Learn Precalculus unit by unit

Connect functions, trigonometry, and analytic geometry before calculus.

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Units at a glance

Every unit in this course, what you will be able to do once you finish it, and roughly when it lands in a 14-week schedule.

Precalculus course map: 7 units
#	Unit	What you can do after it	Weeks
01	Function analysis	You can take functions apart, glue them together, and undo one with its inverse.	Weeks 1-2
02	Polynomial and rational models	You can match a polynomial or rational formula to a real situation and read what its graph is telling you.	Weeks 3-4
03	Exponential and logarithmic functions	You can work with e, natural logs and change of base, and solve equations that mix them together.	Weeks 5-6
04	Trigonometric functions	You can find sine, cosine and tangent for any angle and sketch the waves they trace.	Weeks 7-8
05	Conic sections	You can recognise a circle, ellipse, parabola or hyperbola from its equation and find its centre, vertices and foci.	Weeks 9-10
06	Sequences	You can define a sequence with a direct formula or a step-by-step rule, and add geometric terms exactly.	Weeks 11-12
07	Limits preview	You can say what a function is heading toward near a point, even when it never actually gets there.	Weeks 13-14

Precalculus course units

Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.

Function analysis

Weeks 1-2

You can take functions apart, glue them together, and undo one with its inverse. You can also describe every shift, stretch and flip you see in a graph.

Find f(g(x)) when f(x) = x squared and g(x) = x - 3.
Find the inverse of f(x) = 2x + 7 by swapping x and y.
Say how y = -f(x + 2) moves and flips the graph of f.

Worked example

Find the inverse of f(x) = 2x + 7

Write y = 2x + 7, then swap the letters: x = 2y + 7
x - 7 = 2y

AnswerThe inverse is (x - 7)/2

Most common mistakeMistaking the inverse for the reciprocal: writing 1/(2x + 7), which sends 3 to about 0.077 instead of back to -2.

Explain function analysis Practice function analysis

Polynomial and rational models

Weeks 3-4

You can match a polynomial or rational formula to a real situation and read what its graph is telling you. You can also find holes, asymptotes and turning behaviour.

Find the horizontal asymptote of (3x squared + 1)/(x squared - 5) by comparing lead terms.
Spot a hole where the same factor cancels top and bottom.
Build a cubic volume model for a box made by cutting corners out of a sheet.

Worked example

Find the horizontal asymptote of y = (3x squared + 1)/(x squared - 5)

Top and bottom are both degree 2, so the degrees tie
Divide the leading coefficients: 3/1

Answery = 3

Most common mistakeSetting the numerator to zero instead: solving 3x squared + 1 = 0, finding nothing, and concluding there is no asymptote when the graph clearly levels off at y = 3.

Explain polynomial and rational models Practice polynomial and rational models

Exponential and logarithmic functions

Weeks 5-6

You can work with e, natural logs and change of base, and solve equations that mix them together. You can also model compound interest and half-life.

Solve e to the power 2x = 15 using natural logs.
Rewrite log base 7 of 200 in a form a calculator will accept.
Find the half-life from a decay model such as A = A0 times e to the power -0.05t.

Worked example

Solve e to the power 2x = 15

Take the natural log of both sides: 2x = ln 15
2x = 2.708

Answerx is about 1.354

Most common mistakeCancelling e with a base-10 log: writing 2x = log 15 = 1.176 and answering x = 0.588, less than half the true value.

Explain exponential and logarithmic functions Practice exponential and logarithmic functions

Trigonometric functions

Weeks 7-8

You can find sine, cosine and tangent for any angle and sketch the waves they trace. You can also read amplitude, period and shift straight from a formula.

Convert 150 degrees into radians.
Find the period of y = 4 sin(3x).
Give the exact value of cos(5 pi / 6) without a calculator.

Worked example

Find the amplitude and period of y = 4 sin(3x)

Amplitude is the size of the 4 out front
Period = 2 pi / 3

AnswerAmplitude 4, period 2 pi / 3

Most common mistakeMultiplying by the 3 rather than dividing: reporting a period of 6 pi, nine times too long, so the sketch shows one wave where there should be nine.

