Interpret Exponential Functions and Growth Rate - Algebra
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What exponent property justifies rewriting $a^{\frac{t}{10}}$ as $(a^{\frac{1}{10}})^t$?
What exponent property justifies rewriting $a^{\frac{t}{10}}$ as $(a^{\frac{1}{10}})^t$?
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$a^{mn}=(a^m)^n$ with $m=\frac{1}{10}$. Fractional exponent can be moved inside parentheses.
$a^{mn}=(a^m)^n$ with $m=\frac{1}{10}$. Fractional exponent can be moved inside parentheses.
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What is the percent rate of change for $y=(0.97)^t$ per 1 unit of $t$?
What is the percent rate of change for $y=(0.97)^t$ per 1 unit of $t$?
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$3%$ decrease per unit $t$. Base $0.97 = 1 - 0.03$, so decay rate is $3%$.
$3%$ decrease per unit $t$. Base $0.97 = 1 - 0.03$, so decay rate is $3%$.
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Classify $y=(0.5)^{2t}$ as exponential growth or decay.
Classify $y=(0.5)^{2t}$ as exponential growth or decay.
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Exponential decay. Base $0.5 < 1$, so represents decay.
Exponential decay. Base $0.5 < 1$, so represents decay.
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What is the percent rate of change for $y=a(1)^t$?
What is the percent rate of change for $y=a(1)^t$?
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$0%$ change. Base equals 1, so no growth or decay occurs.
$0%$ change. Base equals 1, so no growth or decay occurs.
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What is the percent change per unit $t$ for $y=(0.9)^{\frac{t}{3}}$ in terms of a factor?
What is the percent change per unit $t$ for $y=(0.9)^{\frac{t}{3}}$ in terms of a factor?
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Factor per $t$ is $(0.9)^{\frac{1}{3}}$. Rewrite as $((0.9)^{1/3})^t$ to find factor per $t$.
Factor per $t$ is $(0.9)^{\frac{1}{3}}$. Rewrite as $((0.9)^{1/3})^t$ to find factor per $t$.
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What is the factor $b$ for a $3.5%$ increase per period in $y=a\cdot b^t$?
What is the factor $b$ for a $3.5%$ increase per period in $y=a\cdot b^t$?
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$b=1.035$. Increase: $b = 1 + 0.035 = 1.035$.
$b=1.035$. Increase: $b = 1 + 0.035 = 1.035$.
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Use exponent rules to rewrite $y=2\cdot(3^t)^4$ as $y=2\cdot( )^t$.
Use exponent rules to rewrite $y=2\cdot(3^t)^4$ as $y=2\cdot( )^t$.
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$y=2\cdot(3^4)^t$. Power rule: $(a^t)^4 = a^{4t} = (a^4)^t$.
$y=2\cdot(3^4)^t$. Power rule: $(a^t)^4 = a^{4t} = (a^4)^t$.
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What exponent property is used to rewrite $a^{m+n}$ as $a^m\cdot a^n$?
What exponent property is used to rewrite $a^{m+n}$ as $a^m\cdot a^n$?
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Product of powers: $a^{m+n}=a^m\cdot a^n$. Combines powers with same base by adding exponents.
Product of powers: $a^{m+n}=a^m\cdot a^n$. Combines powers with same base by adding exponents.
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Identify the percent decrease per period for $y=a(0.75)^t$.
Identify the percent decrease per period for $y=a(0.75)^t$.
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$25%$ decrease per unit $t$. Rate $r = 1 - 0.75 = 0.25 = 25%$ decrease.
$25%$ decrease per unit $t$. Rate $r = 1 - 0.75 = 0.25 = 25%$ decrease.
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Identify whether $y=(0.97)^t$ represents exponential growth or decay.
Identify whether $y=(0.97)^t$ represents exponential growth or decay.
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Exponential decay. Base $0.97 < 1$, indicating decay.
Exponential decay. Base $0.97 < 1$, indicating decay.
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Use exponent rules to rewrite $y=a\cdot b^{\frac{t}{c}}$ as $y=a\cdot( )^t$.
Use exponent rules to rewrite $y=a\cdot b^{\frac{t}{c}}$ as $y=a\cdot( )^t$.
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$y=a\cdot\left(b^{\frac{1}{c}}\right)^t$. Fractional exponent moved inside parentheses as power.
$y=a\cdot\left(b^{\frac{1}{c}}\right)^t$. Fractional exponent moved inside parentheses as power.
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Use exponent rules to rewrite $y=a\cdot b^{ct}$ as $y=a\cdot( )^t$.
Use exponent rules to rewrite $y=a\cdot b^{ct}$ as $y=a\cdot( )^t$.
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$y=a\cdot(b^c)^t$. Power rule applied to exponential functions.
$y=a\cdot(b^c)^t$. Power rule applied to exponential functions.
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Use exponent rules to simplify $(5^{2t})^3$ to a single exponent on base $5$.
Use exponent rules to simplify $(5^{2t})^3$ to a single exponent on base $5$.
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$5^{6t}$. Power rule: $(a^m)^n = a^{mn}$.
$5^{6t}$. Power rule: $(a^m)^n = a^{mn}$.
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What is the percent rate of change per unit $t$ for $y=a(1.04)^{\frac{t}{2}}$ in factor form?
What is the percent rate of change per unit $t$ for $y=a(1.04)^{\frac{t}{2}}$ in factor form?
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Factor per $t$ is $(1.04)^{\frac{1}{2}}$. Rewrite as $((1.04)^{1/2})^t$ to find factor per $t$.
Factor per $t$ is $(1.04)^{\frac{1}{2}}$. Rewrite as $((1.04)^{1/2})^t$ to find factor per $t$.
