Exponential Function Context and Data Modeling - AP Precalculus
Card 1 of 30
Find the value of $f(2)$ for $f(x) = 2 \times 5^x$.
Find the value of $f(2)$ for $f(x) = 2 \times 5^x$.
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- $f(2) = 2 \times 5^2 = 2 \times 25 = 50$.
- $f(2) = 2 \times 5^2 = 2 \times 25 = 50$.
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For $f(x) = 6 \times 0.5^x$, what is the decay factor?
For $f(x) = 6 \times 0.5^x$, what is the decay factor?
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0.5. The base represents the decay factor.
0.5. The base represents the decay factor.
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What is the formula for finding the rate $r$ in $A = P e^{rt}$?
What is the formula for finding the rate $r$ in $A = P e^{rt}$?
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$r = \frac{1}{t} \text{ln}(\frac{A}{P})$. Solved from $A = Pe^{rt}$ for rate $r$.
$r = \frac{1}{t} \text{ln}(\frac{A}{P})$. Solved from $A = Pe^{rt}$ for rate $r$.
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Calculate the future value $A$ for $P = 1000, r = 0.08, t = 2$.
Calculate the future value $A$ for $P = 1000, r = 0.08, t = 2$.
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$A = 1000 e^{0.16}$. $A = 1000e^{0.08 \times 2} = 1000e^{0.16}$.
$A = 1000 e^{0.16}$. $A = 1000e^{0.08 \times 2} = 1000e^{0.16}$.
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What is the horizontal asymptote of $f(x) = 4 \times (0.75)^x$?
What is the horizontal asymptote of $f(x) = 4 \times (0.75)^x$?
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$y = 0$. Exponential functions approach but never reach zero.
$y = 0$. Exponential functions approach but never reach zero.
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What transformation is applied in $f(x) = a \times b^{x} + d$?
What transformation is applied in $f(x) = a \times b^{x} + d$?
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Vertical shift by $d$. Adding $d$ moves the graph up or down.
Vertical shift by $d$. Adding $d$ moves the graph up or down.
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What is the effect of a negative exponent in $f(x) = a \times b^{-x}$?
What is the effect of a negative exponent in $f(x) = a \times b^{-x}$?
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Reflect across y-axis. Negative exponent reverses the function horizontally.
Reflect across y-axis. Negative exponent reverses the function horizontally.
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What is the exponential growth factor if $b = 1.1$?
What is the exponential growth factor if $b = 1.1$?
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1.1. Growth factor is the base $b$ when $b > 1$.
1.1. Growth factor is the base $b$ when $b > 1$.
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State the formula for continuous exponential growth.
State the formula for continuous exponential growth.
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$A = P e^{rt}$. Uses natural base $e$ for continuous compounding.
$A = P e^{rt}$. Uses natural base $e$ for continuous compounding.
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What is the effect of $a$ in $f(x) = a \times b^x$ on the graph?
What is the effect of $a$ in $f(x) = a \times b^x$ on the graph?
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Vertical stretch/compression. Parameter $a$ scales the function vertically.
Vertical stretch/compression. Parameter $a$ scales the function vertically.
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What is the horizontal asymptote of $f(x) = 4 \times (0.75)^x$?
What is the horizontal asymptote of $f(x) = 4 \times (0.75)^x$?
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$y = 0$. Exponential functions approach but never reach zero.
$y = 0$. Exponential functions approach but never reach zero.
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What transformation is applied in $f(x) = a \times b^{x} + d$?
What transformation is applied in $f(x) = a \times b^{x} + d$?
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Vertical shift by $d$. Adding $d$ moves the graph up or down.
Vertical shift by $d$. Adding $d$ moves the graph up or down.
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Determine the effect of $f(x) = a \times (b^x) + c$ when $c > 0$.
Determine the effect of $f(x) = a \times (b^x) + c$ when $c > 0$.
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Vertical shift up by $c$. Positive $c$ shifts the entire graph upward.
Vertical shift up by $c$. Positive $c$ shifts the entire graph upward.
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Calculate decay rate if $b = 0.9$ in $f(x) = a \times b^x$.
Calculate decay rate if $b = 0.9$ in $f(x) = a \times b^x$.
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10%. Decay rate is $(1 - 0.9) = 0.1 = 10%$.
10%. Decay rate is $(1 - 0.9) = 0.1 = 10%$.
