Grade 12 · secular · Common Core-aligned
This is the year that takes everything your child learned about functions in Algebra II and trig in Geometry and welds it into one toolkit. Instead of learning "the quadratic chapter" then "the exponential chapter" as separate worlds, they learn to ask: given this data or this real situation, which kind of function actually fits, and how would I prove it? Along the way they pick up the unit circle version of trig (so sine and cosine work for any angle, not just triangle corners), vectors and matrices for quantities with more than one part, polar and parametric ways to describe curves, and they finish with conic sections and a first taste of limits, the exact ideas a calculus class opens with. By June they should be able to look at a new problem and reason out loud about which tool fits, not just execute a memorized recipe.
The whole year, in plain English. Tap any unit to see every skill inside, nothing is hidden.
This unit doesn't teach any new function type -- your child already met linear, quadratic, exponential, rational, and log functions in Algebra II. What's new is the skill of choosing among them: given a table of numbers or a real situation described in words, figuring out which family actually fits, by testing the underlying assumption (constant add, constant multiply, limited number of turning points, or a value that breaks the function) rather than just eyeballing the shape.
Given a table, graph, or equation for an unfamiliar data set, execute the correct family-specific move (compute first differences, compute ratios, count roots, or check for a vertical asymptote) fluently and correctly on request.
Classify a data set presented as a table with unevenly spaced or noisy values into its most likely function family by selecting and running the appropriate structural test.
Explain, using two contrasted data sets that produce visually similar graphs over a limited window, which structural assumption (constant rate vs. constant ratio vs. bounded root count) each data set actually satisfies and why the other family's assumption fails it.
Given a real-world scenario described only in words (no equation, table, or graph provided), infer which function family's structural assumption plausibly matches the described process and justify the inference by naming the real-world condition that corresponds to the mathematical assumption.
State, for a given function family and a stated real-world domain restriction (e.g., time cannot be negative, population cannot be negative), what the family's end behavior predicts about the modeled quantity, and judge whether that prediction is physically plausible.
Given a verbal description of a real multi-step process (e.g., a unit conversion or a multi-stage pricing rule), construct in words the sequence of steps that undoes the process in the correct reverse order, before writing any symbolic inverse.
Given a novel two-step real process not used in instruction, construct the symbolic inverse function and verify it undoes a specific numeric example correctly.
Given three unfamiliar real data sets, select a function family for each, justify the choice by naming the structural assumption the data satisfies, identify a specific alternative family that was considered and rejected, and state what future data would falsify the chosen model.
Compare two function families' rate-of-change behavior (constant, linearly increasing, or multiplicatively increasing) by summarizing how each family's output changes as input increases across equal intervals.
This unit takes the factoring and rational-expression skills your child already has and turns them into a prediction tool: given just the equation of a polynomial or rational function, predict the entire graph -- every zero, every asymptote, every hole -- before plotting a single point. Division of polynomials and the remainder theorem show up here as tools for finding zeros, not as separate arithmetic drills.
Given a polynomial in factored form, state the multiplicity of each zero and predict whether the graph crosses or bounces at that zero.
Given the leading term of a polynomial (degree and sign of leading coefficient), state the end behavior of the function using correct limit-style notation (as x approaches +/- infinity).
Apply synthetic division and the remainder theorem to determine whether a given linear expression is a factor of a polynomial and to find remaining factors.
Given a higher-degree polynomial and one or more known rational zeros, find all remaining real zeros by successive division and factoring, then use those zeros and their multiplicities to sketch a complete graph without a calculator.
Explain, using the behavior of 1/(x-a) numerically for x-values approaching a from both sides, why division by an expression approaching zero produces unbounded output, and connect this to the definition of a vertical asymptote.
Given a rational function, factor numerator and denominator, cancel common factors, and correctly classify each remaining zero of the denominator as a vertical asymptote and each canceled factor's zero as a removable discontinuity (hole), including finding the hole's coordinates.
Determine the horizontal asymptote (or absence of one) of a rational function by comparing the degrees of numerator and denominator and reasoning about end behavior via substitution of large x-values, rather than by rote application of a memorized rule.
Given only a function's algebraic equation (polynomial or rational, in a form not previously encountered), predict its complete graphical behavior, zeros with crossing/bounce behavior, end behavior, and any asymptotes/holes, entirely from structure, without graphing or a calculator, and justify each prediction using the specific structural feature that produces it.
Construct a rational function equation that satisfies a specified combination of structural constraints (e.g., exactly one hole at a given x-value, one vertical asymptote, a specified horizontal asymptote), reasoning backward from graphical behavior to algebraic structure.