Explain trigonometric functions Practice trigonometric functions

Conic sections

Weeks 9-10

You can recognise a circle, ellipse, parabola or hyperbola from its equation and find its centre, vertices and foci. You can also complete the square to tidy a messy equation up.

Sort an equation by looking at the signs and coefficients of the squared terms.
Complete the square on x squared + 6x + y squared = 7 to find the centre.
Find the foci of an ellipse using c squared = a squared - b squared.

Worked example

Find the centre and radius of x squared + 6x + y squared = 7

Complete the square: add 9 to both sides
(x + 3) squared + y squared = 16

AnswerCentre (-3, 0), radius 4

Most common mistakeAdding the 9 to the left side only: writing (x + 3) squared + y squared = 7 and reporting a radius of about 2.65 instead of 4.

Explain conic sections Practice conic sections

Sequences

Weeks 11-12

You can define a sequence with a direct formula or a step-by-step rule, and add geometric terms exactly. You can also tell whether an endless geometric sum settles on a number.

Turn the recursive rule a(n) = 2a(n-1) + 1 with a(1) = 3 into a list of terms.
Add the infinite series 8 + 4 + 2 + 1 and so on.
Write the sum of the first 12 terms using sigma notation.

Worked example

Add the infinite series 8 + 4 + 2 + 1 + ...

First term a = 8, ratio r = 1/2
S = a/(1 - r) = 8/0.5

Answer16

Most common mistakeUsing S = a/(1 - r) when the ratio is bigger than 1: for 2 + 4 + 8 + ... it returns -2, a negative total for a sum of positive numbers that never stops growing.

Explain sequences Practice sequences

Limits preview

Weeks 13-14

You can say what a function is heading toward near a point, even when it never actually gets there. You can also clear a 0/0 form by factoring first.

Read a one-sided limit off a graph.
Factor and cancel to evaluate a limit that starts out as 0/0.
Decide whether a piecewise function is continuous where its pieces meet.

Worked example

Find the limit of (x squared - 9)/(x - 3) as x approaches 3

Substituting gives 0/0, which tells you nothing yet
Factor: (x - 3)(x + 3)/(x - 3) cancels to x + 3

Answer3 + 3 = 6

Most common mistakeCalling the limit undefined because x = 3 breaks the fraction: the graph has a single missing dot at height 6, and the limit is 6.

Explain limits preview Practice limits preview

Prepare for Precalculus practice

Start with the earliest uncertain prerequisite

Before timing yourself, check whether you can explain Function analysis from a blank page. Then connect it to Polynomial and rational models. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.

Use tools to verify, not to choose the method for you

The Precalculus calculator can test calculations and representations used in Precalculus. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.

Know when to move to the full test

Move from Precalculus practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.

Before and after the syllabus

Learn the ideas, then practise them

The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.

GuidesPrecalculus explainedConcepts, worked methods and checks behind this course.

PracticeMixed Precalculus questionsAnswer review after every attempt.

ReferenceFormulas for this courseConditions and a worked example for each one.

Assess

Take the complete Precalculus test

Begin test →

Reference

Review essential formulas

Open library →

Calculate

Use the Precalculus calculator

Open tool →

Plan

Prepare around your exam date

Build plan →

Questions about the Precalculus course

Where should I start?

Start with Function analysis if you are following the full sequence. If that unit feels automatic, open the Precalculus practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.

How do I know I am ready for the course test?

You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Precalculus test only after you can correct practice errors from a blank page.

Which calculator supports this course?

The Precalculus calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.

Learn Precalculus unit by unit

Units at a glance

Precalculus course units

Function analysis

Worked example

Polynomial and rational models

Worked example

Exponential and logarithmic functions

Worked example

Trigonometric functions

Worked example

Conic sections

Worked example

Sequences

Worked example

Limits preview

Worked example

Prepare for Precalculus practice

Start with the earliest uncertain prerequisite

Use tools to verify, not to choose the method for you

Know when to move to the full test

Learn the ideas, then practise them

Other high school courses

Trigonometry

Probability & Statistics

Mathematical Modeling

Take the complete Precalculus test

Review essential formulas

Use the Precalculus calculator

Prepare around your exam date

Questions about the Precalculus course