So the line, despite its appearance, never goes below the x-axis into negative y-values the graph of y = 2 x is always actually above the x-axis, even if only by a vanishingly-small amount. And a positive 2 cannot turn into a negative number by raising it to a power. Can you ever turn 2 into 0 by raising it to a power? Of course not. On the left-hand side of the x-axis, the graph appears to be on the x-axis. ![]() ![]() You can see this behavior in any basic exponential function, so we'll use y = 2 x as representative of the entire class of functions: This trait - that there is a fixed halving or doubling time - is basic to exponential functions. If the base of the exponential is not 2, then don't expect the doubling/halving time to be emphasized - or else don't expect that doubling/halving time to be a nice neat whole number.) (Note: Many exponentials will have a messy doubling/halving time, so it is often more useful to work in terms of tripling, quadrupling, etc, times, or a given exercise may ignore the issue entirely. For instance, a medical isotope that decays to half the previous amount every twenty minutes and a bacteria culture that doubles every day each exhibits exponential behavior, because, in a given set amount of time (twenty minutes and one day, respectively), the quantity has changed by a constant proportion (one-half as much and two times as much, respectively). ![]() Remember that the basic property of exponentials is that they change by a given proportion over a set interval.
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