Exponential Functions - AP Precalculus
Card 1 of 30
What does the parameter $a$ represent in $f(x) = a \times b^x$?
What does the parameter $a$ represent in $f(x) = a \times b^x$?
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Initial value or y-intercept. When $x = 0$, $f(0) = a \times b^0 = a$.
Initial value or y-intercept. When $x = 0$, $f(0) = a \times b^0 = a$.
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If $f(x) = 5 \times (1.2)^x$, is this function growth or decay?
If $f(x) = 5 \times (1.2)^x$, is this function growth or decay?
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Growth. Since $1.2 > 1$, the function increases as $x$ increases.
Growth. Since $1.2 > 1$, the function increases as $x$ increases.
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Find $f(3)$ for $f(x) = 2 \times 5^x$.
Find $f(3)$ for $f(x) = 2 \times 5^x$.
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$250$. Substitute $x = 3$: $f(3) = 2 \times 5^3 = 2 \times 125 = 250$.
$250$. Substitute $x = 3$: $f(3) = 2 \times 5^3 = 2 \times 125 = 250$.
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For $f(x) = 7 \times 1.2^x$, identify the initial value.
For $f(x) = 7 \times 1.2^x$, identify the initial value.
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$7$. The coefficient 7 is the initial value when $x = 0$.
$7$. The coefficient 7 is the initial value when $x = 0$.
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Determine the asymptote of $f(x) = 100 \times (0.99)^x$.
Determine the asymptote of $f(x) = 100 \times (0.99)^x$.
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$y = 0$. Exponential functions approach 0 but never reach it.
$y = 0$. Exponential functions approach 0 but never reach it.
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What is the y-intercept of $f(x) = 1.5 \times 10^x$?
What is the y-intercept of $f(x) = 1.5 \times 10^x$?
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$1.5$. Y-intercept occurs when $x = 0$: $f(0) = 1.5 \times 10^0 = 1.5$.
$1.5$. Y-intercept occurs when $x = 0$: $f(0) = 1.5 \times 10^0 = 1.5$.
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Determine if $f(x) = 2 \times (0.7)^x$ shows growth or decay.
Determine if $f(x) = 2 \times (0.7)^x$ shows growth or decay.
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Decay. Since $0.7 < 1$, the function decreases as $x$ increases.
Decay. Since $0.7 < 1$, the function decreases as $x$ increases.
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What does $b > 1$ indicate in an exponential function $f(x) = a \times b^x$?
What does $b > 1$ indicate in an exponential function $f(x) = a \times b^x$?
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Exponential growth. Base greater than 1 means output increases as $x$ increases.
Exponential growth. Base greater than 1 means output increases as $x$ increases.
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Determine if $f(x) = 8 \times (1.01)^x$ models growth or decay.
Determine if $f(x) = 8 \times (1.01)^x$ models growth or decay.
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Growth. Since $1.01 > 1$, the function shows exponential growth.
Growth. Since $1.01 > 1$, the function shows exponential growth.
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Identify the vertical stretch in $f(x) = 10 \times 2^x$.
Identify the vertical stretch in $f(x) = 10 \times 2^x$.
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Factor of 10. The coefficient 10 stretches the graph vertically by factor 10.
Factor of 10. The coefficient 10 stretches the graph vertically by factor 10.
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Find $f(2)$ for $f(x) = 4 \times (0.5)^x$.
Find $f(2)$ for $f(x) = 4 \times (0.5)^x$.
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$1$. Substitute $x = 2$: $f(2) = 4 \times (0.5)^2 = 4 \times 0.25 = 1$.
$1$. Substitute $x = 2$: $f(2) = 4 \times (0.5)^2 = 4 \times 0.25 = 1$.
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Find the reflection of $f(x) = 2^x$ across the x-axis.
Find the reflection of $f(x) = 2^x$ across the x-axis.
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$f(x) = -2^x$. Multiplying by -1 reflects the function across the x-axis.
$f(x) = -2^x$. Multiplying by -1 reflects the function across the x-axis.
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Identify the base in the exponential function $f(x) = 3 \times 2^x$.
Identify the base in the exponential function $f(x) = 3 \times 2^x$.
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$2$. The base is the constant being raised to the power $x$.
$2$. The base is the constant being raised to the power $x$.
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What is the initial value in $f(x) = 15 \times 1.3^x$?
What is the initial value in $f(x) = 15 \times 1.3^x$?
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$15$. The coefficient 15 is the value when $x = 0$.
$15$. The coefficient 15 is the value when $x = 0$.
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If $b = 1$, what type of function is $f(x) = a \times b^x$?
If $b = 1$, what type of function is $f(x) = a \times b^x$?
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Constant function. When $b = 1$, $f(x) = a \times 1^x = a$ for all $x$.
Constant function. When $b = 1$, $f(x) = a \times 1^x = a$ for all $x$.
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Identify the base in $f(x) = 6 \times e^{0.5x}$.
Identify the base in $f(x) = 6 \times e^{0.5x}$.
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$e$. The base is the constant $e$ being raised to the power.
