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Follow the authored sequence, or switch to recap mode for a faster review of the same path.

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Starter track6 concepts5 checkpointsMath139 min

Functions and Change

Not started

Keep the first math path compact: read parent-curve moves first, then rational asymptotes and domain breaks, then exponential growth and decay, local slope, visible limit behavior, and finally accumulation so change stays graph-first all the way through.

Use this track when the growing math slice should feel like one coherent launch path instead of isolated pages. Start with one parent curve and its transformed version so shifts, reflections, and vertical scale stay visible on the graph, then keep one shifted reciprocal family honest so vertical and horizontal asymptotes, intercepts, domain breaks, and an optional removable hole stay tied to the same bench, then keep one exponential bench honest so growth, decay, doubling or half-life, and logarithmic target time stay tied to the same curve, then carry that same graph-reading habit into a moving point, secant line, tangent line, and derivative graph, pause on one target point to compare left-hand approach, right-hand approach, holes, jumps, and blow-up behavior, and finally widen into signed area and accumulation without leaving the same graph-first language.

Parent-curve movesShifts and reflectionsRational asymptotes and holesExponential growth and decayTarget time from logsTangent slopeOne-sided limits and continuityLocal rate of changeSigned area and accumulation

Entry diagnostic

Decide where to enter this path without opening a second testing system.

Reuse the graph-transformations quick test, the rational domain-break checkpoint, and the exponential target-time quick test to decide whether to start from parent-curve moves or jump straight into local slope.

Start from beginning0 / 3 probes ready

Check the graph-reading bridge first

Start from beginning

No saved diagnostic checks are available yet, so the opening concept is still the best place to start.

Uses the same local-first quick tests, checkpoint challenges, and track history already saved in this browser.

Begin with Graph Transformations
  1. Quick testNot started3 questions

    Graph-transformation quick test

    Check whether shifts, reflections, and landmark tracking already feel stable on the shared graph.

    No saved quick-test result yet.

    Graph transforms
    Open Graph Transformations quick test
  2. ChallengeNot started8 checks

    Rational domain-break checkpoint

    Use the rational-functions challenge to verify that asymptotes, removable holes, and branch placement still stay tied to one graph family.

    No saved checkpoint attempt yet.

    Rational functions
    Open Domain-break checkpoint

Why this order

The sequence is authored to keep the model honest.

Graph Transformations comes first because it trains the eye to treat changes in an equation as visible changes on one shared graph. Rational Functions / Asymptotes and Behavior follows while that shift language is still fresh, so vertical and horizontal asymptotes stay attached to one shifted reciprocal family instead of becoming detached algebra. Exponential Change / Growth, Decay, and Logarithms then adds a second non-polynomial family while the graph-first habit is still active, so multiplicative change and inverse target time stay visual instead of symbolic. Derivative as Slope / Local Rate of Change builds on that same graph-reading habit by asking how the curve changes at one point. Limits and Continuity / Approaching a Value follows while that limiting language is already on the table, so one-sided approach, removable holes, jumps, and blow-up behavior stay attached to the same honest graph rather than turning into edge-case vocabulary. Integral as Accumulation / Area closes the path by turning local-rate and limit language into a running-total view, so slope, approach, and accumulation stay connected instead of becoming separate shelves.

Shared concept pages

Each step opens the same simulation-first framework.

Compare mode, prediction mode, quick test, worked examples, guided overlays, challenge mode, and read-next cues stay on the concept pages. The track only decides the guided order and the next recommended stop.

Guided path

Follow the concepts and checkpoint moments in order.

Checkpoint cards reuse the authored challenge entries already living on the concept pages.

  1. 1Not startedMastery: NewStart here

    Graph Transformations

    Move one parent curve with honest controls so shifts, vertical scale, and reflections stay tied to the same overlaid graph and landmark points.

    Start here before moving into Rational Functions / Asymptotes and Behavior.

    FunctionsIntro20 min
    Start concept
  2. 2Not startedMastery: New

    Rational Functions / Asymptotes and Behavior

    Vary one shifted reciprocal family so domain breaks, vertical and horizontal asymptotes, intercepts, and removable-hole behavior stay tied to the same graph.

    Builds on Graph Transformations before setting up Exponential Change / Growth, Decay, and Logarithms.

