Exponential Functions & Growth

Calculus Readiness · Axiom Academy

When each step multiplies instead of adds: growth that explodes, decay that fades toward a floor, and the one base that nature keeps choosing — e . 1. The Variable Lives in the Exponent In the variable rides upstairs , in the exponent — not in the base like x^ 2 . Two numbers run the whole show: the starting value a and the base b . Because raising to a power is repeated multiplication, every time x goes up by 1 the output is multiplied by b . Watch the point climb below: it starts at the y -intercept , and each unit step doubles the height — — so the curve barely moves at first and then rockets upward. the general form — a is where it starts, b is the per-step multiplier one step right always multiplies the output by b 2. Growth, Decay, and the Floor It Never Reaches The base b alone decides the shape. If b>1 each step multiplies by more than one and the curve grows ; if 0<b<1 each step multiplies by a fraction and the curve decays . Below, halves at every step — — sliding toward the dashed line y = 0 . It gets arbitrarily close but never touches : that line is the horizontal asymptote . (The faint blue curve is the growth case from Step 1, mirrored, for contrast.) Each step multiplies by more than 1 , e.g. 2^ x doubles. The output climbs without bound. Each step multiplies by a fraction, e.g. halves. The output shrinks toward zero. A pure b^ x stays positive forever; it approaches the x -axis but never reaches it.

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