Retrieve and correctly classify a function's family (polynomial, rational, exponential, or other, based on Unit 1 vocabulary) when given an equation or graph, distinguishing it from the polynomial/rational structural features taught this unit.
This unit nails down the difference between constant-rate growth (polynomial) and constant-ratio growth (exponential), then builds the logarithm as the function that undoes an exponential. Your child already knows exponent rules and probably saw logs briefly before; the new work is deriving log properties from exponent rules instead of memorizing them, solving equations with the unknown in the exponent, and understanding where the number e comes from.
Given a table of values, determine whether the underlying process has a constant additive rate (linear/polynomial) or a constant multiplicative ratio (exponential) and justify the classification using the table structure.
Explain why a percent-growth context (e.g., population, investment) produces an exponential function rather than a linear one, referencing the constant-ratio structure.
Execute the algorithm for rewriting an exponential expression a·b^(ct) into an equivalent form revealing a different (e.g., monthly rather than annual) percent rate, using exponent rules.
Given the graph and rule of an exponential function, construct its inverse function's graph and rule and explain why the inverse cannot be expressed using previously known algebraic operations (roots, polynomials, rational expressions).
Recall the definition of log_b(x) as the exponent to which b must be raised to produce x, and convert fluently between exponential and logarithmic form.
Derive the product, quotient, and power properties of logarithms from the corresponding exponent rules, rather than merely applying them.
Execute multi-step procedures to solve exponential equations (unknown in exponent) using logarithms, and logarithmic equations using exponentiation, including checking for extraneous solutions.
Solve exponential/logarithmic equations that require first combining multiple terms using log properties before isolating the variable, selecting the correct property sequence.
Explain why continuous compounding produces the constant e, comparing discrete compounding at increasing frequency to the continuous limit.
Given a real-world scenario never discussed in class (e.g., radioactive contamination spreading through an untaught decay-rate context, or a novel financial instrument), determine whether an exponential, logarithmic, or polynomial model applies and justify the choice using structural features of the data, not surface vocabulary.
Construct a counterexample or valid proof to evaluate a peer's or textbook's claimed logarithm 'shortcut' (e.g., log(a+b) = log a + log b) and generalize what structural feature of logs makes such shortcuts invalid.
Right-triangle trig only works for angles between 0 and 90 degrees. This unit extends the idea using a circle: an angle becomes a rotation, and the point where that rotation lands gives you (cos θ, sin θ) directly as coordinates. That single move explains why sine and cosine can be negative, why they repeat, and why their graphs are smooth waves. From there your child builds sine, cosine, and tangent graphs from the circle itself, learns to shift and stretch them, and defines arcsine, arccosine, and arctangent as true inverse functions.
Given an angle measure in degrees, convert to radians and vice versa, and explain radian measure as arc length on a unit circle.
State the exact sine, cosine, and tangent values for all unit circle angles that are multiples of π/6 and π/4, without a calculator.
Explain why the unit-circle definitions of sine and cosine agree with right-triangle definitions for acute angles but extend to angles beyond 0-90 degrees and to negative angles.
Given an angle's reference angle and quadrant, determine the sign and exact value of its sine, cosine, or tangent using symmetry of the unit circle.
Sketch the graph of y = sin x or y = cos x directly from unit-circle coordinates, identifying period, midline, and key points without plotting from a table of decimal values.
Given values for amplitude, period, phase shift, and vertical shift, write and graph a transformed sine or cosine function, and given a graph, determine those four parameters.
Explain why the tangent function has vertical asymptotes and a period of π, in terms of the ratio sin θ / cos θ on the unit circle.
Restrict the domain of sine, cosine, and tangent to define arcsine, arccosine, and arctangent as functions, and evaluate them at exact values.
Given a real-world periodic scenario (tide height, Ferris wheel height, sound wave) with a partial data description, construct a sine or cosine model and justify parameter choices against the context.
Given a novel periodic pattern in an unfamiliar context (e.g. a rotating machine part's coordinate over time, or an angle-vs-shadow-length relationship not resembling a standard textbook Ferris wheel), determine whether a sine, cosine, or non-periodic function best models it, and defend the choice using structural features rather than surface similarity.
Determine algebraically why sin(π − x) = sin x and cos(π − x) = −cos x hold for every real x, using unit circle symmetry rather than checking specific numeric cases.
