Which function type has a rate of change proportional to its value?

The function type that has a rate of change proportional to its value is the exponential function. This characteristic means that as the value of the function increases, the rate at which it grows also increases relative to its current value.

For example, if you consider the function ( y = a \cdot e^{kt} ), where ( a ) is a constant, ( e ) is the base of natural logarithms, and ( k ) is a constant rate, the derivative of this function (which represents the rate of change) is also proportional to the value itself. As ( y ) increases, the rate of growth ( dy/dt ) increases too, indicating a compounding effect.

In contrast, linear functions have a constant rate of change regardless of their value, while quadratic and cubic functions have higher-order rates of change that depend on the square or cube of the variable, thus not adhering to the proportionality criterion as seen in exponential functions.