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.

Growth: b > 1, function increases | Decay: 0 < b < 1, function decreases.

Vertical Shifts: Affect the y-values, change the horizontal asymptote | Horizontal Shifts: Affect the x-values, do not change the horizontal asymptote.

Exponential: Variable in the exponent, constant percentage change | Linear: Variable in the base, constant additive change.

Growth Rate: Positive value, increases the function value | Decay Rate: Negative value (expressed in base), decreases the function value.

Both exponential growth and decay functions are concave up. However, growth increases rapidly, while decay decreases rapidly but remains above the x-axis.

Growth: $\lim_{x \to \infty} f(x) = \infty$ | Decay: $\lim_{x \to \infty} f(x) = 0$

Population Growth: Uses a positive growth rate, increases over time | Radioactive Decay: Uses a negative decay rate, decreases over time.

Domain: All real numbers | Range: $(0, \infty)$ for basic exponential functions without vertical shifts.

Simple Interest: Interest calculated only on the principal amount | Compound Interest: Interest calculated on the principal and accumulated interest.

$y = 2^x$: Exponential growth, increasing | $y = (1/2)^x$: Exponential decay, decreasing.

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