Unit 4 was about finding a trig value. This unit is about proving two trig expressions are secretly the same function (an identity), and separately, solving trig equations that are only true for specific angles. The identity-manipulation work reuses the same moves as simplifying rational expressions in Unit 2, now with sin, cos, and tan instead of polynomials.
Given a value of sin θ (or cos θ) and the quadrant of θ, find the remaining trig function values by applying the Pythagorean identity sin^2θ + cos^2θ = 1.
Simplify a given trigonometric expression (e.g., (1-cos^2x)/sin x) to a single term by substituting an equivalent Pythagorean or reciprocal identity, using the same expression-manipulation moves as simplifying rational expressions in Unit 2.
Explain why a valid identity-verification proof may only transform one side of the equation using known-equal substitutions, while a valid equation-solving proof may perform the same operation to both sides, and identify which situation a given problem is.
Verify a given trigonometric identity by producing a chain of algebraically equivalent expressions from one side to the other, using sum/difference, double-angle, or Pythagorean identities as needed, and justify each substitution step.
Derive the double-angle formula for cosine in the three equivalent forms (in terms of cos only, sin only, or both) starting from the sum formula cos(A+B), by substituting A = B.
Solve a trigonometric equation of the form a·trig(bx)=c over a specified restricted domain (e.g., [0, 2π)), finding all solutions.
Solve a trigonometric equation over all real numbers, expressing the complete solution set using the correct period-based general-solution notation (+2πk or +πk).
Given an unfamiliar trigonometric equation involving a squared term (e.g., 2sin^2x - cos x - 1 = 0), decide and justify which identity substitution transforms it into a solvable algebraic form, in a context not explicitly modeled in class.
Given two trigonometric expressions that are claimed to be equivalent, determine whether the claim is a true identity or false by testing strategically chosen values and, if true, construct a proof; if false, produce a counterexample.
Derive the half-angle formulas for sine and cosine from the double-angle cosine identity by solving for cos^2(x) or sin^2(x) and substituting x/2, and determine the correct sign from the quadrant of the half-angle.
Two new ways to hold more than one number in a single object. Vectors carry a magnitude and a direction (used for velocity, force); matrices carry organized rows and columns of data. Vectors literally reuse the (cos θ, sin θ) coordinates from Unit 4 as their x- and y-parts. Matrices extend the 2-equation, 2-unknown systems your child already solved in Algebra II into a compact notation.
Given a directed line segment or a real-world description (velocity, force), state the vector's magnitude and direction and represent it with correct vector notation.
Resolve a vector given as magnitude and direction into horizontal and vertical (x, y) components using cos θ and sin θ.
Add two or more vectors both graphically (tip-to-tail) and algebraically (component-wise), and explain why the magnitude of the sum is generally not the sum of the magnitudes.
Given a real-world scenario with two or more forces or velocities acting on an object (e.g., wind and airspeed, current and rowing), construct a vector model, compute the resultant, and interpret its magnitude and direction in context.
Given a completely novel multi-part-quantity scenario never framed as vectors during instruction (e.g., a recipe-scaling or ranking context with no direction language at all), determine whether a vector representation is appropriate and justify the decision using the definition of a vector.
Organize a multi-category data set (e.g., inventory across stores, scores across categories) into a matrix and identify entries by row and column position.
Add, subtract, and scalar-multiply matrices, and compare these operations to the equivalent operations on vectors and real numbers to identify what stays the same and what changes.
Multiply two matrices where defined, and explain, using a constructed counterexample, why matrix multiplication is not commutative in general.
Compute the determinant of a 2x2 and 3x3 matrix and use it to determine whether the matrix has a multiplicative inverse.
Set up and solve a system of two or three linear equations using matrix methods (inverse-matrix or row-reduction), and verify the solution satisfies all original equations.
Given an unfamiliar multi-constraint scenario (e.g., a network of three interdependent prices/quantities never modeled during instruction), translate it into a matrix equation, solve it, and justify why a matrix approach was more efficient than solving the constraints one at a time.
Two more ways to describe the same curve. Polar coordinates locate a point by distance and angle from a center instead of by x and y. Parametric equations describe x and y as both depending on a hidden third variable, usually called t, often standing for time. This unit is as much about combining Units 4 and 6 as it is about new material -- the conversion formulas are literally the unit-circle coordinates, and motion problems use the vector-direction idea directly.
Given a point in rectangular coordinates, compute its polar coordinates (r, θ) using r = sqrt(x^2+y^2) and tan θ = y/x, and given a point in polar coordinates, compute its rectangular coordinates using x = r cos θ, y = r sin θ.
