SYSTEMATIC MATHEMATICS

Precalculus

This course contains 37 visible knowledge chapters across eight content-sized units of lengths 4, 6, 4, 5, 5, 4, 5, and 4. It is the final bridge before calculus: analyze function domains, transformations, complex geometry, verified division, polar powers and roots, rational signs, asymptotes and partial fractions, exponential time scales, logarithmic equations, trigonometric branches, conics, polar-curve tracing, finite and infinite sequences, limits, squeeze arguments, rates, optimization, regression residuals, and sensitivity. Every new word and symbol is read before use; every formula is tied to a graph, table, domain, exact derivation, complete example, and first boundary where the model or shortcut stops working.

Before this course: Mathematical Foundations, Algebra & Functions, and Geometry & Trigonometry, including exact algebra, coordinate geometry, radians, and general triangle solving.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

A function family separates structure from parameters

Objective: Why must the domain be transformed algebraically rather than shifted by visual memory?

A function family uses parameters to select one member from a shared structural rule. In f(x)=a·g(b(x−h))+k, the parent g supplies qualitative shape while a,b,h,k control vertical scale or reflection, horizontal scale or reflection, and translations. The graph changes can be derived by asking which original input produces a desired transformed input, not by memorizing a list of directions.

Transformations act on domain and range as well as appearance. If g has domain D, then g(b(x−h)) requires b(x−h)∈D. A vertical shift changes range, while a horizontal shift changes domain boundaries. Zeros, extrema, asymptotes, and symmetry transform predictably only after parameter signs and excluded values are handled. A graphing window can hide these facts but cannot change them.

For b≠0, the graph of y=g(b(x−h))+k is obtained from y=g(x) by horizontal scaling factor 1/|b|, reflection across the vertical line x=h when b<0, then translations h and k.

A parent point (u,g(u)) appears in the transformed graph when b(x−h)=u, so x=h+u/b.

Differences in u are divided by |b| in the x-coordinate, establishing horizontal scale 1/|b|; a negative b reverses their left-right order.

The output becomes g(u)+k, adding vertical translation k. The input solution includes h, giving horizontal translation.

From g(x)=√x, analyze f(x)=−2√(3(x−4))+5: domain, range, starting point, reflections, and scale.

  1. The radicand requires 3(x−4)≥0, so x≥4. Parent input zero occurs at x=4, giving starting point (4,5).
  2. Factor 3 compresses horizontal distances by 1/3. Outer factor −2 reflects across the horizontal line y=5 and doubles vertical distances.
  3. Because √· is nonnegative, −2√·≤0 and adding 5 gives range (−∞,5].

Result: Domain [4,∞), range (−∞,5], starting point (4,5).