Mathematical Architects

Pre-Calculus

The bridge year. Scholars assemble every tool calculus will demand — the function families, the unit circle, vectors, and the analytic geometry of \(\sin\theta\), \(e^{x}\), and \(r = a + b\cos\theta\) — then meet the idea of a limit at the edge.

Pre-Calculus is where the whole library of functions earns its keep. We compose and invert functions, name their symmetry, and read their families fluently; we build trigonometry from the unit circle, prove identities, and solve equations and triangles; and we close on analytic geometry, polar and parametric curves, and the sequences and series that foreshadow the infinite. Every idea is treated the way a working mathematician treats a structure: defined symbolically, drawn graphically, tabulated numerically, and described verbally — from \((f\circ g)(x)\) to \(\displaystyle\sum_{k=1}^{\infty} ar^{k-1} = \frac{a}{1-r}\). The course is organized on the Texas Essential Knowledge and Skills for Precalculus (§111.42).

5 Modules
2 Semesters
60 / 40 Major / Minor Grades
No EOC Bridge to Calculus
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Concepts in Action

Pre-Calculus is a year of moving objects — functions chained together, a point traveling around a circle, a curve traced by an angle. Each module opens with an interactive Visual Lab built to make the motion visible.

Visual Labs — one per module

Every module page hosts a hands-on explorer. In Module 1 you compose two parents and watch \((f\circ g)\) and \((g\circ f)\) diverge side by side, then reflect a one-to-one function across \(y=x\) to build its inverse. Later modules animate the unit circle, sweep the sine and cosine waves, and trace polar and parametric curves in real time. Start with the lab that matches what you're studying — or just explore.

M1 · Composition & Inverse Explorer M3 · Unit-Circle & Wave Tracer M4 · Vectors & Identities M5 · Polar & Parametric Plotter
Why it matters. Calculus studies how things change — slopes of curves, areas under them, motion along them. Before you can take a derivative or an integral you have to see the curve: know its family, its symmetry, its period, its asymptotes, and how a parameter moves a point along it. Pre-Calculus builds that fluency so the limit, when it arrives, lands on familiar ground.
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Module by Module

Five modules across two semesters, sequenced from functions to the edge of calculus. The mathematical-process standards P.1A–P.1G are embedded in every module, every day.

Module 01

Functions — Composition, Inverses & Symmetry

Semester 1

Big idea: functions are machines you can chain and reverse — composition feeds one into another, and an inverse undoes the work, reflected across \(y=x\).

Topics

Composition of Functions — \((f\circ g)(x)=f(g(x))\), domain of a composite, and decomposing a function into a chain.

Inverse Functions — the one-to-one test, finding \(f^{-1}\), and the reflection across \(y=x\).

Symmetry — even (\(f(-x)=f(x)\)) and odd (\(f(-x)=-f(x)\)) functions and their graphs.

You'll be able to…

  • Evaluate and simplify \((f\circ g)(x)\) and state the domain of the composite.
  • Determine whether a function is one-to-one and find its inverse algebraically.
  • Classify a function as even, odd, or neither from its rule or graph.
  • Verify that two functions are inverses using composition.

Worked example

Verify an inverse by composition For \(f(x)=2x-6\), claim \(f^{-1}(x)=\dfrac{x+6}{2}\).
Check: \(f\!\left(f^{-1}(x)\right)=2\!\left(\dfrac{x+6}{2}\right)-6\). \[ = (x+6)-6 = x \;\checkmark \]
Test for symmetry For \(f(x)=x^3-x\): \(f(-x)=(-x)^3-(-x)=-x^3+x=-f(x)\), so \(f\) is odd (symmetric about the origin).
P.1A P.2A P.2B P.2C P.2D P.2E
Filled chip marks a standard frequently assessed as readiness for calculus.
Visualize this module → 🧱 Shaky on the basics? → Foundations
Module 02

Analyzing Function Families

Semester 1

Big idea: every curve belongs to a family — polynomial, rational, radical, exponential, logarithmic — and each family has a signature you can read at a glance.

Topics

Polynomial & Rational Functions — end behavior, zeros and multiplicity, vertical/horizontal/slant asymptotes, holes.

Exponential & Logarithmic Functions — growth and decay, the inverse pairing \(b^{y}=x \iff y=\log_b x\), and the laws of logarithms.

Transformations & Piecewise Models — \(a\,f(b(x-h))+k\) across families, plus piecewise-defined functions.

You'll be able to…

  • Sketch polynomial and rational graphs from zeros, multiplicity, and asymptotes.
  • Convert between exponential and logarithmic form and apply the log laws.
  • Predict how \(a\), \(b\), \(h\), and \(k\) reshape any parent function.
  • Evaluate and graph piecewise-defined functions.

