Grade 12 Mathematics

Students analyze the relationship between a polynomial's degree, leading coefficient, and its end behavior, sketching graphs based on these characteristics.

Students investigate the connection between polynomial factors, their roots, and the behavior of the graph at the x-axis, including multiplicity.

Students practice synthetic and long division of polynomials to find factors and apply the Remainder and Factor Theorems.

Students use the Rational Root Theorem to find potential rational roots and explore the nature of complex conjugate roots.

Students identify and graph vertical, horizontal, and oblique asymptotes of rational functions.

Students locate holes, x-intercepts, and y-intercepts of rational functions and sketch complete graphs.

Students solve rational equations algebraically and graphically, paying attention to extraneous solutions and domain restrictions.

Students apply rational functions to model real-world scenarios involving rates, concentrations, and inverse relationships.

Students explore the characteristics of exponential growth and decay functions, including domain, range, and asymptotes.

Students define logarithms as the inverse of exponential functions and graph basic logarithmic functions.

Students apply the product, quotient, and power rules of logarithms to expand and condense logarithmic expressions.

Students solve exponential equations using logarithms, including those with different bases.

Students solve logarithmic equations, checking for extraneous solutions due to domain restrictions.

Students apply exponential functions to model real-world scenarios such as population growth, radioactive decay, and compound interest.

Students explore the significance of the natural base 'e' in continuous compounding and natural growth/decay processes.

Students analyze how transformations affect the graphs of exponential functions, including shifts, reflections, and stretches.

Students define angles in standard position, convert between degrees and radians, and understand radian measure as arc length.

Students use the unit circle to define trigonometric ratios for any angle and evaluate exact values for special angles.

Students graph sine and cosine functions, identifying amplitude, period, phase shift, and vertical shift.

Students graph tangent, cotangent, secant, and cosecant functions, identifying their unique characteristics and asymptotes.

Students prove and apply fundamental identities, including reciprocal, quotient, and Pythagorean identities.

Students use sum and difference identities to find exact trigonometric values and simplify expressions.

Students apply double and half-angle identities to simplify expressions and solve trigonometric equations.

Students solve trigonometric equations algebraically over a given interval and for general solutions.

Students explore the concept of a limit by analyzing the behavior of functions as they approach a specific value from both sides, using graphs.

Students use algebraic techniques (direct substitution, factoring, rationalizing) to evaluate limits.

Students define continuity, identify types of discontinuities, and apply the conditions for continuity.

Students distinguish between average and instantaneous rates of change and calculate average rates from graphs and tables.

Students define the derivative as the limit of the difference quotient and interpret it as the slope of a tangent line.

Students apply basic differentiation rules to find derivatives of polynomial and simple power functions.

Students apply the product and quotient rules to differentiate more complex functions.

Students master the chain rule for differentiating composite functions.

Students define vectors, represent them in component form, and calculate magnitude and direction in two and three dimensions.

Students perform vector addition, subtraction, and scalar multiplication geometrically and algebraically.

Students calculate the dot product and use it to find the angle between two vectors and determine orthogonality.

Students calculate the cross product of two vectors and use it to find a vector orthogonal to both and the area of a parallelogram.

Students represent lines in 2D and 3D space using vector and parametric equations.

Students convert between different forms of line equations and find intersection points of lines.

Students represent planes in 3D space using vector, parametric, and Cartesian (scalar) equations.

Students apply the fundamental counting principle and permutation formulas to count arrangements where order matters.

Students apply combination formulas to count selections where order does not matter.

Students define probability, sample space, and events, calculating probabilities of simple events.

Students calculate conditional probabilities and determine if events are independent.

Students analyze discrete random variables and their probability distributions, including expected value.

Students apply the binomial probability formula to scenarios with a fixed number of independent trials.

Students explore the properties of the normal distribution, calculate z-scores, and find probabilities using the standard normal table.

Students apply the normal distribution to real-world problems, including approximating binomial distributions.

Students use the first derivative to determine intervals of increasing/decreasing and locate local extrema.

Students use the second derivative to determine concavity and locate inflection points.

Students apply derivatives to solve real-world optimization problems, finding maximum or minimum values.

Students solve problems involving rates of change of two or more related variables.

Students define antiderivatives and learn basic integration rules to find indefinite integrals.

Students approximate the area under a curve using Riemann sums (left, right, midpoint, trapezoidal).

Students define the definite integral as the limit of Riemann sums and apply the Fundamental Theorem of Calculus.

Students apply definite integrals to find areas between curves, displacement, and total change.

Students learn and apply the technique of u-substitution for integrating composite functions.