In order to be successful in this course, students are expected to have mastered all prior algebra topics such as: graphing and solving linear equations, solving systems of equations, solving quadratic equations with factoring and the quadratic formula, and graphing and interpreting functions. Students will extend their knowledge and understanding of algebraic functions and relations including, but not limited to, polynomials, rationals, exponentials, and logarithms. Students will also study infinite series, complex numbers, vectors, parametric functions, and limits. Topics from the trigonometry unit include circular functions, identities, polar coordinates, graphing, and solving equations. Applications to other areas of mathematics and to the real world will be an important part of this class. Assessments will require students to apply what they have learned to new situations and to demonstrate strong critical thinking ability

In this chapter, students develop a foundational understanding of functions as tools for representing relationships, analyzing change, and modeling real-world situations. The unit begins with linear relationships, where students explore slope as a measure of rate of change. They determine slope from graphs, tables, and coordinate pairs, write equations of lines using given information, and graph linear functions in slope-intercept form. Students also use slope to identify and interpret parallel and perpendicular relationships, building connections between algebraic equations, graphical representations, and real-world contexts.

Students then extend their understanding from linear relationships to the broader concept of a function. They determine whether relations represent functions, use function notation, evaluate functions, and identify appropriate domains. Using tools such as the Vertical Line Test, students analyze whether a relation is a function and interpret domain and range in both mathematical and contextual settings. This work emphasizes functions as representations of relationships between variables.

As the chapter progresses, students analyze function behavior by examining rates of change and key features of graphs. They evaluate difference quotients to understand average rate of change and identify intervals where functions are increasing, decreasing, or constant. Students determine relative minimum and maximum values and interpret these features within real-world contexts. Emphasis is placed on describing change both quantitatively and qualitatively using multiple representations.

Students also explore special types of functions and their structures, including step functions, piecewise-defined functions, and even and odd functions. They analyze symmetry and investigate parent functions to build a framework for understanding how different types of functions behave. This structural perspective supports students in recognizing patterns that extend beyond linear models.

A major focus of the chapter is transformations of functions. Students investigate how graphs change through translations, reflections, and stretches or compressions. They connect algebraic changes in equations to graphical transformations, allowing them to efficiently predict and sketch new functions based on known parent functions.

Students then explore operations and compositions of functions, interpreting these combinations as multiple processes acting on inputs. They apply these concepts to model real-world scenarios that involve sequential or combined relationships. Additionally, students study inverse functions, determining whether functions are one-to-one, verifying inverses graphically and algebraically, and interpreting inverse relationships as reversing processes.

The chapter concludes with data analysis and modeling. Students construct scatter plots, analyze correlation, and use graphing technology to generate linear models. They interpret these models in context and use them to make predictions, reinforcing the role of functions as practical tools for understanding and analyzing real-world data.

Throughout the chapter, students develop the understanding that functions describe relationships between quantities, provide insight into how values change, and serve as powerful tools for modeling, interpreting, and solving real-world problems.

In this chapter, students deepen their understanding of functions by examining quadratic, polynomial, and rational functions as tools for analyzing patterns, interpreting graphs, and modeling real-world phenomena. The chapter begins with quadratic functions, where students analyze graphs and write equations in standard form. They identify key features such as vertex, axis of symmetry, and direction of opening to sketch and interpret graphs. Students apply quadratic models to real-world contexts, determining and interpreting maximum and minimum values within meaningful situations.

Building on this foundation, students extend their study to higher-degree polynomial functions. They investigate how transformations impact graphs and use the Leading Coefficient Test to determine end behavior. Students connect zeros of polynomial functions to x-intercepts and use them as key features for sketching graphs. The Intermediate Value Theorem is introduced to support reasoning about the existence of real zeros, reinforcing connections between algebraic expressions and graphical behavior.

Students then develop a set of algebraic tools to analyze and solve polynomial equations efficiently. They perform polynomial division using long division and synthetic division, and apply the Remainder and Factor Theorems to evaluate and factor polynomials. Additional strategies, including the Rational Zero Test, Descartes’s Rule of Signs, and Upper and Lower Bound Rules, are used to identify and analyze possible real solutions.

