12th Grade Mathematics

Reviewing definitions of functions, domain, range, and various representations (graphical, algebraic, tabular).

Investigating how adding or subtracting constants and multiplying by negative values transform parent functions.

Analyzing the impact of multiplying by constants on the vertical and horizontal scaling of functions.

Analyzing how nested functions interact and the conditions required for a function to be reversible.

Investigating the intuitive concept of a limit by observing function behavior from graphs and tables.

Investigating how functions behave as they approach specific values or infinity.

Defining continuity and classifying different types of discontinuities (removable, jump, infinite).

Applying theorems to guarantee the existence of specific function values or extrema within an interval.

Calculating and interpreting the average rate of change over an interval for various function types.

Transitioning from average rate of change to the concept of the derivative at a point.

Understanding the formal definition of the derivative and applying power, constant, and sum rules.

Using derivatives to model and analyze motion in physics contexts.

Solving real-world problems by finding maximum or minimum values of functions using derivatives.

Reviewing the properties of exponential functions and their application to growth and decay models.

Understanding logarithms as the inverse of exponential functions and their basic properties.

Applying the product, quotient, and power rules of logarithms to simplify expressions and solve equations.

Using logarithms to linearize data and solve complex growth equations.

Developing strategies to solve equations involving exponential and logarithmic functions.

Investigating the origin and applications of the constant e in continuous compounding and growth.

Applying differentiation rules to functions involving e and natural logarithms.

Solving real-world problems involving continuous growth, decay, and related rates.

Modeling growth that is constrained by environmental or physical factors.

Analyzing the characteristics of logistic curves and fitting them to data.

Solving problems where multiple quantities are changing with respect to time and are related by an equation.

Differentiating equations that are not explicitly solved for y in terms of x.

Using derivatives to evaluate limits that result in indeterminate forms (0/0, ∞/∞).

Understanding angles in standard position and converting between degrees and radians.

Connecting geometric rotation to algebraic coordinates and the logic of radians.

Defining sine, cosine, and tangent for angles beyond the first quadrant using the unit circle.

Analyzing the characteristics (amplitude, period, phase shift, vertical shift) of sinusoidal graphs.

Exploring the graphs of tangent, cotangent, secant, and cosecant functions, including asymptotes.

Introducing reciprocal, quotient, and Pythagorean identities and their basic applications.

Using algebraic manipulation to prove equivalence between complex trigonometric expressions.

Applying identities for the sum and difference of angles to simplify expressions and solve equations.

Using identities to find trigonometric values for double or half an angle.

Developing strategies to solve trigonometric equations over specific intervals and generally.

Defining and evaluating inverse trigonometric functions and their restricted domains.

Applying sine and cosine functions to model sound waves, tides, and pendulums.

Applying differentiation rules to sine, cosine, and other trigonometric functions.

Defining vectors, their components, magnitude, and direction in 2D and 3D space.

Performing operations on vectors to solve physics based problems involving force and velocity.

Calculating the dot product and using it to find the angle between two vectors and determine orthogonality.

Understanding how to project one vector onto another and decompose vectors into orthogonal components, with applications in physics.

Defining matrices, their dimensions, and performing basic operations like addition, subtraction, and scalar multiplication.

Understanding the rules and process of multiplying matrices and its non-commutative nature.

Using matrices to scale, rotate, and reflect geometric figures in a coordinate plane.

Calculating determinants for 2x2 and 3x3 matrices and finding inverse matrices.

Using inverse matrices to solve systems of linear equations.

Solving large systems of linear equations using matrix row reduction techniques.

Understanding the steps and significance of transforming matrices into row echelon and reduced row echelon forms.

Using systems of inequalities and matrices to solve optimization problems with constraints.

Revisiting permutations, combinations, and fundamental probability rules.

Calculating the probability of events based on prior knowledge of related conditions.

Introducing discrete and continuous random variables and their associated probability distributions.

Calculating and interpreting the expected value and standard deviation for discrete random variables.

Applying the binomial distribution to model scenarios with a fixed number of independent trials.

Understanding the properties of the normal distribution and standardizing data using z-scores.

Analyzing binomial and normal distributions to determine the likelihood of outcomes.

Exploring the concept of sampling distributions and the foundational Central Limit Theorem.

Constructing and interpreting confidence intervals to estimate population parameters.

Introducing the framework of hypothesis testing and performing z-tests for population means.

Using p values and confidence intervals to evaluate the validity of experimental claims.

Performing t-tests for population means when the population standard deviation is unknown.

Using chi-square tests to analyze relationships between categorical variables (goodness-of-fit, independence).

Defining sequences and series, and using summation notation.

Identifying arithmetic sequences, finding the nth term, and calculating sums of arithmetic series.

Identifying geometric sequences, finding the nth term, and calculating sums of finite geometric series.

Finding sums of finite and infinite sequences and applying them to financial models.

Using arithmetic and geometric series to model loans, investments, and annuities.

Proving that a statement holds true for all natural numbers using a recursive logic structure.

Exploring the patterns in Pascal's Triangle and its connection to binomial coefficients.

Expanding binomial expressions using Pascal's Triangle and combinatorics.

Applying combinations to calculate probabilities in scenarios where order does not matter.

Exploring the concept of convergence and divergence for infinite sequences and series.

Applying exponential and logarithmic functions to model real-world phenomena such as population growth, decay, and compound interest.