Worked example

Asymptotes of a rational function For \(f(x)=\dfrac{2x}{x-3}\): vertical asymptote at \(x=3\); since the degrees match, the horizontal asymptote is the leading-coefficient ratio \[ y = \tfrac{2}{1} = 2. \]
Law of logarithms Condense \(\log_b M - \log_b N\): \[ \log_b\!\frac{M}{N} = \log_b M - \log_b N, \] so \(\log_2 24 - \log_2 3 = \log_2 8 = 3\).
P.1B P.2F P.2G P.2I P.2J P.2K P.2L P.5A P.5B P.5C
Filled chip marks a standard frequently assessed as readiness for calculus.
Visualize this module → 🧱 Shaky on the basics? → Foundations
Module 03

Trigonometric Functions & the Unit Circle

Semester 1–2

Big idea: wrap the number line around a circle of radius \(1\) and a point's coordinates become \((\cos\theta,\sin\theta)\) — trigonometry is the geometry of going around.

Topics

Angles & the Unit Circle — radians and degrees, arc length \(s=r\theta\), and exact values at the special angles.

The Six Functions — sine, cosine, tangent and their reciprocals defined on the circle and on right triangles.

Graphs of Sine & Cosine — amplitude, period \(\dfrac{2\pi}{b}\), phase shift, and midline; inverse trig functions.

You'll be able to…

  • Convert between radians and degrees and compute arc length and sector area.
  • State exact values of the six trig functions at the special angles from the unit circle.
  • Graph \(y=a\sin(b(x-h))+k\) and read amplitude, period, phase shift, and midline.
  • Evaluate inverse trig functions over their restricted domains.

Worked example

Read a value off the unit circle At \(\theta=\dfrac{2\pi}{3}\) the point is \(\left(-\tfrac12,\tfrac{\sqrt3}{2}\right)\), so \[ \cos\tfrac{2\pi}{3}=-\tfrac12, \qquad \sin\tfrac{2\pi}{3}=\tfrac{\sqrt3}{2}. \]
Period of a sine graph For \(y=3\sin(2x)\): amplitude \(3\) and period \(\dfrac{2\pi}{b}=\dfrac{2\pi}{2}=\pi\) — one full wave every \(\pi\) units.
P.1C P.4A P.4B P.4C P.4D P.4E P.4F P.4G P.4H
Filled chip marks a standard frequently assessed as readiness for calculus.
Visualize this module → 🧱 Shaky on the basics? → Foundations
Module 04

Trig Identities, Equations, Laws & Vectors

Semester 2

Big idea: the identities are the algebra of the circle — they let you rewrite, solve, and measure, and they extend into vectors that carry both size and direction.

Topics

Identities — Pythagorean, sum/difference, double- and half-angle identities; proving and simplifying.

Trigonometric Equations — solving over \([0,2\pi)\) and giving general solutions.

Laws & Vectors — the Law of Sines and Law of Cosines for any triangle; vector components, magnitude, and the dot product.

You'll be able to…

  • Prove identities and simplify trig expressions using fundamental identities.
  • Solve trigonometric equations on an interval and write general solutions.
  • Solve oblique triangles with the Law of Sines and Law of Cosines.
  • Add vectors, find magnitude and direction, and compute a dot product.

Worked example

Solve a trig equation Solve \(2\sin x - 1 = 0\) on \([0,2\pi)\): \(\sin x = \tfrac12\), so \[ x = \tfrac{\pi}{6} \quad\text{or}\quad x = \tfrac{5\pi}{6}. \]
Law of Cosines With \(a=7,\ b=8,\ C=60^\circ\): \[ c^2 = 7^2+8^2-2(7)(8)\cos 60^\circ = 113-56 = 57, \] so \(c=\sqrt{57}\approx 7.55\).
P.1D P.1E P.5G P.5H P.5I P.5J P.4K P.4L
Filled chip marks a standard frequently assessed as readiness for calculus.
Visualize this module → 🧱 Shaky on the basics? → Foundations
Module 05 · Capstone

Analytic Geometry, Polar & Series — The Edge of Calculus

Semester 2

Big idea: conics, polar curves, and infinite series are the doorway — describing motion and accumulation with the very language a first limit is about to formalize.

Topics

Conic Sections — standard forms of the parabola, ellipse, and hyperbola, and their key features.

Polar & Parametric — \((r,\theta)\) coordinates, polar curves, and parametric \((x(t),y(t))\) with conversions to and from rectangular form.

Sequences, Series & Limits — arithmetic and geometric series, sigma notation, the Binomial Theorem, and an intuitive first look at limits.

You'll be able to…

  • Write and graph conics from their standard forms.
  • Convert between polar and rectangular coordinates and plot polar curves.
  • Sum arithmetic and geometric series and evaluate an infinite geometric series.
  • Describe the limit of a sequence or function informally — the gateway to calculus.

