Use parent-curve moves, a shifted reciprocal family, and one exponential bench so graph moves, asymptotes, domain breaks, growth versus decay, and target-time questions stay tied to the same visual branch before the math path widens into local and accumulated change.
This page stays deliberately compact, but it now points toward a fuller math slice. Graph Transformations keeps one parent curve, one transformed curve, and a small set of landmark points in view so horizontal and vertical shifts, vertical scale, and reflections can be read as actual motion on the graph before the symbols get abstract. Rational Functions / Asymptotes and Behavior then keeps one shifted reciprocal family bounded on the same graph so vertical and horizontal asymptotes, domain breaks, intercepts, and an optional removable hole stay honest. Exponential Change / Growth, Decay, and Logarithms keeps one start value, one rate, one target, and one logarithmic inverse question on the same bench so doubling, half-life, and target time do not drift into detached algebra.
Best first concepts
The topic page keeps these starts in their own compact row so the first screen is about orientation and next action, not stacked feature cards.
Move one parent curve with honest controls so shifts, vertical scale, and reflections stay tied to the same overlaid graph and landmark points.
Parent-function moves
Strong first stop for getting into this topic without scanning the whole library.
Vary one shifted reciprocal family so domain breaks, vertical and horizontal asymptotes, intercepts, and removable-hole behavior stay tied to the same graph.
Rational functions and asymptotes
Strong first stop for getting into this topic without scanning the whole library.
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.
Exponential growth and decay
Strong first stop for getting into this topic without scanning the whole library.
Specific learning goals
These goal cards stay authored and transparent. They reuse the current topic page, starter tracks, guided collections, concept bundles, and progress cues instead of adding a separate recommendation system on top of this branch.
Use the functions topic route, the new lesson set, the compact math starter track, and the calculus topic route so graph moves, rational asymptotes, exponential change, local slope, and accumulation stay on one coherent bench.
Primary move
Open topic route
No saved progress yet inside Functions.
Entry diagnostic
Start from the opening step
No saved diagnostic checks are available yet, so the opening step is still the best entry into the collection.
Reuses the guided collection entry for Functions and Change Lesson Set, with 0 of 3 probes already ready.
No saved progress yet inside Functions.
Open the functions topic route is the next guided collection step.
Graph Transformations opens this track and sets up the rest of the path.
No saved progress yet inside Calculus.
Grouped concept overview
Each group is authored in the topic catalog, but the actual concepts, progress badges, and track cues still come from the canonical concept metadata and shared progress model.
Group 01
Start with one parent curve and keep the transformed version overlaid so every parameter change still reads as a specific move on the same graph.
用誠實的控制項移動同一條母函數曲線，讓平移、垂直伸縮與反射都綁在同一張疊圖與標誌點上。
Strong first stop for getting into this topic without scanning the whole library.
Group 02
Keep one shifted reciprocal family on the same graph so the forbidden x-value, the horizontal level, intercepts, and an optional removable hole all stay visible together.
改變同一個平移後的倒數函數族，讓定義域缺口、垂直與水平漸近線、截距與可去間斷都綁在同一張圖上。
Strong first stop for getting into this topic without scanning the whole library.
Group 03
Then keep one start value, one rate, and one target on the same bench so growth, decay, and the logarithmic target question stay visually linked.
改變同一條曲線的初值、變化率與目標值，讓增長、衰減、倍增／半衰期與對數求時都留在同一個即時圖上。
Strong first stop for getting into this topic without scanning the whole library.