Transformations of Exponential Functions

Pre-Calculus · Axiom Academy

LESSON Transformations of Exponential Functions Shift, stretch, and reflect the curve b^x — and watch its horizontal asymptote move with it. Vertical stretch / compression; flips over the x -axis. The growth ( ) or decay ( ) rate. Replaces x with x-h ; shifts right for (opposite sign!). Moves the curve up by k — and the asymptote to y=k . 1. Vertical Shift — the Asymptote Moves to y=k Add a constant k and every point of f(x)=2^x rises by k . The curve's flat tail — its horizontal asymptote — rises with it, from y=0 to y=k . This is the one transformation that relocates the asymptote. 2. Horizontal Shift — Mind the Opposite Sign Replacing x with x-h slides the curve sideways — and the direction is the opposite of the sign you see. The animation carries the anchor point (0,1) along: sends it to (1,1) , a shift right , even though the formula shows a minus. 3. Vertical Stretch & Compression Multiplying by a scales every height by that factor. Heights pinned to the asymptote y=0 stretch away from it: at x=2 the parent's value 4 becomes 12 when a=3 , and 2 when . The animation pulls the curve outward from the axis and back in. The curve grows taller; has y -intercept 3 instead of 1 . The curve flattens toward the axis; has y -intercept . 4. Reflections — Two Different Flips

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