Math
Exponential Function Graph (y = a·bˣ + c)
Enter coefficient a, base b, and shift c to graph y = a·bˣ + c. Automatically computes y-intercept, horizontal asymptote, and doubling increment.
Tips
- When b > 1 the function grows (increases to the right). When 0 < b < 1 it decays (decreases to the right).
- The horizontal asymptote is y = c. When c = 0, the graph approaches the x-axis as x → ±∞.
- A negative a flips the graph vertically — even with b > 1, the function will decrease as x increases.
- The doubling x-increment is the value of x for which b^x = 2, i.e. ln(2)/ln(b). For b = 2, y doubles every time x increases by 1.
FAQ
When b = 1, y = a·1^x + c = a + c, a constant. This is not an exponential function, so b = 1 is not allowed.
If b ≤ 0, b^x is not defined for all real x. For example, (−2)^0.5 is not a real number.
Set b = 2.71828 (or more precisely 2.718281828), a = 1, and c = 0. The graph will closely approximate y = eˣ.
Side Note — Compound interest and the number e
Exponential functions are the mathematics of compound interest. Investing principal P at annual rate r for n years gives P·(1+r)ⁿ — exactly an exponential with a = P, b = 1+r, and c = 0.
The natural base e ≈ 2.71828 emerges from continuously compounded interest: (1 + 1/n)ⁿ → e as n → ∞. The function y = eˣ is unique because it is its own derivative, making it appear throughout calculus, physics, and probability theory.