A function of the form $f(x) = ab^x$, where a is the initial value, b is the base (b > 0, b ≠ 1), and x is the exponent.

Exponential growth occurs when the base $b > 1$. The function increases rapidly as x increases.

Exponential decay occurs when $0 < b < 1$. The function decreases rapidly as x increases.

The initial value is 'a', which represents the y-intercept of the function.

The domain of an exponential function is all real numbers, $(-\infty, \infty)$.

A transformation of the form $g(x) = f(x) + k$, which shifts the graph of $f(x)$ vertically by k units.

Concavity describes the curvature of the graph. Exponential functions are either always concave up or always concave down.

Extrema are maximum or minimum values of a function. Exponential functions do not have extrema on an open interval.

The base 'b' is a positive number not equal to 1 that determines whether the function represents growth or decay.

'a' represents the initial amount or starting value of the exponential function at x = 0.

It indicates exponential growth. The function's values increase rapidly as x increases.

It indicates exponential decay. The function's values decrease rapidly as x increases, approaching zero.

A vertical shift of 'k' units changes the horizontal asymptote from y = 0 to y = k.

The steepness indicates the rate of growth or decay. A steeper graph implies a faster rate.

It represents the initial value of the function at x = 0.

The graph increases rapidly as x increases, and it is always concave up.

The graph decreases rapidly as x increases, approaching the x-axis, and it is always concave up.

A horizontal line (asymptote) indicates the limit of the function as x approaches infinity or negative infinity.

A larger 'b' (b > 1) results in a steeper growth curve. A smaller 'b' (0 < b < 1) results in a faster decay curve.

Compare the horizontal asymptote of the given graph with the standard exponential function's asymptote (y=0). The difference is the vertical shift.

$f(x) = ab^x$

$f(x) = ab^x$, where $b > 1$

$f(x) = ab^x$, where $0 < b < 1$

$g(x) = f(x) + k$

$P(t) = P_0(1 + r)^t$, where $P_0$ is the initial population, r is the growth rate, and t is the time.

$A = P(1 + \frac{r}{n})^{nt}$, where A is the final amount, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years.

$N(t) = N_0e^{-\lambda t}$, where $N(t)$ is the amount remaining after time t, $N_0$ is the initial amount, and $\lambda$ is the decay constant.

$\lim_{x \to \infty} ab^x = \infty$ (when b > 1)

$\lim_{x \to \infty} ab^x = 0$ (when 0 < b < 1)

$\lim_{x \to -\infty} ab^x = 0$ (when b > 1)