The chapter also extends students’ understanding of number systems through the introduction of complex numbers. Students perform operations with complex numbers and use complex conjugates to simplify expressions. They connect complex numbers to polynomial equations by finding complex solutions, applying the Fundamental Theorem of Algebra, and recognizing conjugate pairs of complex zeros. Students synthesize algebraic and graphical approaches to determine all zeros of polynomial functions and interpret these zeros as meaningful features of graphs and models.

Students then explore rational functions, focusing on domain, asymptotic behavior, and end behavior. They identify vertical, horizontal, and slant asymptotes and use these features, along with intercepts, to sketch graphs. Emphasis is placed on interpreting the behavior of rational functions in real-world contexts and understanding how discontinuities and asymptotes reflect constraints within models.

The chapter concludes with modeling using data. Students analyze scatter plots, use graphing technology to generate quadratic models, and compare different models to determine the best fit. They evaluate the appropriateness of models based on context and data, strengthening their ability to make informed, data-driven decisions.

Throughout the chapter, students develop the understanding that the structure of algebraic expressions determines the behavior of function graphs. By connecting algebraic, graphical, and contextual representations, students build the skills needed to model, analyze, and predict real-world phenomena using quadratic, polynomial, and rational functions.

In this chapter, students develop a deep understanding of exponential and logarithmic functions as interconnected tools for modeling change, analyzing patterns, and solving real-world problems. Students begin by exploring exponential functions with various bases, including the natural base (e), and investigate their key characteristics such as growth and decay behavior, rate of change, and long-term trends. Through multiple representations—tables, graphs, and equations—students build fluency in evaluating and graphing exponential functions and distinguish exponential relationships from linear and polynomial models. Emphasis is placed on applying exponential models to real-world situations involving repeated multiplicative change.

Students then extend their understanding to logarithmic functions as the inverses of exponential functions. They evaluate and graph logarithmic and natural logarithmic functions while interpreting essential features such as domain, range, and asymptotic behavior. Students develop symbolic fluency by rewriting logarithmic expressions using different bases and applying the properties of logarithms to expand, condense, and simplify expressions. This work strengthens connections between algebraic structure and function behavior.

Building on these foundations, students solve exponential and logarithmic equations using a variety of strategies, including algebraic methods and graphical approaches. They leverage the inverse relationship between exponential and logarithmic functions to isolate variables and interpret solutions within context. Problems are grounded in real-world scenarios to reinforce meaning and application.

A major focus of the chapter is mathematical modeling. Students investigate and compare different types of models, including exponential growth and decay, logistic growth, and Gaussian models. They analyze how each model represents patterns of change and determine the most appropriate model based on context and data. Students interpret key parameters and evaluate the reasonableness of models in representing real-world phenomena.

The chapter concludes with data analysis and model selection. Students analyze scatter plots, use graphing technology to generate models, and compare the fit of different functions. They classify data patterns and select appropriate models, using technology to refine models and interpret results.

Throughout the chapter, students integrate algebraic reasoning, graphical analysis, and real-world application to understand that exponential and logarithmic functions provide powerful ways to describe, interpret, and predict patterns of growth, decay, and change.

In this chapter, students deepen their understanding of angles and extend prior knowledge of right-triangle trigonometry to a comprehensive study of trigonometric functions defined for all real numbers. The chapter begins with a precise description of angles, including standard position, coterminal angles, and the development of radian measure. Students convert fluently between degree and radian measures and apply both systems to model and solve contextual problems involving rotation, arc length, and angular velocity.

Building on this foundation, students explore the unit circle as the central structure that connects real numbers, angle measure, and trigonometric function values. Through analysis of the unit circle, students evaluate sine, cosine, and tangent for acute angles and for angles of any measure. They identify reference angles, determine function values across all four quadrants, and evaluate trigonometric functions of real numbers using both exact values and calculator approximations. Fundamental trigonometric identities are introduced to strengthen algebraic reasoning and support accurate evaluation and simplification of expressions.

Students then investigate the graphical behavior of trigonometric functions. Emphasis is placed on identifying and interpreting domain, range, amplitude, period, phase shift, and asymptotes. Students sketch and analyze the graphs of sine, cosine, tangent, cotangent, secant, and cosecant functions, including transformations and damped trigonometric models. They extend their understanding to inverse trigonometric functions, evaluating and graphing inverse sine, cosine, and tangent functions while interpreting restricted domains. Students also evaluate compositions of trigonometric functions to reinforce connections between algebraic and graphical representations.

