Define exponential growth.

Growth increasing at an increasing rate; b > 1 in $f(x) = ab^x$ .

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Define exponential growth.

Growth increasing at an increasing rate; b > 1 in $f(x) = ab^x$ .

Define exponential decay.

Decay decreasing at a decreasing rate; 0 < b < 1 in $f(x) = ab^x$ .

What is the general form of an exponential function?

$f(x) = ab^x$ , where 'a' is the initial value and 'b' is the base.

Define the natural base 'e'.

A mathematical constant approximately equal to 2.71828, used for continuous growth/decay.

What is exponential regression?

A method to fit an exponential function to a set of data points.

Define the correlation coefficient $R^2$ in exponential regression.

A measure of how well the exponential model fits the data.

What are residuals in exponential regression?

The difference between the observed data and the predicted values from the exponential model.

What is 'ln'?

The natural logarithm, the inverse function of $e^x$ .

What is an equivalent form of an exponential function?

A different representation of the same function that reveals different properties.

What does the initial value 'a' represent in $f(x) = ab^x$ ?

The value of the function when x = 0.

How do you find an exponential function given two points?

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

How do you determine if a data set represents exponential growth or decay?

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.

How do you solve for 't' in the equation $P(t) = P_0 b^t$ ?

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: $t = ln(P(t)/P_0) / ln(b)$

How do you convert an exponential function from base 'b' to base 'e'?

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

How do you determine the annual growth rate from a function like $P(t) = 500(1.08)^t$ ?

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

How to find the equation of an exponential function given the initial value and growth rate?

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

How do you use exponential regression on a calculator?

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

How do you determine the doubling time of an exponential function?

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

How do you determine the half-life of an exponential function?

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

How do you analyze the effect of adding a constant k to the function $f(x) = ab^x$ ?

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

What are the differences between exponential growth and exponential decay?

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

What are the differences between linear and exponential functions?

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

What are the differences between using 'e' and another base 'b' in exponential models?

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

What are the differences between $f(x) = e^x$ and $f(x) = e^{-x}$ ?

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

Compare and contrast exponential and polynomial functions.

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

Compare and contrast exponential growth and logistic growth.

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

Compare and contrast exponential functions with positive and negative exponents.

Positive: Increasing function | Negative: Decreasing function.

Compare and contrast exponential functions with different initial values.

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

Compare and contrast exponential regression and linear regression.

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

Compare and contrast the effects of adding a constant inside vs. outside the exponential function.

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