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Dr. Shreyank Educare
Curriculum-aligned learning support — not an official school, university, or ministry-approved course, and not a replacement for official credit.

Grade 12 · Mathematics · Advanced

Pre-Calculus 12

Functions, polynomials, exponential and logarithmic functions, geometric sequences and series, and trigonometry for calculus.

Contents

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1

Transformations of Functions

3 of 4 published

  1. 1.1
    Translations

    Move a graph without changing its shape, so you can read f(x−h)+kf(x-h)+k at a glance.

    Published

  2. 1.2
    Stretches

    Scale a graph and predict exactly where its key points land.

    Published

  3. 1.3
    Inverses

    Undo a function by swapping input and output — where inverses come from.

    Published

  4. 1.4
    Composition and operations on functions

    Combining two functions by composition or by adding, subtracting, multiplying and dividing them.

    In preparation

2

Polynomial & Rational Functions

3 of 3 published

  1. 2.1
    Polynomial graphs

    End behaviour and roots.

    Published

  2. 2.2
    Factor theorem

    Finding roots.

    Published

  3. 2.3
    Rational functions

    Asymptotes.

    Published

3

Exponential & Logarithmic Functions

3 of 3 published

  1. 3.1
    Exponential functions

    Growth and decay.

    Published

  2. 3.2
    Logarithms

    Inverse of exponentials.

    Published

  3. 3.3
    Log laws

    Solving log equations.

    Published

4

Geometric Sequences & Series

0 of 2 published

  1. 4.1
    Geometric sequences and series

    Common ratio, first term, general term, and the link to exponential functions.

    In review

  2. 4.2
    Infinite geometric series and sigma notation

    Summing an infinite geometric series and writing sums in sigma notation.

    In review

5

Trigonometry

3 of 4 published

  1. 5.1
    Radians

    Angle measure.

    Published

  2. 5.2
    Identities

    Proving identities.

    Published

  3. 5.3
    Trig equations

    Solving over intervals.

    Published

  4. 5.4
    Graphing trig functions with transformations

    Amplitude, period, phase shift and midline, and sketching a transformed sine or cosine graph.

    In review

6

ExtensionExtension, not counted toward course totals

3 of 3 published

  1. 6.1
    Counting principles

    Fundamental counting.

    Published

  2. 6.2
    Permutations

    Ordered arrangements.

    Published

  3. 6.3
    Binomial theorem

    Expanding powers.

    Published

What is Pre-Calculus 12 in British Columbia?

Short answer

Pre-Calculus 12 is the BC senior-math course that completes the preparation for calculus. It covers transformations of functions, polynomial and rational functions, exponential and logarithmic functions, deeper trigonometry (functions, identities, and equations), and permutations and combinations. It is the expected prerequisite for Calculus 12 and AP Calculus.

Course overview

Pre-Calculus 12 is about functions as objects you can transform, compose, and invert. You study how graphs shift and stretch, work with polynomial, rational, exponential, and logarithmic functions, and take trigonometry to the level of identities and equations. A unit on permutations and combinations introduces systematic counting.

What you will learn

  • Transform, combine, and invert functions and read their graphs
  • Analyse polynomial and rational functions
  • Work with exponential and logarithmic functions and solve related equations
  • Prove and apply trigonometric identities and solve trigonometric equations
  • Count systematically with permutations and combinations

Who this course is for

  • • BC students heading into Calculus 12 or AP Calculus
  • • Students who want to solidify functions and trigonometry before calculus
  • • Anyone who finds logarithms, trig identities, or function transformations difficult

Commonly difficult topics

  • • Function transformations — predicting how a change to the equation moves the graph
  • • Logarithms — treating them as the inverse of exponentials rather than a new rule set
  • • Trigonometric identities — choosing a productive first step in a proof
  • • Rational functions — asymptotes and domain restrictions

How Dr. Shreyank Educare teaches this course

  1. 01 · Diagnose

    Find the exact skill causing difficulty — not the chapter, the skill.

  2. 02 · Understand

    Learn it through explanations, worked examples and visual reasoning.

  3. 03 · Practice

    Work problems that adapt to how you are actually doing.

  4. 04 · Fix

    A wrong answer becomes targeted remediation for the misconception behind it.

  5. 05 · Prove

    Solve new, unseen problems independently before mastery is awarded.

Course curriculum

1Transformations of Functions4 topics
  • — Translations
  • — Stretches
  • — Inverses
  • — Composition and operations on functions
2Polynomial & Rational Functions3 topics
  • — Polynomial graphs
  • — Factor theorem
  • — Rational functions
3Exponential & Logarithmic Functions3 topics
  • — Exponential functions
  • — Logarithms
  • — Log laws
4Geometric Sequences & Series2 topics
  • — Geometric sequences and series
  • — Infinite geometric series and sigma notation
5Trigonometry4 topics
  • — Radians
  • — Identities
  • — Trig equations
  • — Graphing trig functions with transformations
6Extension3 topics
  • — Counting principles
  • — Permutations
  • — Binomial theorem

A sample learning experience

Solve 2ˣ = 16 for x using logarithms, and explain why the answer makes sense.

  1. 1. Recognize 16 as a power of 2: 16 = 2⁴.
  2. 2. So 2ˣ = 2⁴, and since the bases match, x = 4.
  3. 3. Check with logs: x = log₂ 16 = 4, because a logarithm answers “what exponent gives this value?”
  4. 4. It makes sense: doubling four times (2·2·2·2) gives 16.

Frequently asked questions

Is Pre-Calculus 12 harder than Pre-Calculus 11?
It is more abstract — you work with more function types and with trigonometric identities and equations. Students who are fluent with Pre-Calculus 11 algebra usually manage the step up well.
Do I need Pre-Calculus 12 for Calculus 12 or AP Calculus?
Yes. Both assume fluency with functions, logarithms, and trigonometry from Pre-Calculus 12.
How much trigonometry is in Pre-Calculus 12?
A significant amount: trigonometric functions, graphs, identities, and equations — well beyond the introductory trig of Pre-Calculus 11.
What is the difference between Pre-Calculus 12 and Calculus 12?
Pre-Calculus 12 prepares the functions and algebra; Calculus 12 then studies limits, derivatives, and integration built on those functions.
What is the best way to study Pre-Calculus 12?
Practise recognizing which function family and which technique a problem needs, then drill the exact step you find hardest — often log manipulation or an identity proof.

Related courses

Pre-Calculus 11 →Calculus 12 →AP Calculus AB/BC →

Explore all Mathematics courses →

Curriculum & sources

Content source: original lessons, worked examples and diagrams created for Dr. Shreyank Educare.

Curriculum alignment: Grade 12 Mathematics (BC curriculum).

Curriculum-aligned learning support — not an official school, ministry, or College Board course, and not a replacement for official credit.

Last updated . Authored and maintained in-house by the Dr. Shreyank Educare teaching team — see our editorial standards, methodology and source policy.