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.

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