Take any exponential y = a·bx and compare one output to the one before it. Divide:
Do not take that on faith — watch the factors cancel. Nothing here is a rule, it is just counting how many b's are on top and on the bottom:
The exponent x does not have to be a whole number for this to hold, but whole numbers are where you can literally see it. And notice the a cancelled too — the starting value has no say in the step ratio. Whatever a is, the jump from each output to the next is a multiplication by b, and only by b. Switch the figure above to Step by step to watch those multiplications drawn between the dots.
This is now a two-line argument rather than something to memorize. Every output a·bx is a positive number (proved below), and the next output is that positive number times b. So ask the only question that matters: does multiplying a positive number by b make it bigger or smaller?
Multiplying an inequality by the positive number f(x) is what lets each line go through, and that is legal precisely because outputs are never zero or negative. So the sorting rule is not arbitrary: compare b to 1 and you have already decided the shape of the graph. Nothing about a enters into it.
Start two sequences at 2. One adds 3 each step; the other multiplies by 3 each step. For a moment they look comparable:
At step 1 the two are 5 and 6 — practically a tie. Four steps later one is at 14 and the other at 162. Push it further and the linear side never recovers: at step 10 adding gives 32, multiplying gives 118,098. This is why the base is worth reading carefully before anything else.
The base is required to be positive, so for a whole number n, bn is a product of positive numbers and is positive. Negative exponents are the reciprocals of those, so they are positive too:
So bx > 0 for every real x, and with a > 0 the whole output stays above the x-axis forever. As x marches toward −∞ on a growth curve, the outputs keep halving, or thirding, or whatever b dictates — they shrink without limit but never arrive at 0. That is what a horizontal asymptote is: a line the graph closes in on and never reaches. For y = a·bx that line is always y = 0, the x-axis itself. A decay curve does the same thing on the right-hand side instead.
One consequence worth stating outright: an exponential of this form has no x-intercept. Setting a·bx = 0 forces bx = 0, which nothing satisfies. Do not go hunting for one.
Put x = 0 into y = a·bx and you get y = a·b0 = a·1 = a. The reason b0 = 1 is the same cancelling as before, read in one extra direction:
So the base changes how steeply the curve leaves the point (0, a), and which direction it leaves in, but it cannot move that point. Drag b across its whole range above and watch the gold dot sit perfectly still. Only a lifts it. That also gives you a free reading in reverse: whatever height a graph crosses the y-axis at, that height is a.
Bases b and 1/b are not two separate stories. Watch:
Replacing x with −x is exactly what reflects a graph across the y-axis, because the height that used to sit at x = 3 now sits at x = −3. The table makes it plain — the second row is the first row read backwards:
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y = 2x | ¼ | ½ | 1 | 2 | 4 |
| y = (½)x | 4 | 2 | 1 | ½ | ¼ |
Set the figure above to Mirror to see both curves at once. It also gives you a sanity check you can run in your head: if a base and its reciprocal do not produce mirror-image graphs, something is wrong in the arithmetic.
Most real exponentials arrive described as a percent per period, not as a base. The conversion is one line, and it comes from the fact that keeping what you had and adding r of it means multiplying by 1 + r:
| described as | r | b = 1 + r | behavior |
|---|---|---|---|
| up 8% per year | +0.08 | 1.08 | growth |
| up 100% per year | +1.00 | 2 | doubling |
| down 8% per year | −0.08 | 0.92 | decay |
| down 50% per year | −0.50 | 0.5 | halving |
Two traps live in that table. First, a loss of 8% is not the reciprocal of a gain of 8%. Check: 1.08 × 0.92 = 0.9936, not 1. Gain 8% then lose 8% and you are down to 99.36% of where you started — the loss is taken on a bigger number than the gain was. Second, a 100% increase means b = 2, not b = 1. The base counts what you end with, not what you added.
b ≠ 1. With b = 1, y = a·1x = a for every single x, because 1 multiplied by itself any number of times is still 1. The graph is a horizontal line. There is no growth, no decay, no asymptote to approach — it is a constant function wearing an exponential costume, so it is excluded by definition. Drag the base slider to exactly 1 and watch every reading above go flat.
b > 0. Zero fails first: y = 0x gives 0 at x = 3, but at x = −3 it demands 1 ÷ 03 = 1 ÷ 0, which is undefined, and 00 is undefined as well. Negative bases fail differently and more interestingly. Try b = −4:
Some negative-base powers do exist — (−8)1/3 = −2, since odd roots of negatives are fine. That is what makes this worse, not better: the legal inputs would be scattered among illegal ones with gaps everywhere between them, so there is no continuous curve to draw at all. Requiring b > 0 is what buys you a smooth graph over every real x.
And a ≠ 0 — the trap. If a = 0 then y = 0·bx = 0 everywhere, flattening onto the x-axis no matter how dramatic the base is. A negative a is perfectly legal, incidentally: it reflects the whole curve below the x-axis, so the range becomes y < 0 while the asymptote y = 0 and the growth-or-decay verdict from b stay exactly as they were.
Domain: all real numbers. With b > 0 there is nothing to exclude — no denominator holding x, no even root of a negative. Write −∞ < x < ∞. This is different from the rational functions of the last chapter, where excluded values were the first thing to hunt for.
Range: y > 0 when a > 0, and y < 0 when a < 0. Never “all real numbers,” and never including 0. Asymptote: y = 0 always, for this form.
On extraneous solutions. They do not arise from the graphing here, but they will the moment you start solving. Two habits protect you. When you solve bx = k, a solution exists only if k > 0 — asking 3x = −5 or 3x = 0 has no answer at all, since the left side is stuck above zero. And if an exponential equation was reached by squaring, or by clearing a denominator, substitute your answer back into the original statement before writing it down, because those moves can invent roots that were never legal. The graph shows you every value the function actually takes; the original problem decides which ones you are allowed to report.