Precalculus

What you will learn

• Analyze and combine functions.
• Use radians and the unit circle.
• Graph trigonometric functions.
• Solve exponential, logarithmic, and trigonometric equations.
• Model change with several function families.

Common challenges

• Radians

  Radians connect an angle to arc length rather than degrees. Practice familiar angles in both units and check calculator mode.

• Trigonometric identities

  An identity is true throughout its allowed domain. Rewrite one side using known identities instead of testing just one angle.

• Function notation

  Composition and multiplication mean different things. Write out the inner function before substituting it into the outer one.

Key concepts

Unit circle

On a circle of radius 1, the point at angle theta has coordinates (cos(theta), sin(theta)).

Radians

Radian measure is arc length divided by radius. A full turn is 2 pi radians.

Periodicity

A periodic function repeats its outputs over a fixed input interval. Sine and cosine repeat every 2 pi radians.

Composition

For f(g(x)), calculate g first and then apply f. Check that the intermediate output is an allowed input.

Inverse functions

An inverse reverses a one-to-one function. Restricting a domain can make an inverse possible.

Average rate of change

The difference in outputs divided by the difference in inputs measures change across an interval.

How to prepare

• Review factoring and rational expressions.
• Practice graphing quadratic and exponential functions.
• Review right-triangle sine, cosine, and tangent.
• Make a degree-to-radian chart for 0, 30, 45, 60, and 90 degrees.

Study habits that help

• Sketch the unit circle from memory, then check it.
• Compare a function's equation, table, and graph.
• Label every trigonometry problem with degrees or radians.
• Practice a few algebra problems alongside each new topic.

What to expect

Typical workload

Moderate

Time outside class

Typically about 4 to 6 hours outside class; your school and starting skills may change this.

Pace

Units often connect algebraic rules, graphs, and applications at a steady pace.

How it is usually graded

Tests, quizzes, problem sets, and sometimes modeling projects commonly contribute to grades.

Skills that help

• Algebra fluency
• Graph reading
• Exact-value calculations
• Connecting representations

What teachers usually expect

• Give exact values when asked.
• Check angle units before calculating.
• Explain how a graph supports your answer.

Before you start

• Typical Algebra 2 skills
• Geometry and right-triangle relationships
• Comfort solving equations and using function notation

These are general descriptions of a typical course. Workload, pace and prerequisites differ by school and instructor, so check with your counselor, advisor or the course catalog.

Where it can lead

• Engineering
• Physics
• Mathematics
• Economics
• Computer science

Flashcards

Practice questions

Vocabulary

Amplitude

Half the distance between the maximum and minimum of a sinusoidal graph.

Period

The smallest positive input shift over which a periodic function repeats.

Radian

The angle unit for which an arc of one radius has measure 1.

Identity

An equation true for every input where both sides are defined.

One-to-one

Describes a function in which different inputs give different outputs.

End behavior

How a function behaves as the input increases or decreases without bound.

Secant line

A line through two points on a graph.

Phase shift

A horizontal shift of a periodic graph.