Free online graphing tool
Exponential Graph Calculator (y = a·bˣ)
An exponential function has the variable in the exponent, not the base. That one difference makes it grow far faster than any polynomial — the reason compound interest, population growth and viral spread are all modelled this way.
Plot y = a·bˣ below. Bases greater than 1 give growth; bases between 0 and 1 give decay. Every curve of this family passes through (0, a) and approaches the x-axis without ever touching it.
Key points of y = 2^x inside the visible window (-4 ≤ x ≤ 4). Pan or zoom to search a different range.
Roots (x-intercepts)
No root in this window
y-intercept
- y = 1
Turning points
No maximum or minimum here
Intersections
- (0, 1)
Enter a value of x
What is a exponential graph calculator?
An exponential graph calculator plots y = a·bˣ, where the variable sits in the exponent. When b > 1 the curve rises ever more steeply (growth); when 0 < b < 1 it falls towards zero (decay). The x-axis is a horizontal asymptote, so the curve never reaches y = 0, and every such curve passes through (0, a).
Key facts at a glance
| General form | y = a·bˣ, with b > 0 and b ≠ 1 |
|---|---|
| Natural exponential | y = eˣ, where e ≈ 2.71828 |
| Growth when | b > 1 |
| Decay when | 0 < b < 1 |
| Always passes through | (0, a) — because b⁰ = 1 |
| Horizontal asymptote | y = 0; the curve never touches the x-axis |
| Range | y > 0 for a > 0 — an exponential is never zero or negative |
| Inverse | The logarithm — y = bˣ reflects into y = log_b x across y = x |
How to plot exponential graphs — step by step
- 1
Plot a growth curve
Type y = 2^x. The ^ symbol raises to a power, so the exponent can be any expression.
- 2
Compare bases
Add y = 3^x and y = e^x. All three meet at (0, 1); the bigger the base, the steeper the climb.
- 3
Model decay
Type y = (1/2)^x or the equivalent y = e^(-0.7x). The curve falls towards zero without ever reaching it.
- 4
Zoom out to see the asymptote
Scroll to zoom out on the left-hand side — the curve flattens against the x-axis but stays strictly above it.
Exponential examples you can plot right now
Copy any equation into the calculator above, or open it in the full calculator with one click.
Natural growth
y = e^x
The exponential whose gradient equals its own height at every point.
Plot this →Radioactive decay
y = 100 e^(-0.3x)
Starts at 100 and decays with a half-life of ln(2)/0.3 ≈ 2.3 units.
Plot this →Compound interest
y = 1000 (1.08)^x
₹1,000 growing at 8% a year — read the balance after x years off the curve.
Plot this →Understanding the exponential graph
- Why exponentials beat polynomials
- Zoom out far enough and y = 2ˣ overtakes y = x², y = x³ and every other power. Exponential growth multiplies by a constant factor each step, while a polynomial only adds.
- The meaning of e
- e is the base for which the curve's gradient equals its own height everywhere. That property makes eˣ the natural choice in calculus and in any continuously compounding process.
- Half-life and doubling time
- For y = a·e^(kx), the doubling time is ln(2)/k when k > 0 and the half-life is ln(2)/|k| when k < 0 — both independent of where you start.
Exponential graph — frequently asked questions
How do I graph an exponential function?
Type y = 2^x into the calculator above, using ^ for the power. For the natural exponential type y = e^x, and for a general model use y = a b^x to get sliders for both parameters.
What is the difference between exponential growth and decay?
Growth happens when the base is greater than 1, so the curve rises to the right. Decay happens when the base is between 0 and 1, so the curve falls towards the x-axis. y = 2^x grows; y = 0.5^x decays.
Why do all exponential graphs pass through (0, 1)?
Because any non-zero number raised to the power zero equals 1. For the general form y = a·bˣ the curve passes through (0, a) instead, since b⁰ = 1 leaves a behind.
Does an exponential curve ever touch the x-axis?
No. The x-axis is a horizontal asymptote — the curve gets arbitrarily close but never reaches zero, because a positive base raised to any real power stays positive.
How do I plot compound interest?
Use y = P (1 + r)^x, where P is the principal and r the rate as a decimal. For ₹1,000 at 8% a year, type y = 1000 (1.08)^x and read the balance after x years off the graph.
How is the exponential related to the logarithm?
They are inverses. Plot y = e^x, y = ln(x) and y = x together: the first two curves are mirror images in the straight line, which is the visual signature of inverse functions.
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