1. Substitute the points into $y = ab^x$. 2. Solve the resulting system of equations for 'a' and 'b'.

1. Check if the y-values have a constant ratio for equally spaced x-values. 2. If the ratio is > 1, it's growth; if < 1, it's decay.

1. Divide by $P_0$: $P(t)/P_0 = b^t$. 2. Take the logarithm of both sides: $ln(P(t)/P_0) = t * ln(b)$. 3. Solve for t: $t = ln(P(t)/P_0) / ln(b)$

1. Write $b = e^{ln(b)}$. 2. Substitute: $ab^x = a(e^{ln(b)})^x = ae^{ln(b)x}$

The annual growth rate is (1.08 - 1) * 100% = 8%.

1. Identify the initial value (a). 2. Calculate b = 1 + growth rate (for growth) or b = 1 - decay rate (for decay). 3. Write the equation: $f(x) = ab^x$

1. Enter data into lists. 2. Select exponential regression function. 3. Store the regression equation.

1. Set $2 = b^t$. 2. Solve for t using logarithms: $t = \frac{ln(2)}{ln(b)}$

1. Set $0.5 = b^t$. 2. Solve for t using logarithms: $t = \frac{ln(0.5)}{ln(b)}$

The new function is $f(x) = ab^x + k$. This shifts the horizontal asymptote from y = 0 to y = k.

Shifts the y-intercept, allowing proportional growth to be observed from a non-zero baseline.

Represents the constant proportion by which the output value is multiplied at each step.

Models continuous growth/decay and is the base of the natural logarithm.

Changing the base or exponent can reveal the growth rate over different time intervals.

Set up a system of equations and solve for 'a' (initial value) and 'b' (base).

Fits an exponential function to a data set by minimizing the difference between observed and predicted values.

The model is a good fit for the data.

If b > 1, the graph increases exponentially. If 0 < b < 1, the graph decreases exponentially.

It simplifies many calculations, especially those involving derivatives and integrals of exponential functions.

Constant ratio between successive y-values for equally spaced x-values, horizontal asymptote, no x-intercept (unless a constant is added).

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

Linear: Constant rate of change | Exponential: Constant proportional rate of change.

e: Models continuous growth/decay, related to natural logarithm | b: Models growth/decay over discrete intervals.

$e^x$: Exponential growth | $e^{-x}$: Exponential decay.

Exponential: Variable in exponent, rapid growth/decay | Polynomial: Variable in base, growth depends on degree.

Exponential: Unrestricted growth | Logistic: Growth limited by carrying capacity.

Positive: Increasing function | Negative: Decreasing function.

Different y-intercepts, but same growth/decay rate if 'b' is the same.

Exponential: Fits exponential model | Linear: Fits linear model. Use $R^2$ to determine best fit.

Inside: $f(x) = a b^{(x+c)}$: Horizontal shift | Outside: $f(x) = ab^x + c$: Vertical shift.