees ↔ radians: multiply/divide by π/180.
- Sine, cosine, and tangent relate sides of right triangles.
- Reciprocal identities: csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ.
- Pythagorean identities: sin²θ + cos²θ = 1.
- Amplitude = (max - min)/2, period = 2π/frequency; midline = (max + min)/2.
- Quadrant rules: signs of trig functions vary by quadrant.
Transformations of Functions
- Vertical shift: f(x) → f(x) + k.
- Horizontal shift: f(x) → f(x - h).
- Reflections over axes: f(x) → -f(x) or f(-x).
- Dilation: Multiply input or output by a constant.
- Use HDRV (Horizontal, Dilation, Reflection, Vertical) to describe transformations.
- Even functions: symmetric about y-axis; Odd: symmetric about origin.
Sequences & Series
- Arithmetic sequence: difference d; series sum = n[(a₁ + a_n)/2].
- Geometric sequence: ratio r; series sum = a₁(1 - rⁿ)/(1 - r) for r ≠ 1.
- Explicit formula uses term number; recursive formula uses previous term.
Statistics & Inference
- Types: survey, observational study, controlled experiment.
- Normal curve: mean = median = mode; 68–95–99 rule for standard deviations.
- Standard deviation measures spread; z-score = (x - mean)/std dev.
- Confidence interval = mean ± z*(std dev/√n).
- Margin of error = z*√[p(1-p)/n].
Set Theory & Probability
- Universal set: all possible elements; subsets and complements are key terms.
- Probability rule: 0 ≤ P(A) ≤ 1; P(A') = 1 - P(A).
- Mutually exclusive: no overlap, P(A ∩ B) = 0.
- Conditional probability: P(A|B) = P(A ∩ B)/P(B).
- For independent events: P(A ∩ B) = P(A) × P(B).
- "Or" probability: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Applications of Logarithms & Regression
- Compound interest: A = P(1 + r/n)^(nt); continuous: A = Pe^(rt).
- Exponential growth: y = a(1 + r)^t; decay: y = a(1 - r)^t.
- Use TI-Nspire CX for regression models and data fit.
Key Terms & Definitions
- GCF — Greatest Common Factor, largest factor shared by terms.
- Domain — All permissible input values.
- Range — All output values produced by a function.
- Multiplicity — How often a root appears in a polynomial.
- Amplitude — Height from the midline to a peak/trough in a trig graph.
- Z-score — Number of standard deviations a value is from the mean.
- Confidence Interval — Range where a population parameter likely falls.
Action Items / Next Steps
- Memorize all formulas and key properties.
- Practice factoring, solving equations, and graph transformations.
- Complete assigned Regents practice exams.
- Review calculator operations for regression, normalCDF, and statistical functions.