    FunctionsIntro23 min
    Open concept
  3. Checkpoint 1LockedNot started

    Domain-break checkpoint

    Pause once the graph-language from parent curves still holds while the reciprocal family separates a true asymptote from a removable hole.

    Finish Rational Functions / Asymptotes and Behavior first. This checkpoint ties together Graph transforms and Rational functions through Domain-break checkpoint.

    Pause here after Rational Functions / Asymptotes and Behavior before moving into Exponential Change / Growth, Decay, and Logarithms.

    Graph transformsRational functions8 checksCoreGraph-linkedGuided start
    Return to Rational functionsOpen concept
  4. 3Not startedMastery: New

    Exponential Change / Growth, Decay, and Logarithms

    Change one starting value, one rate, and one target so growth, decay, doubling or half-life, and logarithmic target time all stay tied to the same live curve.

    Builds on Rational Functions / Asymptotes and Behavior before setting up Derivative as Slope / Local Rate of Change.

    FunctionsIntro22 min
    Open concept
  5. Checkpoint 2LockedNot started

    Growth-and-decay checkpoint

    Close the functions branch by landing a genuine target-time case where multiplicative change still reads off the same graph before calculus starts asking about local rate.

    Finish Exponential Change / Growth, Decay, and Logarithms first. This checkpoint ties together Graph transforms, Rational functions, and Exponential change through Quarter-target checkpoint.

    Pause here after Exponential Change / Growth, Decay, and Logarithms before moving into Derivative as Slope / Local Rate of Change.

    Graph transformsRational functionsExponential change7 checksCoreGraph-linkedGuided start
    Return to Exponential changeOpen concept
  6. 4Not startedMastery: New

    Derivative as Slope / Local Rate of Change

    Slide a point along one curve, tighten a secant into a tangent, and connect local steepness to the derivative graph without leaving the same live bench.

    Builds on Exponential Change / Growth, Decay, and Logarithms before setting up Limits and Continuity / Approaching a Value.

    CalculusIntro25 min
    Open concept
  7. Checkpoint 3LockedNot started

    Slope checkpoint

    Pause the path where the tangent is nearly flat and the secant has tightened around the same local rate before the track widens into accumulation.

    Finish Derivative as Slope / Local Rate of Change first. This checkpoint ties together Graph transforms, Rational functions, Exponential change, and Derivative as slope through Catch the flat tangent.

    Pause here after Derivative as Slope / Local Rate of Change before moving into Limits and Continuity / Approaching a Value.

    Graph transformsRational functionsExponential changeDerivative as slope8 checksCoreGraph-linkedGuided start
    Return to Derivative as slopeOpen concept
  8. 5Not startedMastery: New

    Limits and Continuity / Approaching a Value

    Approach one target point from the left and right, compare the limiting height with the actual function value, and contrast continuous, removable, jump, and blow-up behavior on one honest graph.

    Builds on Derivative as Slope / Local Rate of Change before setting up Integral as Accumulation / Area.

    CalculusIntro24 min
    Open concept
  9. Checkpoint 4LockedNot started

    Continuity checkpoint

    Hold onto the difference between agreeing one-sided values and true continuity before the track widens into signed area and accumulation.

    Finish Limits and Continuity / Approaching a Value first. This checkpoint ties together Derivative as slope and Limits and continuity through Continuity classification checkpoint.

    Pause here after Limits and Continuity / Approaching a Value before moving into Integral as Accumulation / Area.

    Derivative as slopeLimits and continuity6 checksCoreGraph-linkedGuided start
    Return to Limits and continuityOpen concept
  10. 6Not startedMastery: New

    Integral as Accumulation / Area

    Move one upper bound across a source curve and watch signed area build into a running total so accumulation stays visual instead of symbolic.

    Capstone step after Limits and Continuity / Approaching a Value.

    CalculusIntro25 min
    Open concept
  11. Checkpoint 5LockedNot started

    Accumulation checkpoint

    Finish the path by holding onto the difference between local height and total accumulation when the source has turned negative but the running total still stays positive.

    Finish Integral as Accumulation / Area first. This checkpoint ties together Derivative as slope, Limits and continuity, and Integral as area through Negative height, positive total.

    Final checkpoint that closes the authored track after Integral as Accumulation / Area.

    Derivative as slopeLimits and continuityIntegral as area7 checksCoreGraph-linkedGuided start
    Return to Integral as areaOpen concept