AP Precalculus Flashcards: Exponential Functions

What this deck covers

This deck focuses on Exponential Functions, 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: What is the initial value in $f(x) = 15 \times 1.3^x$?

Answer: $15$. The coefficient 15 is the value when $x = 0$.

Flashcard 2: What is the y-intercept of $f(x) = 1.5 \times 10^x$?

Answer: $1.5$. Y-intercept occurs when $x = 0$: $f(0) = 1.5 \times 10^0 = 1.5$.

Flashcard 3: Find the reflection of $f(x) = 2^x$ across the x-axis.

Answer: $f(x) = -2^x$. Multiplying by -1 reflects the function across the x-axis.

Flashcard 4: Identify the transformation: $f(x) = 2 \times 3^{x-1}$.

Answer: Right shift by 1 unit. The $(x-1)$ in the exponent shifts the graph right by 1.

Flashcard 5: What is the general form of an exponential function?

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

Flashcard 6: Determine the asymptote of $f(x) = 100 \times (0.99)^x$.

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

Flashcard 7: What is the effect of a negative exponent in $f(x) = 3 \times b^{-x}$?

Answer: Reflection across y-axis. Negative exponent transforms $b^{-x}$ to $(1/b)^x$, reflecting over y-axis.

Flashcard 8: What is the decay rate in $f(x) = 12 \times (0.8)^x$?

Answer: $0.2$ or 20\text{%}. Since $0.8 = 1 - 0.2$, the decay rate is $0.2$.

Flashcard 9: If $f(x) = 5 \times (1.2)^x$, is this function growth or decay?

Answer: Growth. Since $1.2 > 1$, the function increases as $x$ increases.

Flashcard 10: What happens to $f(x) = a \times b^x$ as $x \to -\text{∞}$ when $0 < b < 1$?

Answer: $f(x) \to \text{∞}$. When $0 < b < 1$ and $x \to -\infty$, $b^x \to \infty$.

Flashcard 11: What is the value of $f(0)$ for $f(x) = 7 \times 3^x$?

Answer: $7$. Substitute $x = 0$: $f(0) = 7 \times 3^0 = 7 \times 1 = 7$.

Flashcard 12: Determine if $f(x) = 8 \times (1.01)^x$ models growth or decay.

Answer: Growth. Since $1.01 > 1$, the function shows exponential growth.

Flashcard 13: What is $f(x) = 0$ for $f(x) = a \times b^x$?

Answer: Never. Exponential functions have horizontal asymptote at $y = 0$, never equal 0.

Flashcard 14: Identify the horizontal shift in $f(x) = 5 \times 2^{x+3}$.

Answer: Left shift by 3 units. The $(x+3)$ in the exponent shifts the graph left by 3.

Flashcard 15: Identify the growth rate in $f(x) = 10 \times (1.05)^x$.

Answer: $0.05$ or 5\text{%}. From $(1 + r) = 1.05$, so $r = 0.05$.

Flashcard 16: If $b = 1$, what type of function is $f(x) = a \times b^x$?

Answer: Constant function. When $b = 1$, $f(x) = a \times 1^x = a$ for all $x$.

Flashcard 17: State the exponential growth formula.

Answer: $f(x) = a \times (1 + r)^t$. Where $a$ is initial value, $r$ is growth rate, $t$ is time.

Flashcard 18: Identify the base in $f(x) = 6 \times e^{0.5x}$.

Answer: $e$. The base is the constant $e$ being raised to the power.

Flashcard 19: State the exponential decay formula.

Answer: $f(x) = a \times (1 - r)^t$. Where $a$ is initial value, $r$ is decay rate, $t$ is time.

Flashcard 20: For $f(x) = 7 \times 1.2^x$, identify the initial value.

Answer: $7$. The coefficient 7 is the initial value when $x = 0$.

Flashcard 21: What does $0 < b < 1$ indicate in an exponential function $f(x) = a \times b^x$?

Answer: Exponential decay. Base between 0 and 1 means output decreases as $x$ increases.

Flashcard 22: Find $f(2)$ for $f(x) = 4 \times (0.5)^x$.

Answer: $1$. Substitute $x = 2$: $f(2) = 4 \times (0.5)^2 = 4 \times 0.25 = 1$.

Flashcard 23: Express $f(x) = 3 \times (0.9)^x$ in terms of decay rate.

Answer: Decay rate is $0.1$ or 10\text{%}. Since $0.9 = 1 - 0.1$, the decay rate is $0.1$.

Flashcard 24: Identify the base in the exponential function $f(x) = 3 \times 2^x$.

Answer: $2$. The base is the constant being raised to the power $x$.

Flashcard 25: Find the base for the function $f(x) = 3 \times 6^x$.

Answer: $6$. The base is the constant 6 being raised to the power $x$.

Flashcard 26: What does $b > 1$ indicate in an exponential function $f(x) = a \times b^x$?

Answer: Exponential growth. Base greater than 1 means output increases as $x$ increases.

Flashcard 27: What is the y-intercept of the function $f(x) = 4 \times 3^x$?

Answer: $4$. Y-intercept occurs when $x = 0$: $f(0) = 4 \times 3^0 = 4$.

Flashcard 28: Find $f(1)$ for $f(x) = 7 \times (1.1)^x$.

Answer: $7.7$. Substitute $x = 1$: $f(1) = 7 \times 1.1^1 = 7 \times 1.1 = 7.7$.

Flashcard 29: Determine the effect of $f(x) = 4 \times 2^{-x}$ on the graph.

Answer: Reflection over y-axis. The negative exponent $2^{-x}$ reflects the graph over y-axis.

Flashcard 30: Identify the vertical stretch in $f(x) = 10 \times 2^x$.

Answer: Factor of 10. The coefficient 10 stretches the graph vertically by factor 10.

Flashcard 31: What does the parameter $a$ represent in $f(x) = a \times b^x$?

Answer: Initial value or y-intercept. When $x = 0$, $f(0) = a \times b^0 = a$.

Flashcard 32: What is the horizontal asymptote of $f(x) = 5 \times 0.8^x$?

Answer: $y = 0$. Exponential decay functions approach 0 as $x$ approaches infinity.

Flashcard 33: Calculate $f(-1)$ for $f(x) = 3 \times 4^x$.

Answer: $\frac{3}{4}$. Substitute $x = -1$: $f(-1) = 3 \times 4^{-1} = 3 \times \frac{1}{4} = \frac{3}{4}$.

Flashcard 34: Determine if $f(x) = 2 \times (0.7)^x$ shows growth or decay.

Answer: Decay. Since $0.7 < 1$, the function decreases as $x$ increases.

Flashcard 35: Find $f(3)$ for $f(x) = 2 \times 5^x$.

Answer: $250$. Substitute $x = 3$: $f(3) = 2 \times 5^3 = 2 \times 125 = 250$.

Flashcard 36: Find the decay factor for $f(x) = 9 \times (0.75)^x$.

Answer: $0.75$. The decay factor is the base 0.75 in the exponential function.