25 lessons across 4 weeks, 25 verified past-year questions. Free preview below — sign in for the full interactive lessons, AI tutor, and mock exams.
Week 1
- Natural Numbers and Integers — Introduction to counting with natural numbers and extending to integers for subtraction via the number line.
- Rational Numbers — Rational numbers represent ratios of integers and are dense on the number line, unlike the discrete integers.
- Real Numbers and Irrationality — Real numbers extend rationals with irrationals like the square root of 2, filling gaps on the number line.
- Set Theory Foundations — Sets are collections of objects defined by membership, subsets, power sets, and rigorous operations.
- Relations and Functions — Relations define associations between sets via Cartesian products, while functions are specialized relations mapping inputs to unique outputs.
- Primes and Infinite Cardinality — Exploration of the infinite nature of prime numbers and Cantor's diagonalization proving different degrees of infinity.
Week 2
- Rectangular Coordinate System — Introduction to the coordinate plane, quadrants, axes, and identifying point locations using ordered pairs.
- Distance Formula — Using the Pythagorean theorem to calculate the distance between two points in a coordinate plane.
- Section Formula — Calculating the coordinates of a point that divides a line segment into a specific ratio.
- Area of a Triangle — Determining the area of a non-collinear set of points using coordinates and trapezoidal geometry.
- Slope and Inclination — Defining the slope of a line as the 'rise over run' ratio and its relationship to the angle of inclination.
- Line Representation and Forms — Algebraic representations of lines including point-slope, two-point, slope-intercept, and intercept forms.
- Characterization of Lines — Using slopes and general forms to determine if lines are parallel, perpendicular, or intersecting.
- Distance from Point to Line — Calculating perpendicular distances from a point to a line and between two parallel lines.
- Linear Regression Basics — An introduction to best-fit lines and minimizing the sum of squared errors.
Week 3
- Definition and Graphing of Quadratic Functions — Introduction to quadratic functions (f(x) = ax² + bx + c, a ≠ 0), their parabolic shape, and graphing them using the axis of symmetry, vertex, and y-intercept.
- Slope of a Quadratic Function — Generalizing the concept of slope from linear functions to quadratic functions, where the slope becomes a variable function g(x) = 2ax + b.
- Roots of Quadratic Equations — Solving ax² + bx + c = 0 to find roots using graphical techniques, factoring, completing the square, and the quadratic formula.
- Vertex Form and Parabola Geometry — Understanding the vertex form f(x) = a(x-h)² + k as an alternative to the standard form and its utility in identifying the vertex (h, k).
Week 4
- Polynomial Definitions — Polynomials are defined as algebraic expressions with real coefficients and variables raised to non-negative integer exponents, representing functions from the real line to the real line.
- Degrees and Types of Polynomials — The degree of a polynomial is determined by the term with the highest sum of exponents, allowing for classification into linear, quadratic, cubic, quartic, and quintic types.
- Polynomial Algebra — Arithmetic operations on polynomials involve adding, subtracting, and multiplying coefficients of like terms, resulting in new polynomials.
- Polynomial Division — Polynomial division uses an algorithm analogous to long division to find a quotient and remainder, though the result may not always be a polynomial.
- Zeros and Factors — The Factor Theorem relates the roots of a polynomial to its linear factors, facilitating the identification of x-intercepts.
- Graphing Polynomials — Polynomial graphs are smooth, continuous curves whose shapes, intercepts, turning points, and end behavior are determined by the polynomial's degree, leading coefficient, and roots.
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