What does an exponential function look like when its base is between zero and one?

For the parent exponential function f(x) = b^x, a base strictly between 0 and 1 makes the function decrease as x increases. [S1] With f(x) = (1/2)^x, each time x goes up by 1, you multiply the previous output by 1/2 again. Starting at 1, the values halve: 1, then 1/2, then 1/4, then 1/8. That repeated multiplication is why the function is called exponential decay. It is not subtracting the same amount at each step: the amount lost gets smaller as the values shrink.

Here is a comparison with the growing function g(x) = 2^x. Both functions are shown for the same x-values:

x = 0: (1/2)^x = 1; 2^x = 1 x = 1: (1/2)^x = 1/2; 2^x = 2 x = 2: (1/2)^x = 1/4; 2^x = 4 x = 3: (1/2)^x = 1/8; 2^x = 8

At x = 0, both equal 1 because any nonzero base raised to the power 0 is 1. After that, one falls while the other rises. [S1]

Plot the points (0, 1), (1, 1/2), (2, 1/4), and (3, 1/8). Connect them with a smooth curve, not separate straight segments. The curve descends quickly at first and then flattens as it moves to the right. Its outputs stay positive and approach zero, so the curve gets closer to the x-axis without touching it. [S1]

For comparison, plot (0, 1), (1, 2), (2, 4), and (3, 8) for 2^x. That curve rises as x moves right. The two graphs share the point (0, 1), but go in opposite directions on the right side. A useful check when sketching: the fraction-base curve should be above the x-axis, pass through (0, 1), and have smaller positive y-values at successive integer x-values.

The same fraction is applied by multiplication, not subtracted as a fixed amount. For (1/2)^x, the change from 1 to 1/2 is 1/2, but the change from 1/2 to 1/4 is only 1/4; from 1/4 to 1/8 it is 1/8. So the outputs are cut in half each step, while the absolute drop becomes smaller.

A straight-line decrease would have a constant difference between successive outputs. For instance, values 1, 1/2, 0, -1/2 drop by 1/2 each time and lie on a line. That is a different pattern from 1, 1/2, 1/4, 1/8, where each value is half the previous one and the graph curves. The key question is: are you subtracting a fixed amount, or multiplying by a fixed factor?

Use the base to identify the underlying pattern, but check the whole formula before describing the graph. The statements about decreasing, staying positive, and approaching the x-axis apply to the parent function b^x when 0 < b < 1. Multiplying that function by another factor or adding a vertical shift changes its graph. [S1]

For example, compare h(x) = 3(1/2)^x with k(x) = (1/2)^x + 2. The factor 3 makes each output of the parent function three times as large, while the +2 shifts each output up by 2. You can calculate points by applying the outside operation to the parent values: at x = 0, the parent value is 1, so h(0) = 3 and k(0) = 3. At x = 1, the parent value is 1/2, so h(1) = 3/2 and k(1) = 5/2. These are different graphs from the parent, even though the exponential part still uses a base between 0 and 1. The source establishes the parent-function behavior; it does not specify every transformed graph, so calculate points for the formula you actually have.