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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 12-week schedule.

College Algebra course map: 6 units
#	Unit	What you can do after it	Weeks
01	Equations and inequalities	You can solve linear, quadratic, absolute-value and compound problems and write the answers as intervals.	Weeks 1-2
02	Functions	You can use function notation, find domain and range, and build new functions out of old ones.	Weeks 3-4
03	Polynomial models	You can find every root of a polynomial, including repeated ones, and sketch the graph from them.	Weeks 5-6
04	Rational models	You can work with formulas that have variables in the denominator, including work-rate and mixture problems.	Weeks 7-8
05	Exponential and logarithmic models	You can model interest, population and cooling with exponentials, then undo them with logs.	Weeks 9-10
06	Systems	You can solve systems in two or three variables by elimination or matrices, including a line meeting a curve.	Weeks 11-12

College Algebra course units

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

Equations and inequalities

Weeks 1-2

You can solve linear, quadratic, absolute-value and compound problems and write the answers as intervals. You can also solve a formula for whichever letter you need.

Solve an absolute-value equation by splitting it into two cases.
Write -3 < x and x <= 7 as the interval (-3, 7].
Solve a quadratic inequality by testing signs on each side of the roots.

Worked example

Solve the absolute value of (2x - 5) = 9

Case 1: 2x - 5 = 9, so x = 7
Case 2: 2x - 5 = -9, so x = -2

Answerx = 7 or x = -2

Most common mistakeSolving the positive case only: reporting x = 7 alone and losing x = -2, which also sits exactly 9 units away.

Explain equations and inequalities Practice equations and inequalities

Functions

Weeks 3-4

You can use function notation, find domain and range, and build new functions out of old ones. You can also evaluate a piecewise rule on either side of its break.

Find the domain of a square-root function.
Evaluate a piecewise function just below and just above the break point.
Compute (f - g)(2) from two given formulas.

Worked example

Find the domain of f(x) = the square root of (x - 4)

What is under the root cannot be negative: x - 4 is 0 or more
x is 4 or more

AnswerThe interval [4, infinity)

Most common mistakeRequiring the inside to be strictly positive: writing (4, infinity) and throwing out x = 4, even though f(4) = 0 is a perfectly good value.

Explain functions Practice functions

Polynomial models

Weeks 5-6

You can find every root of a polynomial, including repeated ones, and sketch the graph from them. You can also use a polynomial to model a volume or a profit.

List the possible rational roots with the p over q rule.
Explain why a doubled root makes the graph touch the axis and bounce off.
Build a profit polynomial as revenue minus cost.

Worked example

Factor x cubed - 4x squared + 4x and list the roots

Take out the common x: x(x squared - 4x + 4)
x(x - 2)(x - 2)

AnswerRoots x = 0, and x = 2 counted twice

Most common mistakeDividing both sides by x to clear it: that throws away the root x = 0 and leaves only x = 2, so the graph's crossing at the origin disappears.

Explain polynomial models Practice polynomial models

Rational models

Weeks 7-8

You can work with formulas that have variables in the denominator, including work-rate and mixture problems. You can also describe asymptotes and end behaviour.

Solve a work problem where two pipes fill one tank together.
Find a slant asymptote by long division.
Reject any answer that makes a denominator zero.

Worked example

Pipe A fills a tank in 6 hours, pipe B in 3 hours. How long together?

Rates add: 1/6 + 1/3 = 1/6 + 2/6
3/6 = 1/2 tank per hour

Answer2 hours

Most common mistakeAveraging the two times: reporting (6 + 3)/2 = 4.5 hours, which is slower than pipe B managed on its own.

Explain rational models Practice rational models

Exponential and logarithmic models

Weeks 9-10

You can model interest, population and cooling with exponentials, then undo them with logs. You can also compare yearly, quarterly and continuous compounding.

Use A = P(1 + r/n) to the power nt for quarterly compounding.
Solve for the time it takes an amount to double.
Convert between a continuous rate and a yearly percentage.

Worked example

2,000 invested at 5% compounded quarterly. Find the value after 8 years.

A = 2000(1 + 0.05/4) to the power (4 x 8)
A = 2000 x 1.0125 to the power 32 = 2000 x 1.4881

AnswerAbout 2,976.20

Most common mistakeForgetting to divide the rate by 4: using 1.05 to the power 32 and reporting about 9,682, more than three times the true amount.

Explain exponential and logarithmic models Practice exponential and logarithmic models

Systems

Weeks 11-12

You can solve systems in two or three variables by elimination or matrices, including a line meeting a curve. You can also shade the overlap for a system of inequalities.

Solve a three-variable system by eliminating the same variable twice.
Solve a line-and-circle system by substitution.
Shade the region where two inequalities are both true.

Worked example

Solve y = x + 1 together with x squared + y squared = 25

Substitute: x squared + (x + 1) squared = 25
2x squared + 2x + 1 = 25, so x squared + x - 12 = 0

Answer(x + 4)(x - 3) = 0, giving the points (3, 4) and (-4, -3)

Most common mistakeExpanding (x + 1) squared as x squared + 1: the equation collapses to 2x squared = 24 and returns points that are not on the circle at all.

Explain systems Practice systems

Prepare for College Algebra practice

Start with the earliest uncertain prerequisite

Before timing yourself, check whether you can explain Equations and inequalities from a blank page. Then connect it to Functions. 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 Algebra calculator can test calculations and representations used in College Algebra. 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 College Algebra 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.

GuidesAlgebra explainedConcepts, worked methods and checks behind this course.

PracticeMixed Algebra II questionsAnswer review after every attempt.

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

Assess

Take the complete College Algebra test

Begin test →

Reference

Review essential formulas

Open library →

Calculate

Use the Algebra calculator

Open tool →

Plan

Prepare around your exam date

Build plan →

Questions about the College Algebra course

Where should I start?

Start with Equations and inequalities if you are following the full sequence. If that unit feels automatic, open the College Algebra 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 College Algebra test only after you can correct practice errors from a blank page.

Which calculator supports this course?

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

Learn College Algebra unit by unit

Units at a glance

College Algebra course units

Equations and inequalities

Worked example

Functions

Worked example

Polynomial models

Worked example

Rational models

Worked example

Exponential and logarithmic models

Worked example

Systems

Worked example

Prepare for College Algebra 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

Take the complete College Algebra test

Review essential formulas

Use the Algebra calculator

Prepare around your exam date

Questions about the College Algebra course

Watch the worked lesson

Grade 11 Math: Polynomial Models | Build, Evaluate, and Check a Volume Model

Learn College Algebra unit by unit

Units at a glance

College Algebra course units

Equations and inequalities

Worked example

Functions

Worked example

Polynomial models

Worked example

Rational models

Worked example

Exponential and logarithmic models

Worked example

Systems

Worked example

Prepare for College Algebra 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 university courses

Calculus I

Calculus II

Multivariable Calculus

Take the complete College Algebra test

Review essential formulas

Use the Algebra calculator

Prepare around your exam date

Questions about the College Algebra course

Watch the worked lesson

Grade 11 Math: Polynomial Models | Build, Evaluate, and Check a Volume Model