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Precalculus

Master trigonometric functions, identities, polar coordinates, vectors, conic sections, sequences, limits, and parametric equations. Desmos visualizations for every major concept. TEKS §111.42 aligned.

14 Units

40-55 minutes per unit

What You Will Learn

Trigonometry & Analysis

Unit circle, identities, graphs, inverse trig, polar coordinates, and vectors — the full trigonometric toolkit.

Calculus Preview

Limits, rates of change, and area under curves — conceptual foundations that make Calculus click.

TEKS §111.42 Aligned

Covers Texas Essential Knowledge and Skills for Precalculus.

All Units

Functions: Analysis and Transformation

Review and extend function families, master transformations including shifts, reflections, and stretches, and analyze piecewise-defined functions to build the analytical foundation for Precalculus.

•Classify functions by family (linear, quadratic, absolute value, square root, cubic) and describe their key features
•Apply vertical and horizontal shifts, reflections, and stretches/compressions to parent functions
•Write and interpret piecewise-defined functions from graphs and real-world contexts
+1 more objectives

Polynomial and Rational Analysis

Analyze polynomial and rational functions through end behavior, zeros and multiplicity, the Rational Zero Theorem, and asymptotic behavior to develop fluency in curve sketching without a calculator.

•Determine end behavior of polynomial functions using degree and leading coefficient
•Identify zeros, their multiplicity, and the corresponding behavior of the graph at each zero
•Apply the Rational Zero Theorem and synthetic division to factor higher-degree polynomials
+1 more objectives

Exponential and Logarithmic Functions

Develop fluency with exponential and logarithmic functions, their properties and graphs, the natural logarithm, solving equations involving exponentials and logs, and modeling real-world growth and decay.

•Graph exponential and logarithmic functions and describe their key features including domain, range, and asymptotes
•Apply properties of logarithms (product, quotient, power rules) to simplify and expand expressions
•Solve exponential and logarithmic equations using algebraic techniques
+1 more objectives

Trigonometric Functions

Build trigonometric functions from the unit circle, master radian measure, graph sine, cosine, and tangent with transformations, and connect periodic functions to real-world phenomena.

•Define sine, cosine, and tangent using the unit circle and evaluate them at key angles
•Convert fluently between degree and radian measure
•Graph sine, cosine, and tangent functions and identify amplitude, period, phase shift, and vertical shift
+1 more objectives

Trigonometric Identities

Master the fundamental trigonometric identities — Pythagorean, reciprocal, quotient, sum/difference, double-angle, and half-angle — and develop proof-writing skills for verifying identities.

•State and apply the Pythagorean, reciprocal, and quotient identities
•Derive and apply sum/difference identities for sine and cosine
•Use double-angle and half-angle identities to simplify expressions and solve equations
+1 more objectives

Trigonometric Applications

Apply trigonometry to solve oblique triangles using the Law of Sines and Law of Cosines, compute triangle areas with trigonometric formulas, and model real-world measurement problems.

•Apply the Law of Sines to solve triangles given AAS, ASA, or SSA configurations
•Apply the Law of Cosines to solve triangles given SAS or SSS configurations
•Calculate the area of a triangle using the formula A = (1/2)ab sin C
+1 more objectives

Polar Coordinates and Complex Numbers

Explore the polar coordinate system, graph polar equations including roses, limacons, and cardioids, and convert fluently between polar and rectangular representations.

•Plot points and graph equations in the polar coordinate system
•Convert between polar and rectangular coordinates and equations
•Identify and graph standard polar curves: circles, roses, limacons, cardioids, and lemniscates
+1 more objectives

Represent quantities with magnitude and direction using vectors, perform vector operations, compute dot products, and apply vectors to problems in force and motion.

•Represent vectors in component form and perform vector addition, subtraction, and scalar multiplication
•Compute the magnitude and direction of a vector and find unit vectors
•Calculate the dot product of two vectors and use it to determine the angle between them
+1 more objectives

Analyze parabolas, ellipses, and hyperbolas in standard form, graph each conic section, understand eccentricity, and explore applications from satellite orbits to architecture.

•Write and graph equations of parabolas, ellipses, and hyperbolas in standard form
•Identify the key features of each conic section including vertices, foci, axes, and asymptotes
•Classify a conic from its general second-degree equation
+1 more objectives

Sequences and Series

Analyze arithmetic and geometric sequences, compute partial and infinite sums, explore convergence, and encounter mathematical induction as a proof technique.

•Identify and write explicit and recursive formulas for arithmetic and geometric sequences
•Compute partial sums of arithmetic and geometric series using closed-form formulas
•Determine whether an infinite geometric series converges and find its sum when it does
+1 more objectives

Parametric Equations

Describe curves using parametric equations, eliminate the parameter to find rectangular equations, and model motion including projectile trajectories.

•Graph curves defined by parametric equations and describe the direction of motion
•Eliminate the parameter to convert parametric equations to rectangular form
•Write parametric equations for lines, circles, ellipses, and other curves
+1 more objectives

Introduction to Limits

Develop the concept of a limit intuitively and algebraically, evaluate limits using multiple techniques, analyze limits at infinity, and connect limits to continuity.

•Evaluate limits numerically, graphically, and algebraically
•Apply limit laws and algebraic techniques including factoring, rationalizing, and simplifying
•Determine limits at infinity and identify horizontal asymptotes
+1 more objectives

Rates of Change and Area Preview

Distinguish average from instantaneous rate of change, develop the tangent line concept using limits, and preview the area problem through Riemann sums.

•Compute the average rate of change of a function over an interval and interpret it as a secant line slope
•Use limits to transition from average to instantaneous rate of change at a point
•Find the equation of a tangent line to a curve at a given point
+1 more objectives

Precalculus Capstone

Synthesize all Precalculus topics by comparing function families, modeling complex real-world scenarios, and building the conceptual bridges to calculus.

•Compare and contrast polynomial, rational, exponential, logarithmic, and trigonometric function families
•Select and apply appropriate function models for multi-step real-world problems
•Connect rate-of-change and area concepts to the formal definitions of derivatives and integrals
+1 more objectives

Precalculus. Distinguished Level of Achievement. TEKS §111.42 aligned. Prerequisite: Algebra II.