Explain why a single point can be represented by more than one valid polar coordinate pair, using the periodicity of angle measurement and the meaning of negative r.
Given the equation of a common polar curve (circle, cardioid, or rose r = a cos(nθ)), predict qualitative features of its graph (does it pass through the pole, how many petals, symmetry) from the equation's coefficients without plotting every point.
Given a parametric pair x(t), y(t), generate a table of (t, x, y) values and use it to sketch the resulting curve, indicating direction of motion as t increases.
Eliminate the parameter from a parametric pair to produce a rectangular equation, selecting substitution when one equation is invertible and the Pythagorean identity when the pair involves sine and cosine of the same argument.
Given a real-world description of motion with an initial position, direction, and constant rate (e.g. a plane's displacement), write a parametric model x(t), y(t) using vector-direction components from Unit 6.
Given two representations of the same curve, one polar, one parametric, both unfamiliar in exact form, decide and justify which representation is more efficient to convert to rectangular form or graph directly, based on the algebraic structure of each equation.
Convert a simple rectangular equation of a line or circle (e.g. x^2+y^2=9, or y=x) into its polar equation form and vice versa, using substitution of the conversion identities.
The last unit does two things that turn out to share an idea. First, conic sections (circles, parabolas, ellipses, hyperbolas) get defined by distance rules -- an ellipse is every point where the distance to two fixed points adds up to the same total -- rather than by a formula to memorize. Second, limits ask what a function's output approaches as the input gets closer and closer to some value, even if the function is never actually defined there. Both ideas are 'defined by approaching or relating, not by a formula you just plug into.'
Given the center and radius of a circle, derive its standard-form equation using the distance formula/Pythagorean theorem, and given an equation in expanded form, complete the square to find center and radius.
Explain why every conic section (circle, ellipse, parabola, hyperbola) can be produced as a planar cross-section of a double cone, connecting the cross-section angle to which curve results.
Derive the standard-form equation of a parabola from its focus-directrix definition (the set of points equidistant from a fixed point and a fixed line).
Derive the standard-form equation of an ellipse and of a hyperbola from their distance-sum / distance-difference definitions given the foci, and identify foci, vertices, and axes from a standard-form equation.
Given eccentricity values and defining parameters, classify a conic by type and explain how eccentricity controls the shape's elongation, ranging continuously from circle through ellipse to parabola to hyperbola.
Given a conic equation in a real-world modeling context not previously encountered (e.g. a satellite orbit, a whispering gallery, a comet's path), identify the conic type, extract relevant parameters, and answer a question about the physical situation.
Estimate the value a function's output approaches as the input approaches a given value or infinity, using a table of successive values and using a graph, and represent that value using limit notation.
Distinguish between a function's limit at a point and its actual value at that point, and identify from a graph or table cases where a limit exists but the function is undefined or defined differently there (removable discontinuity).
Given a novel piecewise or rational function not previously seen, determine algebraically or graphically whether a limit exists at a specified point, and if the function is discontinuous there, classify the discontinuity as removable or non-removable and justify the classification.
Compute the average rate of change of a function over progressively smaller intervals around a point and explain how this sequence of average rates is used to estimate an instantaneous rate of change as a limit.
Given a real-world scenario described only verbally (e.g., a quantity whose behavior involves both a distance-defined boundary curve and a described rate of change near a critical moment), decide whether a conic model, a limit/rate argument, or both are needed, and justify the choice of representation.
From the parent guide
This is the year that takes everything your child learned about functions in Algebra II and trig in Geometry and welds it into one toolkit. Instead of learning "the quadratic chapter" then "the exponential chapter" as separate worlds, they learn to ask: given this data or this real situation, which kind of function actually fits, and how would I prove it? Along the way they pick up the unit circle version of trig (so sine and cosine work for any angle, not just triangle corners), vectors and matrices for quantities with more than one part, polar and parametric ways to describe curves, and they finish with conic sections and a first taste of limits, the exact ideas a calculus class opens with. By June they should be able to look at a new problem and reason out loud about which tool fits, not just execute a memorized recipe.
Unit 1 · what to expect
This unit doesn't teach any new function type -- your child already met linear, quadratic, exponential, rational, and log functions in Algebra II. What's new is the skill of choosing among them: given a table of numbers or a real situation described in words, figuring out which family actually fits, by testing the underlying assumption (constant add, constant multiply, limited number of turning points, or a value that breaks the function) rather than just eyeballing the shape.
The full guide covers all 8 units: where kids get stuck, what to say, and how to tell it's working. Included with the course.
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