Growth or Decay

Part of the Exponentials, Logs & Sequences section of Coddy's Algebra 2 journey. Lesson 2 of 49.

The multiplier b decides the whole shape. When b is bigger than 1, every step makes the output bigger: y = 2^x gives 1, 2, 4, 8. That is growth. When b is between 0 and 1, every step makes it smaller: y = (1/2)^x gives 1, 1/2, 1/4, 1/8. That is decay.

$The graphs of y = 2^x, rising to the right, and y = (1/2)^x, falling to the right, both through (0, 1)$

y = 2^x grows to the right; y = (1/2)^x, dashed, decays.

Two values of b are left out of the model. With b = 1 nothing changes, since 1^x is 1 for every x: the graph is a flat line. A negative b makes the output flip sign at every step, as in 1, -2, 4, -8, and it has no real value at x = 1/2. So in y = a × b^x, b is positive and not 1.

With a positive start the curve never touches the x-axis: a positive number multiplied by a positive b stays positive, however many times you do it. It does get closer and closer to 0, on one side. For growth that side is the left, where each step divides by b, a number above 1. For decay it is the right, where each step multiplies by b, a number below 1.