Writing Equations from Graphs

Pre-Calculus · Axiom Academy

LESSON Writing Equations from Graphs Run a transformation backwards — read a graph's clues and rebuild the equation that drew it. 1. The Graph Is Telling You h, k, and the Sign Every transformed parent carries its recipe on its face. A U-shaped curve says the parent is y = x^2 . The turning point sits at the vertex (h, k) — here it's (2, 3) . And which way it opens fixes the sign of a : opening downward means a is negative. Watch the parent y = x^2 slide its vertex from the origin out to (2, 3) and flip over — that motion is the transformation you're reading in reverse. Vertex form — every quadratic graph fits this Infinitely many parabolas share the vertex (2, 3) — wide ones, narrow ones, gentle and steep. Only one also passes through the second point we can read off the graph, (3, 1) . Watch the candidate curve sweep through stretch factors: the moment it threads (3, 1) , the stretch locks. That click is the algebra — substitute the point into vertex form and solve for a . At x = 3 the curve has already dropped to 0 — it passes below (3, 1) . At x = 3 the curve sits at exactly y = 1 . This is the only stretch that fits. At x = 3 the curve is still at y = 2 — it passes above the point. Any single point off the vertex turns the unknown a into one equation with one solution. The full equation, recovered from the graph

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