Building a Function from Two Points

An exponential function has two unknowns, a starting value a and a growth factor b. Each point you're given turns into an equation: y1 = a times b raised to x1, and y2 = a times b raised to x2. Two equations, two unknowns, so two points pin the function down completely, the same logic as needing two points to fix a line's slope and intercept. The trick is dividing one equation by the other. That cancels a and leaves b raised to the power (x2 minus x1) equal to y2 divided by y1. Solve for b by taking a root, then plug b back into either original equation to get a.

Work it through for (2, 12) and (5, 96). Dividing gives b³ = 96/12 = 8, so b = 2. Then 12 = a times 2² = 4a, so a = 3. The function is f(x) = 3 times 2^x, and you can check both points satisfy it. Compare this to the line through the same two points: slope = (96 minus 12)/(5 minus 2) = 28, giving y = 12 + 28(x minus 2). Near x = 2 the two curves sit close together, but the exponential pulls away fast as x grows, since it multiplies by 2 every unit while the line only adds 28.

When the two x-values aren't one unit apart, the same ratio method still works, you just divide by the exponent difference when solving for b, as in the worked example above with an exponent of 3. This is the algebraic engine behind every exponential model you'll build from data: once you can extract a and b from any two points, you're ready to fit models to tables, compare average rates of change, and recognize when a data set is better described by a line than a curve.