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Classify $y=8\cdot(0.98)^{\frac{t}{2}}$ as exponential growth or decay.
Classify $y=8\cdot(0.98)^{\frac{t}{2}}$ as exponential growth or decay.
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Exponential decay. Base $0.98 < 1$ gives decay even with fractional exponent.
Exponential decay. Base $0.98 < 1$ gives decay even with fractional exponent.
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Identify the effective factor per 1 unit $t$ for $y=8\cdot(0.98)^{\frac{t}{2}}$.
Identify the effective factor per 1 unit $t$ for $y=8\cdot(0.98)^{\frac{t}{2}}$.
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$(0.98)^{\frac{1}{2}}$ per unit $t$. Rewrite as $((0.98)^{1/2})^t$ to find factor per $t$.
$(0.98)^{\frac{1}{2}}$ per unit $t$. Rewrite as $((0.98)^{1/2})^t$ to find factor per $t$.
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Classify $y=10\cdot(1.03)^{4t}$ as exponential growth or decay.
Classify $y=10\cdot(1.03)^{4t}$ as exponential growth or decay.
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Exponential growth. Base $1.03 > 1$ gives growth when compounded.
Exponential growth. Base $1.03 > 1$ gives growth when compounded.
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Use exponent rules to rewrite $y=(1.01)^{12t}$ in the form $y=\left( \text{____} \right)^t$.
Use exponent rules to rewrite $y=(1.01)^{12t}$ in the form $y=\left( \text{____} \right)^t$.
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$y=\left((1.01)^{12}\right)^t$. Power rule: $(a^m)^n = a^{mn}$ rearranged.
$y=\left((1.01)^{12}\right)^t$. Power rule: $(a^m)^n = a^{mn}$ rearranged.
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What is the growth/decay factor $b$ in $y=a\cdot b^t$ for $y=5(1.08)^t$?
What is the growth/decay factor $b$ in $y=a\cdot b^t$ for $y=5(1.08)^t$?
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$b=1.08$. The base of the exponential term.
$b=1.08$. The base of the exponential term.
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Identify the effective factor per 1 unit $t$ for $y=10\cdot(1.03)^{4t}$.
Identify the effective factor per 1 unit $t$ for $y=10\cdot(1.03)^{4t}$.
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$(1.03)^4$ per unit $t$. Rewrite as $((1.03)^4)^t$ to see factor per $t$.
$(1.03)^4$ per unit $t$. Rewrite as $((1.03)^4)^t$ to see factor per $t$.
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What exponent property rewrites $a^{-t}$ as $\left(\frac{1}{a}\right)^t$?
What exponent property rewrites $a^{-t}$ as $\left(\frac{1}{a}\right)^t$?
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Negative exponent: $a^{-n}=\frac{1}{a^n}$. Converts negative exponent to reciprocal base.
Negative exponent: $a^{-n}=\frac{1}{a^n}$. Converts negative exponent to reciprocal base.
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Classify $y=(1.5)^{-t}$ as exponential growth or decay.
Classify $y=(1.5)^{-t}$ as exponential growth or decay.
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Exponential decay. Negative exponent creates base $< 1$, indicating decay.
Exponential decay. Negative exponent creates base $< 1$, indicating decay.
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Use exponent rules to rewrite $y=(1.5)^{-t}$ with a positive exponent on the base.
Use exponent rules to rewrite $y=(1.5)^{-t}$ with a positive exponent on the base.
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$y=\left(\frac{1}{1.5}\right)^t$. Negative exponent: $a^{-t} = (1/a)^t$.
$y=\left(\frac{1}{1.5}\right)^t$. Negative exponent: $a^{-t} = (1/a)^t$.
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What is the initial value $a$ in $y=a\cdot b^t$ for $y=5(1.08)^t$?
What is the initial value $a$ in $y=a\cdot b^t$ for $y=5(1.08)^t$?
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$a=5$. The coefficient multiplying the exponential term.
$a=5$. The coefficient multiplying the exponential term.
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What does it mean about growth or decay if $b>1$ in $y=a\cdot b^t$?
What does it mean about growth or decay if $b>1$ in $y=a\cdot b^t$?
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Growth. Base greater than 1 means quantity increases.
Growth. Base greater than 1 means quantity increases.
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What does it mean about growth or decay if $0<b<1$ in $y=a\cdot b^t$?
What does it mean about growth or decay if $0<b<1$ in $y=a\cdot b^t$?
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Decay. Base between 0 and 1 means quantity decreases.
Decay. Base between 0 and 1 means quantity decreases.
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What is the percent rate of change if the factor is $b=1.15$ per time period?
What is the percent rate of change if the factor is $b=1.15$ per time period?
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$15%$ increase. Rate $r = b - 1 = 1.15 - 1 = 0.15$.
$15%$ increase. Rate $r = b - 1 = 1.15 - 1 = 0.15$.
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What is the percent rate of change if the factor is $b=0.84$ per time period?
What is the percent rate of change if the factor is $b=0.84$ per time period?
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$16%$ decrease. Rate $r = 1 - b = 1 - 0.84 = 0.16$.
$16%$ decrease. Rate $r = 1 - b = 1 - 0.84 = 0.16$.
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State the formula that converts percent rate $r$ to factor $b$ for growth per period.
State the formula that converts percent rate $r$ to factor $b$ for growth per period.
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$b=1+r$. Add growth rate to 1 to get factor.
$b=1+r$. Add growth rate to 1 to get factor.
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State the formula that converts percent rate $r$ to factor $b$ for decay per period.
State the formula that converts percent rate $r$ to factor $b$ for decay per period.
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$b=1-r$. Subtract decay rate from 1 to get factor.
$b=1-r$. Subtract decay rate from 1 to get factor.
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