Throughout the chapter, students apply trigonometric concepts to authentic contexts. Applications include solving right-triangle problems, analyzing directional bearings, modeling harmonic motion, and interpreting periodic phenomena. By the end of the chapter, students demonstrate fluency in evaluating and graphing trigonometric functions, applying identities, interpreting inverse functions, and using trigonometric models to solve real-world problems.

This chapter emphasizes conceptual understanding through multiple representations—numeric, algebraic, graphical, and geometric—while promoting mathematical reasoning, precision, and application.

In this chapter, students extend their understanding of trigonometric functions by exploring the relationships that connect them and applying those relationships to simplify expressions and solve equations. The chapter begins with the development and recognition of the fundamental trigonometric identities, including the reciprocal, quotient, and Pythagorean identities. Students learn how these identities reveal structural relationships among trigonometric functions and provide tools for evaluating expressions and rewriting functions in equivalent forms.

Students then apply these identities to simplify trigonometric expressions and verify trigonometric identities. Emphasis is placed on strategic reasoning and algebraic manipulation, as students use substitution, factoring, common denominators, and other familiar algebraic techniques to demonstrate that two expressions are equivalent. Through this process, students deepen their understanding of how trigonometric relationships mirror the structure of algebraic expressions while requiring careful attention to domain and equivalence.

Building on this foundation, students use standard algebraic strategies to solve trigonometric equations. They begin with equations that can be solved using direct substitution and known unit circle values and then progress to more complex equations involving quadratic forms, multiple angles, and trigonometric expressions requiring algebraic manipulation. Students also use inverse trigonometric functions to determine solutions and interpret general solutions in terms of periodicity.

The chapter further introduces the sum and difference formulas for sine, cosine, and tangent. Students use these formulas to evaluate trigonometric functions, verify identities, and solve equations involving combinations of angles. Through these applications, students see how angle relationships extend the usefulness of trigonometric functions beyond basic unit circle values.

Throughout the chapter, students integrate algebraic reasoning with trigonometric concepts, developing fluency in transforming expressions, verifying identities, and solving increasingly complex trigonometric equations. By the end of the chapter, students are able to apply identities and formulas strategically to analyze trigonometric relationships and solve problems involving periodic behavior.

In this chapter, students expand their understanding of trigonometry and algebra by applying trigonometric relationships to solve non-right triangles, exploring vectors as tools for modeling quantities with magnitude and direction, and representing complex numbers in multiple forms. The chapter begins with the study of oblique triangles. Students use the Law of Sines to solve triangles given angle–side relationships (AAS or ASA) and explore the ambiguous SSA case, analyzing when one, two, or no triangles exist. Students also apply the Law of Cosines to solve triangles when three sides or two sides and the included angle are known (SSS or SAS). Using these relationships, students determine missing measures and apply trigonometric models to solve contextual problems involving navigation, surveying, and other real-world situations. Students also calculate areas of oblique triangles using trigonometric formulas and Heron’s Area Formula, strengthening their understanding of geometric relationships and measurement.

The chapter then introduces vectors as mathematical objects used to represent quantities with both magnitude and direction. Students represent vectors as directed line segments and write vectors in component form. They perform vector operations algebraically and graphically, express vectors as linear combinations of unit vectors, and determine direction angles. Students extend their understanding by computing the dot product of two vectors and applying its properties to determine the angle between vectors and identify orthogonal relationships. Vectors are further applied to real-world contexts, including decomposing vectors into component parts and modeling work done by a force.

The final section of the chapter introduces complex numbers in the complex plane. Students plot complex numbers, determine their absolute values, and perform operations within the complex plane. They apply distance and midpoint concepts in this coordinate system and explore alternate representations of complex numbers using trigonometric form. Students learn to multiply and divide complex numbers expressed in trigonometric form and use DeMoivre’s Theorem to efficiently compute powers of complex numbers. The chapter concludes with finding the nth roots of complex numbers, connecting algebraic and geometric interpretations of complex solutions.

Throughout the chapter, students integrate geometric reasoning, algebraic techniques, and trigonometric relationships to solve problems and interpret mathematical models. Emphasis is placed on representing mathematical ideas in multiple forms—algebraic, geometric, and graphical—while applying these tools to analyze real-world situations involving triangles, vectors, and complex numbers.

