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.

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.

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