Worked example

Infinite geometric series For \(a=8\) and \(r=\tfrac12\) (\(|r|<1\)): \[ \sum_{k=1}^{\infty} 8\left(\tfrac12\right)^{k-1} = \frac{a}{1-r} = \frac{8}{1-\tfrac12} = 16. \]
Polar to rectangular The point \(\left(r,\theta\right)=\left(4,\tfrac{\pi}{3}\right)\) becomes \[ x = 4\cos\tfrac{\pi}{3}=2, \qquad y = 4\sin\tfrac{\pi}{3}=2\sqrt3. \]
P.1F P.1G P.3A P.3B P.3C P.3D P.3E P.5D P.5E P.5F
Filled chip marks a standard frequently assessed as readiness for calculus.
Visualize this module → 🧱 Shaky on the basics? → Foundations
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What You'll Build

Pre-Calculus is the toolkit calculus will reach for on day one. By the end of the year, every scholar can:

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Read a function's structure

Name the family and call out domain, range, intercepts, asymptotes, symmetry, and end behavior straight from an equation or graph.

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Think on the unit circle

Recall exact trig values, convert radians and degrees, and graph sine and cosine with the right amplitude, period, and phase.

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Prove, solve & measure

Prove identities, solve trig equations, and apply the Laws of Sines and Cosines and vectors to real geometry.

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Stand at the edge of calculus

Handle conics, polar and parametric curves, and series — and describe a limit informally before the formal definition arrives.

The Toolkit

Required Materials

  • Graphing calculator (TI-84 or equivalent) or a school Chromebook running Desmos
  • Interactive notebook (composition or spiral, dedicated to this class)
  • Pencils — mathematics is always drafted in pencil
  • Graph paper for hand-built coordinate, polar, and unit-circle work
Curriculum

Built on the TEKS

This course is organized around the Texas Essential Knowledge and Skills for Precalculus (§111.42): the mathematical-process standards (P.1), Functions (P.2), Relations and geometric reasoning (P.3), Number and measure (P.4), and Algebraic reasoning (P.5). There is no STAAR End-of-Course exam for Pre-Calculus — it is the bridge year into Calculus. With Algebra I, Geometry, and Algebra II as prerequisites, the year paces toward calculus-readiness across two semesters, and mastery is assessed locally through teacher-built unit tests and benchmarks.

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Learning Resources & Supports

Free, vetted places to practice between class — plus the on-site reference sheet you'll lean on all year. Use them when you're stuck, then bring questions to class.

On-site

Pre-Calculus Reference Sheet

The unit circle, the core trig identities, function-family summaries, conic standard forms, sequence and series formulas, and polar/parametric conversions — all on one printable page.

Open the Reference Sheet →
Skill practice

IXL — Precalculus

Targeted skill practice that adapts to you, with instant feedback. Great for drilling a single skill — composition, the unit circle, identities, conics — until it clicks.

ixl.com/math/precalculus →
Video + practice

Khan Academy — Precalculus

Free video lessons and practice sets covering every topic in this course. Best when you want a concept re-taught a different way.

khanacademy.org/math/precalculus →
Graphing tool

Desmos Graphing Calculator

The free graphing calculator we use in class. Plot functions, switch to polar mode, add a parameter \(t\), and watch trig and parametric curves trace live.

desmos.com/calculator →
Coming to the Assessment Center

Module Practice & Calculus-Readiness Checks

Aligned Pre-Calculus practice sets and unit benchmarks will be delivered through the STEM Studio Assessment Center. A scholar progress view and module checkpoints are in development — the math layer is being built now and will go live as modules are released. (No live math analytics or scores exist yet; this is what's coming.)

Stuck on a problem? Try the worked example for that module above, pull up the Reference Sheet, regraph it on Desmos or the module's Visual Lab, and check whether a trig answer falls in the right quadrant before you commit to it. Still stuck? Bring it to class, office hours, or message through ParentSquare — asking precise questions is itself a college-math skill.

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Where to Go Next

Four doors into the course. Start with the syllabus, or jump straight to the reference sheet.

Course Syllabus

Policies, the studio learning environment, grading, expectations, and the full itinerary by grading period. Acknowledgment due on ParentSquare by the second week.

Visit Student Support

Pacing Guide

Every module mapped across the two-semester calendar, with the order each function family, trig topic, and analytic-geometry idea is introduced.

View the Pacing Guide

Reference Sheet

One printable page: the unit circle, trig identities, function families, conic forms, series formulas, and polar/parametric conversions — everything you'll reach for all year.

Open the Reference Sheet

Instructor: Dr. Goodluck Ijezie-Desbois, PharmD · Beta Academy · Room: TBA
Reach out by appointment, at gijezie-desbois@betaacademy.org, or through ParentSquare.

Course organized on the Texas Essential Knowledge and Skills for Precalculus (§111.42). Pre-Calculus is the bridge year into Calculus; prerequisites are Algebra I, Geometry, and Algebra II.