What does the graph of an exponential growth function look like?

Starts near zero and increases rapidly, approaching infinity as x increases.

Flip to see [answer/question]

Revise later

SpaceTo flip

If confident

All Flashcards

What does the graph of an exponential growth function look like?

Starts near zero and increases rapidly, approaching infinity as x increases.

What does the graph of an exponential decay function look like?

Starts at a positive value and decreases rapidly, approaching zero as x increases.

How can you identify the initial value from the graph of an exponential function?

It's the y-intercept of the graph (the value of y when x = 0).

How does the value of 'b' affect the steepness of the exponential graph?

Larger 'b' values (b > 1) result in steeper growth; smaller 'b' values (0 < b < 1) result in slower decay.

What does the horizontal asymptote of an exponential decay function represent?

The limiting value that the function approaches as x goes to infinity.

How does adding a constant 'k' to an exponential function affect its graph?

It shifts the graph vertically by 'k' units.

What does the intersection of two exponential functions represent?

The point where the two functions have the same value for the same input.

How can you visually estimate the growth rate from an exponential graph?

By observing how quickly the y-values increase for a given change in x.

What does the graph of $e^x$ look like?

An increasing exponential function that passes through (0, 1) and has a horizontal asymptote at y = 0 for x approaching negative infinity.

What does the graph of $e^{-x}$ look like?

A decreasing exponential function that passes through (0, 1) and has a horizontal asymptote at y = 0 for x approaching positive infinity.

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.