Hold the base still, let the exponent run: y = aˣ (a > 0, a ≠ 1) — growth when a > 1, decay when 0 < a < 1.
Last stage's power functions kept the exponent fixed and let the base vary — that was x², x³, the family y = xᵅ. Now flip that hinge. Picture a single cell that doubles each generation: 1, 2, 4, 8, 16, … . The generation number is the exponent, and the count is 2 raised to it. That is an exponential function, y = ax, and it grows in a way nothing linear can match. This lesson builds it from a doubling sequence, draws it for growth and for decay, names the three things every exponential curve shares — through (0, 1), always above the axis, hugging the red line y = 0 — and then slides the whole curve around.
Start with the doubling cell. After 0 generations there is 1 cell; after 1 there are 2; after 2 there are 4; after 3 there are 8. Each step to the right multiplies the count by 2. Writing the generation as x and the count as y, the pattern is exactly
y = 2x : 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, …
The variable has moved into the exponent. Last lesson taught us how to read 2x for every real x — including 21/2 ≈ 1.41 and 21.7 ≈ 3.25 — so the dots can join into one smooth curve. Replace the base 2 by any constant a and you have the general shape.
An exponential function is y = ax, where the base a is a fixed constant with a > 0 and a ≠ 1, and the variable x sits in the exponent. Each step right multiplies y by the same factor a.
Why the two restrictions? They are not fussiness — each one rules out a case that would not give an exponential curve at all.
a ≤ 0 is barred. If a were negative, say a = −2, then a1/2 = √(−2) is undefined and ax would skip wildly between positive and negative — no smooth curve exists. And a = 0 gives 0x, which collapses to 0. So we insist a > 0.
a = 1 is barred. Since 1x = 1 for every x, the graph of y = 1x is the flat horizontal line y = 1 — a constant, not an exponential. So we insist a ≠ 1.
When the base is bigger than 1, the curve rises — and not at a steady rate, the way a line does. Each step to the right multiplies the height by a, so a tall value gets multiplied to an even taller one. The steepness itself keeps growing. That compounding is the engine of exponential growth.
Compare y = 2x with y = 3x. Both start at (0, 1), but 3x triples at every step while 2x only doubles — so 3x climbs harder and pulls away fast:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| 2x | 1 | 2 | 4 | 8 | 16 |
| 3x | 1 | 3 | 9 | 27 | 81 |
What makes an exponential explode is a constant ratio, not a constant difference. A line adds the same amount per step; y = ax multiplies by the same factor a per step. Multiplying compounds — that is why exponential growth eventually outruns any line.
Use the slider below to dial the base a above 1 and watch the curve tilt steeper. Notice two anchors that never move: every curve still passes through (0, 1) (because a0 = 1), and the point one step right is (1, a) — the base itself, read straight off the graph.
Drag the base a. Above 1 the curve rises (growth); between 0 and 1 it falls (decay). The red line y = 0 is never crossed.
Now make the base a fraction between 0 and 1. The curve falls toward zero. This is decay: a drug clearing the bloodstream, a hot cup cooling, a radioactive sample fading — half of what was there leaves each interval. The cleanest example is
y = (½)x : at x = 0 it is 1, at x = 1 it is ½, at x = 2 it is ¼, at x = 3 it is ⅛, …
Each step right halves the height. The curve plunges quickly at first, then flattens out as it creeps toward the floor — but it never reaches it. There is a neat way to see decay as growth seen in a mirror. Because ½ = 2−1,
(½)x = (2−1)x = 2−x.
For 0 < a < 1 the exponential y = ax is decreasing — it falls toward the x-axis. And (½)x = 2−x: the decay curve is exactly the growth curve y = 2x reflected across the y-axis. Halving going right is the same as doubling going left.
Whatever the base — soaring like 3x or fading like (½)x — every exponential curve obeys the same three laws. They are worth committing to memory, because they let you sketch any y = ax in seconds.
The only thing that changes with the base is the direction: a > 1 rises, 0 < a < 1 falls. Everything else — the point (0, 1), the strip y > 0, the asymptote — is shared.
| a > 1 (growth) | 0 < a < 1 (decay) | |
|---|---|---|
| shape | rising → | falling → |
| passes through | (0, 1) | (0, 1) |
| range | y > 0 | y > 0 |
| asymptote | y = 0 (left tail) | y = 0 (right tail) |
A common slip is to think y = ax eventually "hits" the x-axis. It does not. As x runs to the far tail, ax shrinks toward 0 but stays strictly positive — there is no x with ax = 0. The red line y = 0 is a wall the curve approaches forever and never crosses.
Because the curve is monotone — always rising, or always falling — its shape answers comparison questions instantly. You do not need a calculator; you need only to know whether the base is above or below 1.
When a > 1 the function is increasing, so the point farther right sits higher:
21.5 < 21.7 (the bigger exponent wins, because 2 > 1).
When 0 < a < 1 the function is decreasing, so the point farther right sits lower — the order flips:
(½)2 = ¼ > (½)3 = ⅛ (the bigger exponent loses, because ½ < 1).
