Exponential Functions and Graphs

Core Idea

An exponential function has the form f(x) = a*b^x where b > 0, b != 1. If b > 1, the function models exponential growth; if 0 < b < 1, exponential decay. Key features: the graph passes through (0, a), has a horizontal asymptote at y = 0, is always positive (for a > 0), and increases/decreases without bound. Exponential functions grow faster than any polynomial for large x. Transformations shift, stretch, and reflect the basic curve.

How It's Best Learned

Graph y = 2^x and y = (1/2)^x as parent functions. Apply transformations. Compare growth rates by graphing y = x² and y = 2^x on the same axes. Discuss real-world contexts: population growth, radioactive decay, compound interest. Introduce the natural base e as a special exponential base.

Common Misconceptions

• Confusing exponential growth (b^x) with polynomial growth (x^b).
• Thinking the graph touches or crosses the x-axis (it approaches but never reaches the horizontal asymptote).
• Not recognizing that (1/2)^x = 2^-x (decay is growth reflected).
• Thinking b must be an integer.

Explainer

You already know how to work with expressions like 2³ or 5⁴ from your study of exponent rules. An exponential *function* takes that idea and lets the exponent be a variable: f(x) = 2^x means the base is fixed at 2 and x — the input — is the exponent. This flip in roles is what makes exponential functions behave so differently from polynomials like x² or x³, where x is the base.

The graph of f(x) = b^x has several features that follow directly from the algebra. When x = 0, any nonzero base raised to the 0 power equals 1, so the graph always passes through (0, 1). For large positive x with b > 1, the function grows rapidly — each unit step multiplies the output by b. For large negative x, the exponents are large negative numbers, so the output becomes a very small positive fraction, approaching zero but never reaching it. This is why y = 0 is a horizontal asymptote: the function gets arbitrarily close but can never equal zero, because b raised to any real power is always positive.

Decay is not a different kind of function — it is the same structure with a base between 0 and 1. The function (1/2)^x gets smaller as x increases, because repeatedly multiplying by 1/2 halves the value each time. Using your exponent rules: (1/2)^x = 2^-x, which means the decay curve is exactly the growth curve reflected across the y-axis. This connection between growth and decay is worth internalizing — it means you only need to understand one shape.

The comparison with polynomial growth is one of the most important conceptual ideas here. For small values of x, x² or x¹⁰⁰ can dwarf 2^x. But eventually the exponential wins, because in 2^x the exponent keeps growing and each step multiplies by a constant factor. The polynomial only adds; the exponential compounds. This is why population growth, compound interest, and viral spread are all modeled exponentially — the pattern of "multiply by a constant ratio each period" is precisely what these phenomena share.

Transformations of exponential functions follow the same rules you have seen with other functions: f(x) = a · b^{x - h} + k shifts the graph horizontally by h, vertically by k, and stretches it by a. The asymptote shifts from y = 0 to y = k. When you encounter logarithms next, you will find that they are defined specifically to answer the question "for what x does b^x equal this value?" — the exponential and logarithmic functions are inverses of each other, which is why understanding the graph of 2^x now will make logarithms much more intuitive.