Exponential Function Calculator (find y = a·bˣ through two points)

Q: How do I tell growth from decay?

b > 1 is growth, 0 < b < 1 is decay.

Understanding the exponential function calculator — concept and principle

This calculator finds the exponential function y = a·bˣ that passes through two given points, then evaluates it at any x. It reports the initial value a, the growth factor b, the percent change per unit of x, the continuous rate k, the same curve with base e, and the doubling time or half-life. Use it for algebra problems about growth and decay.

Over a gap of 3 the value grew 27-fold, so each step multiplies by 3.

How it is calculated

From y₁ = a·b^(x₁) and y₂ = a·b^(x₂):

b = (y₂ ÷ y₁)^(1/(x₂ − x₁)); a = y₁ ÷ b^(x₁)

Change per unit of x = (b − 1) × 100 %

k = ln b; y = a·e^(kx); doubling time (or half-life) = ln 2 ÷ |k|

Both y values must have the same sign, because a·bˣ never changes sign.

The principle

In an exponential function, each step of 1 in x multiplies y by the same factor b. Dividing the two point equations cancels a and leaves only b raised to the gap between the x values; taking the matching root gives b. Once b is known, a follows from either point. a is the value at x = 0, the starting amount, and b > 1 means growth while 0 < b < 1 means decay.

Worked example

Default inputs: points (2, 18) and (5, 486); evaluate at x = 6.

y₂ ÷ y₁ = 486 ÷ 18 = 27, over a gap of 3, so b = 27^(1/3) = 3.
a = 18 ÷ 3² = 2; the function is y = 2 · 3ˣ, growth (b > 1).
f(6) = 2 × 3⁶ = 2 × 729 = 1,458.
Change per unit = (3 − 1) × 100 = 200 %; k = ln 3 = 1.09861, so y = 2 · e^(1.09861x).
Doubling time = ln 2 ÷ ln 3 = 0.6309 units of x.

How to read the result

Here y triples with every unit of x, so it doubles in about 0.63 units. Check the function on both points: 2 × 3² = 18 and 2 × 3⁵ = 486. With real data that only roughly follow an exponential pattern, use exponential regression with more points instead of fitting through two.

Common mistakes

Subtracting y values instead of dividing them.
Taking the root with the wrong index (it must be x₂ − x₁).
Entering y values of opposite sign.
Reading 200 % change as "doubles"; it means triples.

Keep in mind: Two points always define an exponential curve, but that does not prove the data follow one. Check with more observations before predicting.

Frequently asked questions

How do I find an exponential function from two points?

Divide the y values to get b raised to the x gap, take the root, then solve for a.

What does b mean?

The factor by which y is multiplied when x increases by 1.

How do I tell growth from decay?