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.

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': $a = \frac{y}{b^x}$

If a quantity increases at a rate proportional to its size, then it exhibits exponential growth.

If a quantity decreases at a rate proportional to its size, then it exhibits exponential decay.

For $b > 1$, $\lim_{x \to \infty} b^x = \infty$ and $\lim_{x \to -\infty} b^x = 0$. For $0 < b < 1$, $\lim_{x \to \infty} b^x = 0$ and $\lim_{x \to -\infty} b^x = \infty$.

Adding a constant 'k' to an exponential function, $f(x) = ab^x + k$, shifts the graph vertically by 'k' units, changing the horizontal asymptote to y = k.

The domain of $f(x) = ab^x$ is all real numbers, and the range is $(0, \infty)$ if a > 0, or $(-\infty, 0)$ if a < 0, assuming no vertical shifts.

Exponential functions of the form $f(x) = ab^x$ are always concave up if a > 0 and always concave down if a < 0.

The y-intercept of an exponential function $f(x) = ab^x$ is always 'a', which is the value of the function when x = 0.

For $f(x) = ab^x$, the horizontal asymptote is y = 0 if there are no vertical shifts. Vertical shifts will change the horizontal asymptote accordingly.

If b > 1, the function exhibits exponential growth. If 0 < b < 1, the function exhibits exponential decay. The magnitude of 'b' influences the rate of growth or decay.

Transformations such as vertical shifts, reflections, and stretches/compressions can be applied to exponential functions, altering their position, orientation, and shape.