From 4d43a668d18e1e0ae9a8c862a0bed3513d0be6d3 Mon Sep 17 00:00:00 2001 From: Stephen Umunna Date: Wed, 24 Jun 2026 10:12:23 -0700 Subject: [PATCH] feat(content): make the reach-saturation curve interactive (foundations-08) foundations-08 ("Logarithms, growth, and decay") illustrated reach saturation (R = Rmax(1 - e^(-kx))) with a static lineChart that was the section's only visual and no interactive. Swap it for the live responseCurve so the learner drags spend and watches each extra dollar buy less new reach -- the diminishing returns the lesson teaches, made manipulable. The self-contained predict that follows is unchanged. Validator green (5,271 checks). Co-Authored-By: Claude Opus 4.8 --- Content/Lessons/foundations-08.md | 8 +++----- 1 file changed, 3 insertions(+), 5 deletions(-) diff --git a/Content/Lessons/foundations-08.md b/Content/Lessons/foundations-08.md index e21d35b..6a4b6dc 100644 --- a/Content/Lessons/foundations-08.md +++ b/Content/Lessons/foundations-08.md @@ -66,11 +66,9 @@ The same multiply-by-a-fraction logic explains the most important curve in media Adstock, the carryover of advertising over time, is the time-domain cousin of this idea. The geometric adstock model is $A_t = X_t + \lambda A_{t-1}$, where this week's effect $A_t$ is this week's new advertising $X_t$ plus a decayed fraction $\lambda$ of last week's accumulated effect. The decay piece is the exponential fade from the previous section, and the accumulation piece is what builds the saturating curve. Both reach and adstock curves are concave because the next unit always does a bit less than the last, which is diminishing returns made visible. -:::widget lineChart -title: Saturating reach curve, new reach per dollar shrinks as spend rises (illustrative) -labels: 1, 2, 3, 4, 6 -data: 39.35, 63.21, 77.69, 86.47, 95.02 -unit: % +The curve below makes this concrete. Drag spend along it and watch each extra dollar buy less new reach than the one before, the saturation the math predicts. + +:::widget responseCurve ::: :::predict