In this chapter, students develop a deep understanding of systems of equations and extend their algebraic reasoning through the use of matrices and determinants. The chapter begins with solving systems of linear equations in two variables using multiple methods, including graphing, substitution, and elimination. Students interpret solutions graphically, identifying systems with one solution, no solution, or infinitely many solutions, and connect these outcomes to the structure of the equations. Emphasis is placed on modeling real-world situations, where students represent and solve problems using systems of equations and interpret solutions within context.

Students then extend their work to systems involving three or more variables. They explore graphical interpretations of three-variable systems and learn to solve systems using back-substitution and Gaussian elimination. Students also investigate nonsquare systems and apply systems of equations to more complex applications, including partial fraction decomposition and real-world modeling scenarios.

The chapter introduces matrices as an efficient way to organize and solve systems of equations. Students write matrices, determine their dimensions, and perform elementary row operations. They use matrices in conjunction with Gaussian and Gauss-Jordan elimination to solve systems and deepen their understanding of solution structures. Students perform matrix operations, including addition, subtraction, scalar multiplication, and matrix multiplication, and apply these operations to model and solve problems. They also explore how matrices can represent and transform vectors, reinforcing connections between algebra and geometry.

Students further investigate inverse matrices and determinants. They verify whether matrices are inverses, use multiple methods to find inverses, and apply inverse matrices to solve systems of equations. Students compute determinants using minors and cofactors and interpret determinants geometrically by finding areas of triangles and parallelograms and determining whether points are collinear. The chapter also introduces Cramer’s Rule as another method for solving systems and explores applications such as encoding and decoding messages using matrices.

Throughout the chapter, students connect algebraic, graphical, and matrix-based representations to analyze and solve increasingly complex systems. Emphasis is placed on precision, structure, and the use of multiple methods to deepen understanding and support problem solving in both mathematical and real-world contexts.

In this chapter, students develop an understanding of sequences and series as tools for describing patterns, representing change, and modeling real-world situations. The chapter begins with foundational notation, where students use sequence notation to represent terms, apply factorial notation, and write sums using summation notation. Students extend this work to evaluate finite and infinite series, developing an understanding of how sequences and series can model accumulation and long-term behavior in real-world contexts.

Students then focus on arithmetic sequences and series, recognizing patterns of constant rate of change. They write explicit formulas for the (n)th term, find partial sums, and connect these representations to real-life situations involving linear growth and repeated addition. Emphasis is placed on interpreting both individual terms and cumulative totals within context, reinforcing connections between algebraic expressions and real-world meaning.

The chapter then introduces the Binomial Theorem as a structured way to expand powers of binomials. Students calculate binomial coefficients using both the theorem and Pascal’s Triangle, and use these coefficients to write binomial expansions efficiently. This work highlights patterns in algebraic structure and connects to broader concepts of combinatorics.

Students next explore counting techniques as a foundation for probability. They begin with simple counting problems and extend to more complex situations using the Fundamental Counting Principle. Students apply permutations and combinations to count outcomes in situations where order matters or does not matter, building a clear understanding of different counting strategies.

The chapter concludes with probability, where students use counting methods to determine the likelihood of events. They calculate probabilities of simple events, as well as compound events involving mutually exclusive and independent outcomes. Students interpret probabilities in context and connect them to real-world scenarios involving uncertainty and decision-making.

Throughout the chapter, students develop the understanding that sequences, series, and counting principles provide powerful tools for describing patterns, quantifying possibilities, and analyzing chance. By connecting algebraic reasoning with real-world applications, students build a strong foundation for modeling situations involving growth, accumulation, and probability.

In this chapter, students expand their understanding of coordinate systems and geometric relationships by studying conic sections, parametric equations, and polar coordinates. The chapter begins with conic sections as the intersection of a plane and a double-napped cone, providing a geometric foundation for circles, parabolas, ellipses, and hyperbolas. Students write and analyze equations of conics in standard form, beginning with circles and parabolas. They interpret key features such as center, vertex, focus, and directrix, and apply properties—such as the reflective property of parabolas—to solve real-world problems.

Students then extend their study to hyperbolas, writing equations in standard form, identifying asymptotes, and sketching graphs. Emphasis is placed on connecting algebraic equations to graphical features and interpreting how these functions model real-world situations. Students also classify conic sections from general second-degree equations and explore how rotating the coordinate axes can eliminate cross-product terms, strengthening their understanding of how algebraic structure influences graph behavior.

