Overview
This lecture covered key concepts in functions and their transformations, calculus (including differentiation and stationary points), and probability, focusing on foundational methods, important formulas, and strategies for exam preparation.
Functions and Equations
- Linear equations are degree 1 polynomials, generally in the form y = mx + c, where m is the gradient and c is the y-intercept.
- The distance between two points: d = √[(x2 - x1)² + (y2 - y1)²].
- The midpoint formula: ((x1 + x2)/2, (y1 + y2)/2).
- Gradient between two points: m = (y2 - y1)/(x2 - x1); also relates to tan(θ).
- Quadratic, cubic, and quartic polynomials have degrees 2, 3, and 4, respectively.
- Expanded, factorized, and turning point forms provide different information—expanded shows y-intercept, factorized gives x-intercepts, and turning point form reveals the vertex/stationary point.
- Completing the square transforms a quadratic to turning point form.
- The quadratic formula: x = [-b ± √(b² - 4ac)]/(2a).
- The discriminant (b² - 4ac) determines the number of solutions: <0 (none), =0 (one), >0 (two).
Circular Functions
- Angles are measured in degrees (°) or radians; 360° = 2π radians.
- To convert: radians = degrees × (π/180), degrees = radians × (180/π).
- The unit circle has radius 1; coordinates are (cos θ, sin θ).
- Symmetry rules: "All Stations To Central" (ASTC)—which functions are positive in each quadrant.
- Exact values for sin, cos, and tan at key angles can be memorized with tables or triangles.
- Graphing sine, cosine, and tangent functions requires amplitude, period, gap calculation, and identifying intercepts and asymptotes.
Inverse Functions and Transformations
- A one-to-one function passes the horizontal line test (each y-value corresponds to one x-value).
- The inverse function reflects the graph across y = x.
- To find the inverse: swap x and y, then solve for y.
- Transformations include reflection, dilation (stretch/compress), and translation (shift).
- The coordinate method systematically applies transformations to x and y values.
Calculus Basics
- The rate of change (gradient) is average between points: (y2 - y1)/(x2 - x1), or instantaneous at a point (derivative).
- The gradient of a tangent at a point equals the derivative there; normal lines are perpendicular with gradient -1/m.
- The first principles definition of the derivative: f'(a) = lim(h→0) [f(a + h) - f(a)]/h.
- Notation: f '(x), dy/dx, or Newton’s ẏ.
- Differentiation rules: power rule, constant rule (derivative of a constant is zero), sum/difference rule.
- Stationary points occur where the derivative is zero—can be maxima, minima, or points of inflection.
- Use nature tables or second derivative test to classify stationary points.
- Absolute maxima/minima are the highest/lowest function values within a domain, found by checking endpoints and stationary points.
Probability
- Probability = (number of favorable outcomes)/(total outcomes); values are between 0 and 1.
- Sample space: set of all possible outcomes; events: subsets of the sample space.
- Key notation: ∩ (intersection, A and B), ∪ (union, A or B), A' (complement), ∅ (null set).
- Venn diagrams and tree diagrams visually represent probabilities and events.
- The sum of probabilities of all outcomes equals 1.
- For non-mutually exclusive events: P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
- Conditional probability: P(A|B) = P(A ∩ B)/P(B).
- Events are independent if P(A ∩ B) = P(A) × P(B).
- Combinations: nCr = n!/[r!(n - r)!] calculates ways to choose r items from n.
Key Terms & Definitions
- Gradient — Rate of change or slope of a line.
- Discriminant — Expression b² - 4ac in the quadratic formula determining number of solutions.
- Unit Circle — Circle with radius 1, center at the origin.
- Stationary Point — Where the derivative is zero; possible max, min, or inflection.
- Sample Space — Set of all possible outcomes in a probability experiment.
- Conditional Probability — Probability of one event given another has occurred.
- Combination (nCr) — Number of ways to choose r items from n without regard to order.
Action Items / Next Steps
- Memorize key formulas (distance, midpoint, quadratic, combination, etc.).
- Practice converting between radians and degrees and memorizing exact trig values.
- Review differentiation rules and practice classifying stationary points.
- Complete practice problems in probability, especially using Venn/carol/tree diagrams.
- Prepare for exams by organizing summary notes and practicing calculator skills.