Starter track
Follow the authored sequence, or switch to recap mode for a faster review of the same path.
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.
Entry diagnostic
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.
Check the graph-reading bridge first
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.
Check whether shifts, reflections, and landmark tracking already feel stable on the shared graph.
No saved quick-test result yet.
Build a reciprocal family where the true vertical asymptote sits near $x=-1$, the removable hole sits at a positive x-value, and the right branch stays below the horizontal asymptote.
No saved checkpoint attempt yet.
Check whether growth versus decay and the logarithmic target question already feel stable on the same curve.
No saved quick-test result yet.
About this track
Keep the first scan focused on the next lesson. Open the authored rationale and shared-framework notes only when you need them.
Why this order
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
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
Checkpoint cards reuse the authored challenge entries already living on the concept pages.
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.
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.
Build a reciprocal family where the true vertical asymptote sits near $x=-1$, the removable hole sits at a positive x-value, and the right branch stays below the horizontal asymptote.
Finish Rational Functions / Asymptotes and Behavior first. This checkpoint ties together Graph transforms and Rational functions through Keep the true break and the removable break separate.
Pause here after Rational Functions / Asymptotes and Behavior before moving into 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.
Build a decay case where the target is about one quarter of the start, so the curve reaches it in about two half-lives.
Finish Exponential Change / Growth, Decay, and Logarithms first. This checkpoint ties together Graph transforms, Rational functions, and Exponential change through Make the target one quarter of the start.
Pause here after Exponential Change / Growth, Decay, and Logarithms before moving into 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.
Move to the right-hand turning point so the tangent is almost flat, then shrink Δx until the secant slope is almost flat there too.
Finish Derivative as Slope / Local Rate of Change first. This checkpoint ties together Graph transforms, Rational functions, Exponential change, and Derivative as slope through Match the near-flat slopes.
Pause here after Derivative as Slope / Local Rate of Change before moving into Limits and Continuity / Approaching a Value.
Move one target-distance slider inward from both sides, compare the finite-limit guide with the actual point, and separate continuous, removable-hole, jump, and blow-up behavior on one graph-first bench.
Builds on Derivative as Slope / Local Rate of Change before setting up Integral as Accumulation / Area.
Switch to the case where the left-hand and right-hand values agree on one finite limit, but the function is still not continuous at $x = 0$.
Finish Limits and Continuity / Approaching a Value first. This checkpoint ties together Derivative as slope and Limits and continuity through Limit exists, continuity fails.
Pause here after Limits and Continuity / Approaching a Value before moving into Integral as Accumulation / Area.
Drag one upper bound across a source curve, watch signed area and the accumulation graph update together, and separate current source height from the running total.
Capstone step after Limits and Continuity / Approaching a Value.
Move the bound to a place where the source height is negative, but the running total is still 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.
