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Precalculus Study Guide

9 units, each with what to know, how to apply it, the mistakes that cost marks, and a linked practice set.

Nine units

Function families, a full trigonometry sequence, polar coordinates, then sequences and modeling.

Calculus preparation

The course exists to make the transition into calculus routine rather than abrupt.

Prerequisite

Algebra 2 or Algebra and Trigonometry.

Where it leads

AP Calculus, AP Statistics, and college mathematics.

Units

The 9 units

Precalculus revisits every function family with more analytical weight and adds the trigonometric and polar tools that calculus assumes. The recurring demand is to move between representations and justify a claim from whichever one makes it clearest.

1

Quadratic and Polynomial Functions

Zeros, multiplicity, and end behavior. Together these three facts determine the shape of any polynomial graph before a single point is plotted.

2

Radical and Rational Functions

Two families defined by what they exclude. Radicals restrict the domain; rationals introduce asymptotes and holes, and both create extraneous solutions.

3

Exponential and Logarithmic Functions

Growth by constant ratio, and the inverse that recovers the exponent. Most of the unit follows from those two sentences.

4

Unit Circle and Trigonometric Functions

Extending trigonometry beyond right triangles. Structure and symmetry replace memorisation, which is what makes unfamiliar angles tractable.

5

Trigonometric Graphs and Models

Four parameters that describe any sinusoid, and using them to model anything that repeats.

6

Identities and Trigonometric Equations

Rewriting trigonometric expressions into a form that can be evaluated or solved, and reporting every solution the interval demands.

7

Analytic and Applied Trigonometry

Triangles that are not right-angled, and the inverse functions that recover an angle. Choosing the correct law from the given parts is most of the work.

8

Polar Coordinates

A second way to locate a point, and the curves that are far easier to describe in it than in rectangular form.

9

Sequences, Series, and Integrated Modeling

Sequences and series alongside the model-selection judgement that closes the course.

Habits

The habits that run through every unit

Four habits this sequence assesses directly, whichever unit the question comes from.

1. Write what you were given

List the known quantities, the quantity asked for, and any restriction on the domain before starting. Most lost marks come from answering a slightly different question.

2. Choose a representation

Equation, graph, table, or diagram are four views of one relationship. When one stalls, switch; the assessment usually rewards moving between them.

3. Justify the step, not the answer

Name the property or theorem that licenses each move. On written work the justification is frequently worth more than the arithmetic.

4. Test the boundary

Zero, a negative value, an excluded value, an endpoint. A method that works in the middle of a range fails at its edges, and that is where questions are set.

Plan

How to use this guide

Consolidate the function families

The first three units are revision with more rigour. Use them to close Algebra 2 gaps before trigonometry begins.

Make exact values automatic

Four units depend on producing unit circle values without hesitation. This is the highest-leverage memorisation in the course.

Practise choosing the law

Naming the configuration before reaching for the Law of Sines or Cosines prevents most errors in the applied unit.

Treat polar as a change of view

Polar coordinates describe the same plane differently. Converting fluently in both directions is what the unit assesses.

Multiple choice

  • • Identify what kind of object the question is about before computing; the structure usually names the method.
  • • Check for excluded values whenever a denominator, an even root, or a logarithm appears.
  • • When two options differ only at an endpoint, test that endpoint rather than re-reading the question.
  • • Estimate the size of the answer first so an implausible result is caught immediately.
  • • Watch for questions asking which statement is true rather than for a value; those are read carelessly most often.

Written work

  • • Show the steps that carry the reasoning, not every line of arithmetic.
  • • Name the property, theorem, or identity that justifies the key move.
  • • Give units and context in the final sentence when the problem had them.
  • • State any restriction you imposed and why.
  • • Check the answer back in the original statement before moving on.

Scope

What this course covers

Core

  • • Polynomial and rational function analysis
  • • Exponential and logarithmic functions and equations
  • • The unit circle, sinusoidal graphs, and identities
  • • Trigonometric equations and triangle applications
  • • Polar coordinates and polar graphs
  • • Sequences, series, and convergence

Breadth

  • • Model selection and interpretation
  • • Solving graphically where no exact route exists
  • • Preparation for limits and rates of change

Varies by course

  • Limits. Included in honors versions; this standard sequence stops before them.
  • Vectors and matrices. Coverage varies.
  • Conic sections. Sometimes revisited here.