Modeling with Transformations

Pre-Calculus · Axiom Academy

Real data rarely fits a parent function exactly — so we stretch, shift, and flip it until it does. Put three of those models in your hands. A basketball arc, a summer day's temperature, a musical note — none of them are a textbook curve. But each one is a parent function ( x² , cos , sin ) transformed to fit. Build all three by hand. A player releases the ball 2 m off the floor; it peaks at 5 m when it's 4 m downrange. The peak fixes the vertex — drag the stretch a until the curve also passes through the release point. Set the temperature's daily swing A summer day runs 92° at the 3 PM high and 68° at the pre-dawn low. The cosine already peaks at 3 PM — drag its amplitude and midline until the high and low land on the real numbers. A sound wave is just y = A·sin(Bt) : A sets the loudness, B sets the pitch. Start at concert A ( 440 Hz ) and drag B up — double it and you've jumped a full octave to A5 (880 Hz). Three situations, one move: take a parent function and transform it to fit the data. A vertical stretch sets the shape of the arc, the amplitude and midline set the swing of the day, a horizontal compression sets the pitch of the note. Anywhere the world bends a clean curve — projectiles, tides, sound, signals — the same shift-stretch-flip toolkit builds the model.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.