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

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.