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What does the variable $a$ represent in $f(x) = a \times b^x$?
What does the variable $a$ represent in $f(x) = a \times b^x$?
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Initial value. The coefficient $a$ is the starting amount.
Initial value. The coefficient $a$ is the starting amount.
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Identify the base in the exponential function $f(x) = 5 \times 2^x$.
Identify the base in the exponential function $f(x) = 5 \times 2^x$.
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- The base is the number being raised to the power.
- The base is the number being raised to the power.
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What property does the base $b$ have in an exponential function $f(x) = a \times b^x$?
What property does the base $b$ have in an exponential function $f(x) = a \times b^x$?
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$b > 0$ and $b \neq 1$. Base must be positive and not equal to 1.
$b > 0$ and $b \neq 1$. Base must be positive and not equal to 1.
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Find the initial value of the function $f(x) = 3 \times 4^x$.
Find the initial value of the function $f(x) = 3 \times 4^x$.
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- Initial value is coefficient $a$ when $x = 0$.
- Initial value is coefficient $a$ when $x = 0$.
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State the formula for continuous exponential growth.
State the formula for continuous exponential growth.
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$A = P e^{rt}$. Uses natural base $e$ for continuous compounding.
$A = P e^{rt}$. Uses natural base $e$ for continuous compounding.
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Which parameter affects horizontal shifts in $f(x) = a \times b^{x+c}$?
Which parameter affects horizontal shifts in $f(x) = a \times b^{x+c}$?
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$c$. Parameter $c$ controls horizontal translation.
$c$. Parameter $c$ controls horizontal translation.
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What is the value of $f(0)$ for $f(x) = a \times b^x$?
What is the value of $f(0)$ for $f(x) = a \times b^x$?
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$a$. When $x = 0$, $b^0 = 1$, so $f(0) = a$.
$a$. When $x = 0$, $b^0 = 1$, so $f(0) = a$.
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Define an exponential decay function.
Define an exponential decay function.
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$f(x) = a \times b^x$, $0 < b < 1$. Decay occurs when base is between 0 and 1.
$f(x) = a \times b^x$, $0 < b < 1$. Decay occurs when base is between 0 and 1.
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State the natural base $e$ approximately.
State the natural base $e$ approximately.
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2.718. Euler's number, base of natural logarithm.
2.718. Euler's number, base of natural logarithm.
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Identify the effect of $f(x) = -a \times b^x$ on the graph.
Identify the effect of $f(x) = -a \times b^x$ on the graph.
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Reflection over x-axis. Negative coefficient flips graph over x-axis.
Reflection over x-axis. Negative coefficient flips graph over x-axis.
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What is the range for an exponential decay function?
What is the range for an exponential decay function?
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$(0, a)$ if $a > 0$. Decay functions approach zero from above.
$(0, a)$ if $a > 0$. Decay functions approach zero from above.
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Determine the effect of $f(x) = a \times (b^x) + c$ when $c > 0$.
Determine the effect of $f(x) = a \times (b^x) + c$ when $c > 0$.
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Vertical shift up by $c$. Positive $c$ shifts the entire graph upward.
Vertical shift up by $c$. Positive $c$ shifts the entire graph upward.
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Calculate decay rate if $b = 0.9$ in $f(x) = a \times b^x$.
Calculate decay rate if $b = 0.9$ in $f(x) = a \times b^x$.
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10%. Decay rate is $(1 - 0.9) = 0.1 = 10%$.
10%. Decay rate is $(1 - 0.9) = 0.1 = 10%$.
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For $f(x) = 9 \times 1.2^x$, what is the growth rate?
For $f(x) = 9 \times 1.2^x$, what is the growth rate?
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20%. Growth rate is $(b - 1) \times 100%$.
20%. Growth rate is $(b - 1) \times 100%$.
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Is $f(x) = 1 \times (0.5)^x$ growth or decay?
Is $f(x) = 1 \times (0.5)^x$ growth or decay?
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Decay. Base 0.5 is less than 1, indicating decay.
Decay. Base 0.5 is less than 1, indicating decay.
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Determine the growth factor for $f(x) = 3 \times 1.05^x$.
Determine the growth factor for $f(x) = 3 \times 1.05^x$.
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1.05. Growth factor is the base when $b > 1$.
1.05. Growth factor is the base when $b > 1$.
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