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What is the general formula for an exponential function?

$f(x) = ab^x$

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What is the general formula for an exponential function?

$f(x) = ab^x$

Given two points (x1, y1) and (x2, y2) on an exponential function, what are the formulas for 'a' and 'b'?

$a = y1$ , $b = y2 / y1$ (when x1 = 0 and x2 = 1)

What is the formula for continuous growth using the natural base 'e'?

$f(x) = e^x$

What is the formula for continuous decay using the natural base 'e'?

$f(x) = e^{-x}$

How to find 'b' if given two points (x1, y1) and (x2, y2)?

$b = (y2/y1)^{1/(x2-x1)}$

Given f(d) = 2^d, what is the equivalent form showing weekly growth?

$f(d) = (2^7)^{(d/7)}$

How do you calculate the growth factor over a specific interval?

Growth factor = $b^{\Delta x}$ , where $\Delta x$ is the length of the interval.

What is the formula for exponential growth with an initial value of $P_0$ ?

$P(t) = P_0 b^t$

What is the formula for exponential decay with an initial value of $A_0$ ?

$A(t) = A_0 b^t$ , where 0 < b < 1

How do you find the value of 'e' using a limit?

$e = \lim_{n \to \infty} (1 + \frac{1}{n})^n$

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.

Explain how adding a constant to the dependent variable can reveal an exponential relationship.

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

Explain the significance of the base 'b' in an exponential function.

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

Why is the natural base 'e' used in exponential functions?

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

Explain how equivalent forms of exponential functions can provide different interpretations.

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

Describe how to construct an exponential function using two input-output pairs.

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

Explain how exponential regression works.

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

What does $R^2$ close to 1 imply?

The model is a good fit for the data.

How does the value of 'b' affect the graph of $f(x) = ab^x$ ?

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

Why is 'e' important in calculus?

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

What are the key characteristics of exponential functions?

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