Exponential Functions, Growth, and Decay
An exponential function \(f(x) = a\cdot b^x\) multiplies by the same base b every time x increases by 1 — that single rule is what makes it grow or shrink so much faster than anything additive.
By the end you'll be able to tell growth from decay just by looking at the base b, read off the y-intercept and horizontal asymptote, and predict how shifting the graph moves that asymptote.
Predict: if you set b = 0.5, is that growth or decay? Set the base slider below to 0.5 and check.
This plots \(y = b^x + k\). Drag b and watch the curve switch between growth (b > 1) and decay (0 < b < 1) — the dot marks the y-intercept, always at \(x=0\). Drag k to shift the whole graph up or down and watch the dashed asymptote move with it. Hover the curve or the dot for exact readouts.
\(y = 2.0^x\) — growing (each step ×2.00); y-intercept (0, 1); asymptote y = 0
\(f(x) = a\cdot b^x\) multiplies by b every time x goes up by 1 — the base b alone decides whether the graph rockets up (growth) or fades toward zero (decay).
Think of a linear function as adding the same amount every step — here you're multiplying by the same factor every step instead. That's the whole difference, and it's why your exponential eventually blows past any linear function, no matter how big a head start the line gets. Multiply by a number bigger than 1 (b > 1) and things get bigger and bigger — exponential growth. Multiply by a fraction between 0 and 1 and things get smaller and smaller, closer and closer to zero without you ever quite reaching it — exponential decay.
Plug \(x=0\) into \(f(x) = a\cdot b^x\) and you'll see every one of these graphs passes through \((0, a)\) — \(b^0=1\) leaves just a behind. Follow the curve and it hugs the x-axis on one side and shoots away on the other, but it never crosses \(y = 0\): that line is the horizontal asymptote. Add a constant k and you shift the entire curve — asymptote included — up or down by k, exactly like any other function's vertical shift.
Say you put money in an account earning rate r, compounded n times a year for t years — it grows to \(A = P(1+r/n)^{nt}\), exponential in t with base \((1+r/n)^n\), always bigger than 1 because interest only adds money. You'll see the same rule run backwards for radioactive decay and a cooling drink, just with a base between 0 and 1.
Say you've got 200 bacteria that triple every hour: \(f(x) = 200\cdot 3^x\). After 4 hours, you'd find \(f(4) = 200\cdot 3^4 = 200\cdot 81 = \) 16,200. Since \(b = 3 > 1\), this is growth, the y-intercept is \((0, 200)\), and the asymptote stays at \(y=0\).
Say you're tracking a 500 mg dose that halves every 6 hours: \(g(t) = 500\cdot(1/2)^{t/6}\). Since \(b = 1/2\) is between 0 and 1, this is ____. At \(t = 12\) hours, you'd get \(g(12) = 500\cdot(1/2)^{12/6} = 500\cdot(1/2)^2 = 500\cdot 0.25 = \) ____ mg.
Reveal the answer
It's decay — the base 1/2 is between 0 and 1, so the amount shrinks toward (but never reaches) zero. \(g(12) = 500\cdot 0.25 = \) 125 mg. Set b close to 0.5 on the slider above and watch the curve fall the same way.
More info — why b = 1 and negative bases don't count
If \(b=1\), \(f(x) = a\cdot 1^x = a\) for every x — a flat line, not an exponential curve, so \(b \neq 1\) is required by definition. If \(b\) were negative, \(b^x\) would be undefined for plenty of x (like \(x=1/2\), a square root of a negative number), so exponential functions require \(b>0\). Both restrictions are why the base slider above skips straight over \(b=1\) and never goes negative. See the Paul's Online Notes link in Dive deeper for the full definition, and revisit shifting a graph up or down for how the asymptote moves under a vertical shift.
Check your understanding
A pond starts with 50 fish and the population doubles every year (\(f(x) = 50\cdot 2^x\)). How many fish are there after 3 years?
For \(f(x) = 5\cdot(0.8)^x\), is this growth or decay, and where is the horizontal asymptote?
The graph of \(y = 3^x\) is shifted left 2 units and down 4 units. What is the new equation, and what happens to the horizontal asymptote?
For any exponential function \(f(x) = a\cdot b^x\) (with \(b > 0\), \(b \neq 1\)), what is \(f(0)\), and why?
Recap
- \(f(x) = a\cdot b^x\) multiplies by the base b every time x increases by 1; b > 0 and \(b \neq 1\).
- \(b > 1\) is growth (output increases without bound); \(0 < b < 1\) is decay (output shrinks toward zero).
- The graph always passes through \((0, a)\), and the x-axis (\(y=0\)) is a horizontal asymptote it approaches but never crosses.
- Adding k shifts the whole graph — and its asymptote — vertically by k; the base b alone never moves the asymptote.
Dive deeper
- Paul's Online Notes — Exponential Functions Study the definition f(x)=b^x and graph increasing versus decreasing bases.
- Math is Fun — Exponential Growth and Decay See real-world growth, decay, and half-life models.
Sources
- Exponential Functions, Growth, and Decay