The Shape of an Exponential Function

In f(x) = a·b^x, the base b does all the shape work. Each time x increases by 1, the output gets multiplied by b, not added to. That multiplicative step is why the graph never flattens into a straight line and never turns around to come back down: the output keeps getting scaled by the same factor forever, so it either keeps growing faster and faster or keeps shrinking toward zero, but it never reverses direction. A line changes by a fixed amount each step; an exponential changes by a fixed factor, and that factor compounds.

If b > 1, multiplying by b repeatedly makes the output bigger each time, so the graph rises and curves upward more steeply as x increases — growth. If 0 < b < 1, multiplying by b repeatedly makes the output smaller each time (since you are scaling down), so the graph falls and flattens toward the x-axis — decay. The constant a just stretches the graph vertically and sets the y-intercept at (0, a); it does not change whether the curve grows or decays.

This shape is exactly what makes logarithms necessary: because an exponential is always increasing or always decreasing, it is one-to-one, so it has an inverse function, and that inverse is the logarithm. Seeing why the curve never touches the x-axis here also sets up the horizontal asymptote y = 0, a feature you will track carefully when comparing exponential models to linear, quadratic, or logarithmic ones later in the unit.