Graph y = a·b^x, identify key features (intercept, asymptote, increasing/decreasing), and compare exponential to linear growth.
Every exponential function y = a·b^x (with a > 0, b > 0, b ≠ 1) shares the same skeleton: the y-intercept is a (since b^0 = 1), the graph never touches the x-axis - that line, y = 0, is called a horizontal asymptote, a boundary the curve approaches forever without reaching - and the whole graph stays positive. If b > 1 the graph climbs (growth); if 0 < b < 1 it falls toward the asymptote (decay).
Race an exponential against a linear model with a big head start, and the exponential always wins eventually - not usually, ALWAYS, as long as its base is bigger than 1. Linear growth adds the same amount each step; exponential growth multiplies, and multiplication compounds on top of itself in a way addition never does. No matter how far behind it starts, repeated multiplying overtakes repeated adding.
Mastery looks like: They can read intercept/asymptote/growth-or-decay directly from y = a·b^x and explain why exponential growth eventually overtakes any linear model.
Common stumbles: Confusing a (the intercept) with b (the growth factor); thinking a linear head start can outlast exponential growth if it is big enough.