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: 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 4: 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 5: 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 6: 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 7: 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 8: 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 9: 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 10: 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 11: Determine the asymptote of $f(x) = 100 \times (0.99)^x$.

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

Flashcard 12: 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 13: 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 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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 26: 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 27: 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 28: 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 29: 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 30: 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 31: 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 32: 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 33: 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 34: 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 35: 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 36: 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 37: 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 38: 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 39: 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 40: 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 41: 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 42: 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 43: 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 44: 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 45: 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 46: 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 47: 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 48: 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 49: 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 50: 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 51: 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 52: 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 53: 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 54: Determine the asymptote of $f(x) = 100 \times (0.99)^x$.

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

Flashcard 55: 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 56: 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 57: 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 58: 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 59: 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 60: 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 61: 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 62: 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 63: 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 64: 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 65: 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 66: 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 67: 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 68: 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.

Flashcard 69: 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 70: 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 71: 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.