So "which is bigger, am or an?" needs two facts: compare the exponents m and n, and check whether the base is above or below 1. The widget below marks two heights on one curve so you can read the verdict straight off the picture.
Pick a base, then slide the two inputs m and n. The curve tells you which output is larger — and why.
The comparison rule reverses the moment the base crosses 1. For a base above 1, "more exponent" means "more height." For a base below 1, "more exponent" means "less height." Always check which side of 1 the base is on before you decide.
Once you can draw the basic y = ax, you can draw a whole family by sliding it. The same translation rules met in representing functions and the coordinate plane apply here. The general shifted form is
y = ax − h + k.
Subtracting h inside the exponent slides the curve right by h; adding k outside slides it up by k. Two landmarks move with it. The horizontal asymptote rises from y = 0 to y = k, and the anchor point (0, 1) moves to (h, 1 + k).
Sketch y = 2x − 1 + 3. Here h = 1 and k = 3, so the basic curve y = 2x slides 1 right and 3 up. The asymptote moves to y = 3, and the anchor (0, 1) moves to (1, 4) — check: at x = 1, y = 20 + 3 = 1 + 3 = 4. ✓ The range is now y > 3.
Dial h and k below. The slate ghost is the original y = 2x; the blue curve is the shifted one. Watch the red asymptote ride up and down with k, and the green anchor follow.
Step h (right) and k (up). The asymptote moves to y = k; the anchor moves to (h, 1 + k).
y = ax − h + k is the basic curve slid h right and k up. The asymptote moves to y = k, the anchor to (h, 1 + k), and the range to y > k. The shape never changes — only its position.
An exponential function is y = ax with a > 0, a ≠ 1 — the base held still while the exponent runs. For a > 1 it grows (rising, steepening); for 0 < a < 1 it decays (falling toward zero), and (½)x = 2−x is doubling seen in a mirror. Every such curve has domain all real x, range y > 0, passes through (0, 1), and hugs the red asymptote y = 0 without ever touching it. Because the curve is monotone, comparisons follow from the exponents and which side of 1 the base lies. Finally, y = ax − h + k slides the whole picture h right, k up, lifting the asymptote to y = k and the anchor to (h, 1 + k). Next we run the question backward — "what exponent gives this number?" — and meet the logarithm.
Is y = (0.5)x growth or decay? Which way does the curve go as x increases?
Decay. The base 0.5 is between 0 and 1, so the function is decreasing: as x increases the height falls toward the asymptote y = 0 (without reaching it).
What single point does every exponential curve y = ax pass through, no matter the base?
(0, 1). Because a0 = 1 for every allowed base, all of them cross the y-axis at height 1.
What is the horizontal asymptote of y = 3x? And of y = 2x + 5?
For y = 3x the asymptote is y = 0. Adding 5 lifts the whole curve up 5, so for y = 2x + 5 the asymptote is y = 5 (and the range is y > 5).
Which is larger, 23 or 23.2? Explain using the shape of the curve.
23.2 is larger. Since the base 2 > 1, the curve is increasing, so the larger exponent gives the larger value (23 = 8 and 23.2 ≈ 9.19).
Which is larger, (⅓)2 or (⅓)4? Why does the answer flip the usual expectation?
(⅓)2 is larger. The base ⅓ is between 0 and 1, so the function is decreasing — the larger exponent gives the smaller value: (⅓)2 = ⅑ > (⅓)4 = ¹⁄₈₁.
Where is the asymptote of y = 2x − 1 + 4, and what point has the anchor (0, 1) moved to?
Here h = 1, k = 4. The asymptote rises to y = 4, and the anchor moves to (h, 1 + k) = (1, 5). Check: at x = 1, y = 20 + 4 = 5. ✓
Six questions to lock it in. Tap the answer you think is right.
The big idea. An exponential function moves the variable into the exponent: y = ax with a > 0 and a ≠ 1. Its defining feature is a constant multiplicative ratio — each unit step multiplies the output by a — which is exactly what distinguishes it from a line's constant additive difference. Anchor the discussion on the three invariants (passes (0, 1), range y > 0, horizontal asymptote y = 0) and the single variable feature (rising for a > 1, falling for 0 < a < 1). Then translations y = ax − h + k slide the asymptote to y = k.
Misconceptions to watch. Three are common: (1) believing the curve eventually touches the x-axis — it approaches y = 0 forever but never reaches it (ax is always positive); (2) confusing growth and decay — a base just below 1 still produces a falling curve, not a rising one; (3) reading y = ax − h as a left shift — subtracting h inside the exponent slides the curve to the right. The interactive figures are built so the picture and the readout always agree, which lets students test these claims directly.
Common Core. This lesson supports F-IF.C.7e (graph exponential functions, showing intercepts and end behavior, including asymptotes), F-LE.A.1 (distinguish exponential from linear growth — constant ratio vs constant difference), and F-BF.B.3 (identify the effect of the transformations h and k on the graph).