$e$. The base is the constant $e$ being raised to the power.
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Calculate $f(-1)$ for $f(x) = 3 \times 4^x$.
Calculate $f(-1)$ for $f(x) = 3 \times 4^x$.
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$\frac{3}{4}$. Substitute $x = -1$: $f(-1) = 3 \times 4^{-1} = 3 \times \frac{1}{4} = \frac{3}{4}$.
$\frac{3}{4}$. Substitute $x = -1$: $f(-1) = 3 \times 4^{-1} = 3 \times \frac{1}{4} = \frac{3}{4}$.
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What is the decay rate in $f(x) = 12 \times (0.8)^x$?
What is the decay rate in $f(x) = 12 \times (0.8)^x$?
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$0.2$ or $20\text{%}$. Since $0.8 = 1 - 0.2$, the decay rate is $0.2$.
$0.2$ or $20\text{%}$. Since $0.8 = 1 - 0.2$, the decay rate is $0.2$.
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Determine the effect of $f(x) = 4 \times 2^{-x}$ on the graph.
Determine the effect of $f(x) = 4 \times 2^{-x}$ on the graph.
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Reflection over y-axis. The negative exponent $2^{-x}$ reflects the graph over y-axis.
Reflection over y-axis. The negative exponent $2^{-x}$ reflects the graph over y-axis.
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Identify the horizontal shift in $f(x) = 5 \times 2^{x+3}$.
Identify the horizontal shift in $f(x) = 5 \times 2^{x+3}$.
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Left shift by 3 units. The $(x+3)$ in the exponent shifts the graph left by 3.
Left shift by 3 units. The $(x+3)$ in the exponent shifts the graph left by 3.
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Find the decay factor for $f(x) = 9 \times (0.75)^x$.
Find the decay factor for $f(x) = 9 \times (0.75)^x$.
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$0.75$. The decay factor is the base 0.75 in the exponential function.
$0.75$. The decay factor is the base 0.75 in the exponential function.
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Find the reflection of $f(x) = 2^x$ across the x-axis.
Find the reflection of $f(x) = 2^x$ across the x-axis.
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$f(x) = -2^x$. Multiplying by -1 reflects the function across the x-axis.
$f(x) = -2^x$. Multiplying by -1 reflects the function across the x-axis.
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What happens to $f(x) = a \times b^x$ as $x \to -\text{∞}$ when $0 < b < 1$?
What happens to $f(x) = a \times b^x$ as $x \to -\text{∞}$ when $0 < b < 1$?
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$f(x) \to \text{∞}$. When $0 < b < 1$ and $x \to -\infty$, $b^x \to \infty$.
$f(x) \to \text{∞}$. When $0 < b < 1$ and $x \to -\infty$, $b^x \to \infty$.
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Determine the asymptote of $f(x) = 100 \times (0.99)^x$.
Determine the asymptote of $f(x) = 100 \times (0.99)^x$.
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$y = 0$. Exponential functions approach 0 but never reach it.
$y = 0$. Exponential functions approach 0 but never reach it.
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For $f(x) = 7 \times 1.2^x$, identify the initial value.
For $f(x) = 7 \times 1.2^x$, identify the initial value.
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$7$. The coefficient 7 is the initial value when $x = 0$.
$7$. The coefficient 7 is the initial value when $x = 0$.
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Find $f(2)$ for $f(x) = 4 \times (0.5)^x$.
Find $f(2)$ for $f(x) = 4 \times (0.5)^x$.
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$1$. Substitute $x = 2$: $f(2) = 4 \times (0.5)^2 = 4 \times 0.25 = 1$.
$1$. Substitute $x = 2$: $f(2) = 4 \times (0.5)^2 = 4 \times 0.25 = 1$.
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What is the effect of a negative exponent in $f(x) = 3 \times b^{-x}$?
What is the effect of a negative exponent in $f(x) = 3 \times b^{-x}$?
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Reflection across y-axis. Negative exponent transforms $b^{-x}$ to $(1/b)^x$, reflecting over y-axis.
Reflection across y-axis. Negative exponent transforms $b^{-x}$ to $(1/b)^x$, reflecting over y-axis.
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Identify the vertical stretch in $f(x) = 10 \times 2^x$.
Identify the vertical stretch in $f(x) = 10 \times 2^x$.
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Factor of 10. The coefficient 10 stretches the graph vertically by factor 10.
Factor of 10. The coefficient 10 stretches the graph vertically by factor 10.
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Determine if $f(x) = 8 \times (1.01)^x$ models growth or decay.
Determine if $f(x) = 8 \times (1.01)^x$ models growth or decay.
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Growth. Since $1.01 > 1$, the function shows exponential growth.
Growth. Since $1.01 > 1$, the function shows exponential growth.
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Identify the transformation: $f(x) = 2 \times 3^{x-1}$.
Identify the transformation: $f(x) = 2 \times 3^{x-1}$.
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Right shift by 1 unit. The $(x-1)$ in the exponent shifts the graph right by 1.
Right shift by 1 unit. The $(x-1)$ in the exponent shifts the graph right by 1.
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