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.

If b > 1, the function represents exponential growth, increasing rapidly. If 0 < b < 1, the function represents exponential decay, decreasing rapidly.

A vertical shift moves the entire graph up or down. Adding a constant 'k' shifts the graph up if k > 0 and down if k < 0.

Limits describe the behavior of the function as x approaches infinity or negative infinity. Growth functions tend to infinity, while decay functions tend to zero as x approaches infinity.

Exponential functions are always either concave up (growth) or concave down (decay), so their concavity never changes, meaning no inflection points.

Exponential functions are always increasing or always decreasing; therefore, they do not have maximum or minimum values on an open interval.

Exponential functions are used to model situations with rapid growth or decay, such as population growth, compound interest, and radioactive decay.

The larger the base (b > 1), the faster the growth. The smaller the base (0 < b < 1), the faster the decay.

The y-intercept represents the initial value of the function when x = 0. It is the point where the function starts its growth or decay.

Since the domain is all real numbers, the function is defined for all values of x, allowing it to model continuous growth or decay over time.

Exponential decay functions are concave up. This means that the rate of decay decreases as x increases, but the function never changes direction.

Identify the base 'b' in the function $f(x) = ab^x$. If b > 1, it's growth. If 0 < b < 1, it's decay.

Set x = 0 in the function $f(x) = ab^x$. The y-intercept is f(0) = a.

Add a constant 'k' to the function: $g(x) = f(x) + k$. If k > 0, shift up. If k < 0, shift down.

If b > 1, the limit is infinity. If 0 < b < 1, the limit is 0.

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

Set up the equation $f(t) = ab^t = target\\_value$. Use logarithms to solve for t: $t = \frac{\log(\frac{target\\_value}{a})}{\log(b)}$

1. Substitute the points into $f(x) = ab^x$ to get two equations. 2. Solve for 'a' in one equation. 3. Substitute 'a' into the other equation and solve for 'b'. 4. Substitute 'a' and 'b' back into the general form.

Compare the horizontal asymptote of the transformed function with the horizontal asymptote of the original function ($y=0$). The difference is the vertical shift.

1. Set up the equation $y_2 = y_1(1+r)^t$, where $y_1$ and $y_2$ are the data points, and t is the time difference. 2. Solve for r: $r = (\frac{y_2}{y_1})^{1/t} - 1$

1. Substitute the point (x, y) and the base 'b' into the general form $y = ab^x$. 2. Solve for 'a': $a = \frac{y}{b^x}$