Precalculus study guide
This guide is designed to help you understand the core ideas that commonly appear in introductory college Precalculus courses. Read the explanation, follow the example, then try the practice questions without looking back. The goal is to understand the reasoning—not just memorize an answer.
Course requirements vary by school and instructor. Use this as a learning and review resource alongside your assigned textbook, lectures, and instructor guidance.
Functions & transformations
Concept: Functions can be shifted, stretched, reflected, and combined. Understanding transformations helps you predict a graph from its parent function without plotting every point.
Worked example: For y=(x-2)^2+3, the basic parabola y=x^2 moves 2 units right and 3 units up.
Remember: Focus on the central relationship in functions & transformations, define the important terms, and connect each step to the evidence, rule, formula, or process being used.
Common mistakes: Common mistakes include memorizing functions & transformations without understanding when it applies, skipping units or definitions, and giving an answer without showing the reasoning that supports it.
Practice it
- Explain functions & transformations in your own words.
- Use the lesson idea to create or solve one new precalculus example.
- What is one common mistake a student should avoid when working with functions & transformations?
Polynomial & rational functions
Concept: Polynomial functions are built from powers of x, while rational functions are ratios of polynomials. Zeros, end behavior, holes, and asymptotes describe their graphs.
Worked example: For f(x)=(x+1)/(x-2), x=2 is excluded and creates a vertical asymptote.
Remember: Focus on the central relationship in polynomial & rational functions, define the important terms, and connect each step to the evidence, rule, formula, or process being used.
Common mistakes: Common mistakes include memorizing polynomial & rational functions without understanding when it applies, skipping units or definitions, and giving an answer without showing the reasoning that supports it.
Practice it
- Explain polynomial & rational functions in your own words.
- Use the lesson idea to create or solve one new precalculus example.
- What is one common mistake a student should avoid when working with polynomial & rational functions?
Exponential & logarithmic functions
Concept: Exponential functions model repeated multiplication; logarithms undo exponentials. Their inverse relationship is central to growth, decay, pH, and compound-interest problems.
Worked example: 2^x=16 gives x=4; equivalently log_2(16)=4.
Remember: Focus on the central relationship in exponential & logarithmic functions, define the important terms, and connect each step to the evidence, rule, formula, or process being used.
Common mistakes: Common mistakes include memorizing exponential & logarithmic functions without understanding when it applies, skipping units or definitions, and giving an answer without showing the reasoning that supports it.
Practice it
- Explain exponential & logarithmic functions in your own words.
- Use the lesson idea to create or solve one new precalculus example.
- What is one common mistake a student should avoid when working with exponential & logarithmic functions?
Trigonometric functions
Concept: Sine, cosine, and tangent connect angles to side ratios and repeat periodically on the unit circle.
Worked example: At 30 degrees, sin(30°)=1/2 and cos(30°)=√3/2.
Remember: Focus on the central relationship in trigonometric functions, define the important terms, and connect each step to the evidence, rule, formula, or process being used.
Common mistakes: Common mistakes include memorizing trigonometric functions without understanding when it applies, skipping units or definitions, and giving an answer without showing the reasoning that supports it.
Practice it
- Explain trigonometric functions in your own words.
- Use the lesson idea to create or solve one new precalculus example.
- What is one common mistake a student should avoid when working with trigonometric functions?
Sequences & series
Concept: A sequence is an ordered list; a series is a sum of sequence terms. Arithmetic patterns use a constant difference and geometric patterns use a constant ratio.
Worked example: 3,7,11,... is arithmetic with common difference 4.
Remember: Focus on the central relationship in sequences & series, define the important terms, and connect each step to the evidence, rule, formula, or process being used.
Common mistakes: Common mistakes include memorizing sequences & series without understanding when it applies, skipping units or definitions, and giving an answer without showing the reasoning that supports it.
Practice it
- Explain sequences & series in your own words.
- Use the lesson idea to create or solve one new precalculus example.
- What is one common mistake a student should avoid when working with sequences & series?