Algebra II and precalculus: functions and structure
Twelve lessons connect representations, transformations, systems and algebraic function families. Exact calculations and written arguments supply varied practice. This unit does not complete Algebra II or AP Precalculus; trigonometric, polar and broader modeling sequences remain separate work.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation
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Functions and structure: mixed assessment
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Lesson sequence
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Read domains and ranges across representations
Lesson details and objectives
Distinguish allowed inputs, attained outputs and the information a table cannot supply.
- Determine domain and range from finite relations and restricted formulas.
- Identify whether a relation defines a function.
- Distinguish an observed table from a complete function rule.
6 practice tasks.
Transform inputs and outputs in the right order
Lesson details and objectives
Map points through shifts, reflections and stretches while tracking the transformed domain.
- Interpret g(x)=Af(B(x−h))+k using point mappings.
- Distinguish horizontal and vertical changes.
- Compose transformations without assuming they commute.
6 practice tasks.
Choose the correct branch at a boundary
Lesson details and objectives
Build and interpret rules whose formulas depend on input conditions.
- Evaluate piecewise functions including endpoints.
- Check whether branches leave gaps or conflicting outputs.
- Construct a contextual threshold rule and interpret discontinuities.
6 practice tasks.
Connect functions through intermediate values
Lesson details and objectives
Track the order, units and domain of a two-stage calculation.
- Evaluate and simplify compositions in the correct order.
- Find a composition domain from both component domains.
- Model successive operations and disprove commutativity.
9 practice tasks.
Reverse a function on a suitable domain
Lesson details and objectives
Recover inputs, restrict many-to-one rules and distinguish inverses from reciprocals.
- Find and verify inverses with domain/range exchange.
- Restrict a quadratic so that it is one-to-one.
- Distinguish an inverse function from a reciprocal or inverse image.
6 practice tasks.
Find intersections and classify systems
Lesson details and objectives
Connect simultaneous constraints with substitution, elimination and intersection geometry.
- Solve two-variable linear systems and classify dependent/inconsistent cases.
- Reduce a line–curve system to an equation with all candidates.
- Check full ordered pairs in both original constraints.
6 practice tasks.
Divide polynomials and interpret the remainder
Lesson details and objectives
Use the division identity to connect evaluation, remainders and factors.
- Perform polynomial long division with missing-power placeholders.
- Apply the remainder and factor theorems.
- Verify a quotient and distinguish a polynomial identity from a rational domain.
6 practice tasks.
Read roots, multiplicity and end behavior
Lesson details and objectives
Use factors and leading terms to predict a graph without assuming its entire shape.
- Identify distinct zeros and their multiplicities.
- Relate parity of multiplicity to local sign change.
- Use degree and leading coefficient for end behavior and preserve complex-root distinctions.
6 practice tasks.
Separate holes, poles and end behavior
Lesson details and objectives
Factor a rational function while preserving its original domain and directional behavior.
- Classify canceled and uncanceled denominator factors.
- Find intercepts and horizontal/slant asymptotes.
- Explain one-sided signs and why asymptotes are not barriers.
7 practice tasks.
Use exponent laws without losing the base conditions
Lesson details and objectives
Compare multiplicative change, equivalent exponential forms and exact equations.
- Apply integer/rational exponent laws on valid domains.
- Solve exponential equations by matching positive bases.
- Construct and interpret growth/decay models from observations.
6 practice tasks.
Solve logarithmic equations on their real domains
Lesson details and objectives
Use logarithms as inverse exponents, expand valid products and reject forbidden candidates.
- Interpret logarithms with positive bases other than 1.
- Use product/quotient/power laws with real-domain conditions.
- Solve logarithmic equations and check candidates in the original domain.
6 practice tasks.
Distinguish repeated addition from repeated multiplication
Lesson details and objectives
Use explicit and recursive sequence rules and derive finite sums with clear indexing.
- Identify arithmetic and geometric changes from terms.
- Translate between recursive and explicit rules.
- Derive finite sum formulas and handle the ratio-one exception.
6 practice tasks.