What Does Precalculus Cover? MATH 115 Chapter by Chapter

Precalculus covers the algebra, functions, graphing and trigonometry that help you prepare for calculus. It also often includes topics such as systems of equations, vectors and sequences.

What is precalculus math?

Precalculus is a course of study that brings together mathematical tools used in calculus. Rather than focusing only on solving for an unknown, you learn to describe how quantities relate through functions and to interpret those relationships in equations, graphs and real-world situations.

For example, if a population grows by the same percentage each year, an exponential function can model that growth. Understanding its graph and how its inputs and outputs relate gives you a foundation for studying changing quantities in calculus.

Precalculus: chapter summaries and practice answers

The table below shows the full UPI Study Pre-Calculus MATH 115 syllabus, including lessons and hours. The summaries that follow explain what each chapter adds.

1. Algebraic foundations: Review exponents, radicals, factoring, inequalities and coordinate geometry. These skills support the rest of the course.
2. Functions and their behavior: Study domain, range, graphs, transformations, composition and inverses—the language used to describe relationships throughout precalculus.
3. Linear and quadratic functions: Graph and apply straight-line and quadratic models, solve quadratic equations and inequalities, and work with linear regression.
4. Polynomial and rational functions: Analyze zeros and graphs, use polynomial division, work with complex numbers, and solve polynomial and rational equations and inequalities.
5. Exponential and logarithmic functions: Study growth and decay, logarithm properties, graphs and equations.
6. Trigonometric foundations: Learn degrees and radians, the unit circle, the six trigonometric functions, right triangles and periodic models.
7. Graphs of trigonometric functions: Graph and transform trigonometric functions and study inverse trigonometric functions.
8. Trigonometric identities and equations: Use fundamental, sum-and-difference, double-angle and half-angle identities to simplify expressions and solve equations.
9. Applications of trigonometry: Solve oblique triangles with the Laws of Sines and Cosines, and explore polar coordinates, complex numbers in polar form and vectors.
10. Systems of equations and matrices: Solve linear and nonlinear systems, use matrices and Gaussian elimination, and study partial fraction decomposition.
11. Sequences, series and counting: Work with arithmetic and geometric sequences, sums, the binomial theorem and basic probability.

For a quick practice check, let f(x) = x² and g(x) = x + 1. Then f(g(2)) = f(3) = 9. This is function composition: evaluate the inner function first, then use its result as the input to the outer function.

Does precalculus cover trigonometry?

Yes. Trigonometry is a major part of MATH 115, spanning foundations, graphs, identities and applications. You move from angles and right triangles to tools for describing periodic behavior and solving triangles that do not have a right angle.

For example, on the unit circle, sine and cosine describe coordinates associated with an angle. Their repeating values also make them useful for modeling periodic phenomena, such as a simplified cycle of daylight hours.

What is the difference between college algebra and precalculus?

College algebra generally emphasizes algebraic equations, functions and graphs. Precalculus often builds on those ideas and adds more extensive trigonometry, although course titles and content vary by institution.

Compare the actual syllabi rather than relying on the names alone. If your algebra skills feel rusty, review factoring, fractions, exponents and solving equations before starting; this guide to restarting STEM math after time away can help you plan that review.

What is the difference between precalculus and calculus?

Precalculus concentrates on functions and the algebraic and trigonometric tools needed to work with them. Calculus uses those foundations to study concepts such as limits, derivatives and integrals, including rates of change and accumulation.

For instance, precalculus might ask you to graph a quadratic function; calculus can use a derivative to find its instantaneous rate of change. See precalculus versus calculus for a closer comparison.

Do you have to take precalculus before calculus?

Not always. The answer depends on your college’s prerequisites and your prior coursework or placement results. Even when a separate precalculus course is not required, you will need a strong grasp of its core algebra and trigonometry skills.

Check the requirements for the specific calculus course you want to take. If you can solve equations, interpret function graphs and use the unit circle comfortably, you have a useful starting point; if not, a review may be worthwhile. Read more about whether you need precalculus before Calculus 1.

Frequently Asked Questions about Precalculus

Final Thoughts on Precalculus

Start by comparing the chapter list with your current skills and your college’s calculus prerequisites. If functions, algebraic manipulation or trigonometry are unfamiliar, use those areas to guide your review.