2.3 Exponential Functions

What makes a function exponential

An exponential function has the form f(x) = a·b^x, where a ≠ 0, b > 0, and b ≠ 1. The input variable appears in the exponent, not the base. As x increases by a constant amount, output values are multiplied by a constant factor b — this is constant percent change, unlike polynomials which show constant differences.

Growth vs. decay

If b > 1, the function models exponential growth; if 0 < b < 1, it models exponential decay. The value a is the initial value (y-intercept, at x = 0). The graph always has a horizontal asymptote at y = 0 (when no vertical shift), and the function is always positive, increasing (growth) or decreasing (decay), and either concave up throughout.

Identifying from data or equations

Exponential functions can be recognized from tables by checking that consecutive ratios of outputs are constant (for equally spaced x-values), rather than constant differences. This constant ratio equals the base b. This distinguishes exponential change from linear (constant difference) and other polynomial behaviors.

Key terms

Exponential function form

f(x) = a·b^x, where a ≠ 0, b > 0, and b ≠ 1.

Exponential growth

Occurs when b > 1; output values increase as x increases.

Exponential decay

Occurs when 0 < b < 1; output values decrease as x increases.

Horizontal asymptote (basic exponential)

y = 0, since a·b^x approaches but never reaches zero.

Recognizing exponential data

Constant ratios between consecutive y-values for equally spaced x-values indicate an exponential function.

Try an AP-style question

Which describes f(x) = 2(0.5)ˣ?

1. Increasing, with horizontal asymptote y = 0
2. Decreasing, with horizontal asymptote y = 0
3. Increasing, with no asymptote
4. Decreasing, with horizontal asymptote y = 2