The chapter then introduces parametric equations as an alternative way to represent relationships between variables. Students evaluate parametric equations for given parameter values, graph curves defined parametrically, and rewrite parametric equations as single rectangular equations by eliminating the parameter. They also create parametric equations to model motion and other dynamic real-world situations, connecting algebraic representations to graphical behavior.

Students further expand their understanding by exploring the polar coordinate system. They plot points in polar coordinates, recognize multiple representations of the same point, and convert between polar and rectangular forms. Students graph polar equations using point plotting, symmetry, and zeros, and identify common polar graphs.

The chapter concludes by connecting conic sections to polar coordinates through the concept of eccentricity. Students define conics in terms of eccentricity, write equations of conics in polar form, and use these equations to model real-world situations.

Throughout the chapter, students develop the understanding that multiple coordinate systems and representations—rectangular, parametric, and polar—provide different perspectives for analyzing and modeling geometric relationships. By connecting algebraic, graphical, and geometric reasoning, students build a deeper understanding of how mathematical structures describe patterns, motion, and real-world phenomena.

In this chapter, students extend their understanding of coordinate geometry and vectors from the plane to three-dimensional space. The chapter begins with the three-dimensional coordinate system, where students plot points and analyze their positions using ordered triples. They apply distance and midpoint formulas in space, strengthening connections between algebraic formulas and geometric interpretation. Students also write equations of spheres in standard form and examine traces of surfaces, building an understanding of how three-dimensional objects can be represented and analyzed algebraically.

Students then develop a deeper understanding of vectors in space. They write vectors in component form, determine magnitudes, and find unit vectors in a given direction. Using the dot product, students calculate angles between vectors and determine whether vectors are orthogonal or parallel. These concepts are applied to solve real-world problems involving direction, magnitude, and spatial relationships.

The chapter further extends vector analysis through the cross product. Students compute cross products and use their geometric properties to find vectors perpendicular to a plane. They apply the cross product and triple scalar product to determine areas and volumes, including finding the volume of parallelepipeds. This work emphasizes the connection between algebraic operations and geometric meaning in three-dimensional space.

Students then study equations of lines and planes in space. They write parametric and symmetric equations of lines and determine equations of planes using vectors and points. Students sketch planes and analyze their orientation in space, reinforcing spatial reasoning. They also calculate distances between points and planes, applying geometric and algebraic techniques to solve problems in three dimensions.

Throughout the chapter, students develop the understanding that vectors and coordinate systems provide powerful tools for representing and analyzing relationships in three-dimensional space. By connecting algebraic methods with geometric visualization, students build the skills needed to model, interpret, and solve problems involving space, direction, and distance.

In this chapter, students are introduced to the concept of limits as the foundation of calculus and a tool for describing how functions behave as inputs approach specific values. Students begin by developing an intuitive understanding of limits, using numerical tables, graphs, and verbal reasoning to estimate limits and determine whether they exist. They explore the meaning of approaching a value and distinguish between function values and limit values, including situations where limits do not exist.

Students then build procedural fluency by applying properties of limits to evaluate limits algebraically. They use key techniques such as factoring and dividing out, rationalizing expressions, and simplifying complex fractions to evaluate limits that initially appear undefined. Students also evaluate one-sided limits and analyze how these contribute to determining overall limits at a point. Technology is incorporated to approximate limits graphically and numerically, reinforcing connections between multiple representations.

The chapter extends to limits involving difference quotients, providing a bridge to the concept of instantaneous rate of change. Students evaluate these limits to understand how average rates of change approach instantaneous rates, laying the groundwork for derivatives. They also investigate limits at infinity, analyzing end behavior of functions and describing how functions behave as inputs grow without bound.

Students further explore limits in the context of sequences and summations. They determine limits of sequences and examine how summations behave as the number of terms increases. Using rectangular approximations, students connect limits of summations to the concept of area under a curve, developing an early understanding of definite integrals and accumulation.

Throughout the chapter, students develop the understanding that limits provide a precise way to describe change, continuity, and behavior of functions. By connecting numerical, graphical, and algebraic approaches, students build a strong conceptual foundation for future work in calculus, including derivatives and integrals.