AP®︎ Precalculus: Topic 2.3 Practice Test

Essential Knowledge: 2.3.A.1

The general form of an exponential function is $f(x) = ab^x$, with the initial value $a$, where $a \ne 0$, and the base $b$, where $b > 0$, and $b \ne 1$. When $a > 0$ and $b > 1$, the exponential function is said to demonstrate exponential growth. When $a > 0$ and $0 < b < 1$, the exponential function is said to demonstrate exponential decay.

Essential Knowledge: 2.3.A.2

When the natural numbers are input values in an exponential function, the input value specifies the number of factors of the base to be applied to the function’s initial value. The domain of an exponential function is all real numbers.

Essential Knowledge: 2.3.A.3

Because the output values of exponential functions in general form are proportional over equal-length input-value intervals, exponential functions are always increasing or always decreasing, and their graphs are always concave up or always concave down. Consequently, exponential functions do not have extrema except on a closed interval, and their graphs do not have points of inflection.

Essential Knowledge: 2.3.A.4

If the output values of the function $f$ are not proportional over equal-length input-value intervals, but the output values of an additive transformation of $f$ are proportional over equal-length input-value intervals, then $f$ can be modeled by an additive transformation of an exponential function.

Essential Knowledge: 2.3.A.5

For an exponential function in general form, as the input values increase or decrease without bound, the output values will increase or decrease without bound or will get arbitrarily close to zero. That is, for an exponential function in general form, $\lim_{x \to \pm\infty} ab^x = \infty$, $\lim_{x \to \pm\infty} ab^x = -\infty$, or $\lim_{x \to \pm\infty} ab^x = 0$.