AP Precalculus Flashcards: Exponential Function Context And Data Modeling

What this deck covers

This deck focuses on Exponential Function Context And Data Modeling, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: State the formula for continuous exponential growth.

Answer: $A = P e^{rt}$. Uses natural base $e$ for continuous compounding.

Flashcard 2: What is the doubling time formula in exponential growth?

Answer: $T_d = \frac{\text{ln}(2)}{\text{ln}(b)}$. Time required for quantity to double in size.

Flashcard 3: Define an exponential decay function.

Answer: $f(x) = a \times b^x$, $0 < b < 1$. Decay occurs when base is between 0 and 1.

Flashcard 4: Calculate decay rate if $b = 0.9$ in $f(x) = a \times b^x$.

Answer: 10%. Decay rate is $(1 - 0.9) = 0.1 = 10\%$.

Flashcard 5: Determine the effect of $f(x) = a \times (b^x) + c$ when $c > 0$.

Answer: Vertical shift up by $c$. Positive $c$ shifts the entire graph upward.

Flashcard 6: For $f(x) = 6 \times 0.5^x$, what is the decay factor?

Answer: 0.5. The base represents the decay factor.

Flashcard 7: Which parameter affects horizontal shifts in $f(x) = a \times b^{x+c}$?

Answer: $c$. Parameter $c$ controls horizontal translation.

Flashcard 8: What is the inverse of an exponential function?

Answer: Logarithmic function. Exponential and logarithmic functions are inverses.

Flashcard 9: Is $f(x) = 1 \times (0.5)^x$ growth or decay?

Answer: Decay. Base 0.5 is less than 1, indicating decay.

Flashcard 10: For $f(x) = 9 \times 1.2^x$, what is the growth rate?

Answer: 20%. Growth rate is $(b - 1) \times 100\%$.

Flashcard 11: What is the effect of a negative exponent in $f(x) = a \times b^{-x}$?

Answer: Reflect across y-axis. Negative exponent reverses the function horizontally.

Flashcard 12: Calculate the future value $A$ for $P = 1000, r = 0.08, t = 2$.

Answer: $A = 1000 e^{0.16}$. $A = 1000e^{0.08 \times 2} = 1000e^{0.16}$.

Flashcard 13: State the half-life formula for exponential decay.

Answer: $N(t) = N_0 \times (\frac{1}{2})^{\frac{t}{h}}$. Models quantity halving every $h$ time units.

Flashcard 14: Determine the growth factor for $f(x) = 3 \times 1.05^x$.

Answer: 1.05. Growth factor is the base when $b > 1$.

Flashcard 15: Calculate $f(3)$ for $f(x) = 4 \times (0.5)^x$.

Answer: 0.5. $f(3) = 4 \times (0.5)^3 = 4 \times 0.125 = 0.5$.

Flashcard 16: Identify the range of $f(x) = 8 \times (1.5)^x$.

Answer: $(0, \text{inf})$. All exponential functions have range $(0, \infty)$.

Flashcard 17: Identify the base in the exponential function $f(x) = 5 \times 2^x$.

Answer:

1. The base is the number being raised to the power.

Flashcard 18: State the domain of any exponential function $f(x) = a \times b^x$.

Answer: All real numbers. Exponential functions accept all real number inputs.

Flashcard 19: State the natural base $e$ approximately.

Answer: 2.718. Euler's number, base of natural logarithm.

Flashcard 20: What is the horizontal asymptote of $f(x) = 4 \times (0.75)^x$?

Answer: $y = 0$. Exponential functions approach but never reach zero.

Flashcard 21: What property does the base $b$ have in an exponential function $f(x) = a \times b^x$?

Answer: $b > 0$ and $b \neq 1$. Base must be positive and not equal to 1.

Flashcard 22: For $f(x) = 5 \times (0.8)^x$, what is the percentage decay rate?

Answer: 20%. Decay rate is $(1 - b) \times 100\%$.

Flashcard 23: What is the formula for finding the rate $r$ in $A = P e^{rt}$?

Answer: $r = \frac{1}{t} \text{ln}(\frac{A}{P})$. Solved from $A = Pe^{rt}$ for rate $r$.

Flashcard 24: What transformation occurs when $f(x) = a \times b^{x+c}$?

Answer: Horizontal shift left by $c$. Adding $c$ to exponent shifts graph leftward.

Flashcard 25: Determine the y-intercept of $f(x) = 2 \times 3^x$.

Answer:

1. Y-intercept occurs when $x = 0$.

Flashcard 26: What transformation is applied in $f(x) = a \times b^{x} + d$?

Answer: Vertical shift by $d$. Adding $d$ moves the graph up or down.

Flashcard 27: Find the value of $f(2)$ for $f(x) = 2 \times 5^x$.

Answer:

1. $f(2) = 2 \times 5^2 = 2 \times 25 = 50$.

Flashcard 28: Identify the effect of $f(x) = -a \times b^x$ on the graph.

Answer: Reflection over x-axis. Negative coefficient flips graph over x-axis.

Flashcard 29: What is the general form of an exponential function?

Answer: $f(x) = a \times b^x$. Standard form with initial value $a$ and base $b$.

Flashcard 30: What is the value of $f(0)$ for $f(x) = a \times b^x$?

Answer: $a$. When $x = 0$, $b^0 = 1$, so $f(0) = a$.

Flashcard 31: Find the initial value of the function $f(x) = 3 \times 4^x$.

Answer:

1. Initial value is coefficient $a$ when $x = 0$.

Flashcard 32: What does the variable $a$ represent in $f(x) = a \times b^x$?

Answer: Initial value. The coefficient $a$ is the starting amount.

Flashcard 33: What does the parameter $r$ represent in $A = P e^{rt}$?

Answer: Rate of growth or decay. The exponent controls growth or decay rate.

Flashcard 34: What is the exponential growth factor if $b = 1.1$?

Answer: 1.1. Growth factor is the base $b$ when $b > 1$.

Flashcard 35: What is the range for an exponential decay function?

Answer: $(0, a)$ if $a > 0$. Decay functions approach zero from above.

Flashcard 36: What is the effect of $a$ in $f(x) = a \times b^x$ on the graph?

Answer: Vertical stretch/compression. Parameter $a$ scales the function vertically.