Recognize Percent Growth or Decay - Algebra
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What is the decay factor for $3%$ decay per interval?
What is the decay factor for $3%$ decay per interval?
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$0.97$. Decay factor is $1 - 0.03 = 0.97$.
$0.97$. Decay factor is $1 - 0.03 = 0.97$.
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What is the growth factor (multiplier) for $8%$ growth per interval?
What is the growth factor (multiplier) for $8%$ growth per interval?
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$1.08$. Growth factor is $1 + 0.08 = 1.08$.
$1.08$. Growth factor is $1 + 0.08 = 1.08$.
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Identify the model type: “A population doubles every $5$ years.”
Identify the model type: “A population doubles every $5$ years.”
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Exponential growth. Doubling represents multiplication by 2 each interval, which is exponential growth.
Exponential growth. Doubling represents multiplication by 2 each interval, which is exponential growth.
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Identify whether the change is constant percent: values $100,120,140$ at equal intervals.
Identify whether the change is constant percent: values $100,120,140$ at equal intervals.
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No; differences are constant, not ratios. Differences of $20$ are constant, indicating linear (not exponential) change.
No; differences are constant, not ratios. Differences of $20$ are constant, indicating linear (not exponential) change.
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Identify the unit interval in “decays by $2%$ per year.”
Identify the unit interval in “decays by $2%$ per year.”
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Year. The phrase 'per year' identifies year as the time interval unit.
Year. The phrase 'per year' identifies year as the time interval unit.
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What is the multiplier for “triples each interval”?
What is the multiplier for “triples each interval”?
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$3$. Tripling means multiplying by 3 each interval.
$3$. Tripling means multiplying by 3 each interval.
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Identify the model type: “A tank drains $3$ gallons per minute.”
Identify the model type: “A tank drains $3$ gallons per minute.”
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Linear change. A constant rate of change (gallons per minute) creates linear change.
Linear change. A constant rate of change (gallons per minute) creates linear change.
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Which description matches $A(t)=A_0b^t$ when $b>1$?
Which description matches $A(t)=A_0b^t$ when $b>1$?
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Exponential growth. When $b > 1$, each multiplication makes the quantity larger.
Exponential growth. When $b > 1$, each multiplication makes the quantity larger.
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Which description matches $A(t)=A_0b^t$ when $0<b<1$?
Which description matches $A(t)=A_0b^t$ when $0<b<1$?
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Exponential decay. When $0 < b < 1$, each multiplication makes the quantity smaller.
Exponential decay. When $0 < b < 1$, each multiplication makes the quantity smaller.
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Identify the model type: “A car loses $20%$ of its value each year.”
Identify the model type: “A car loses $20%$ of its value each year.”
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Exponential decay. Losing a constant percentage each period creates exponential decay.
Exponential decay. Losing a constant percentage each period creates exponential decay.
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Identify the model type: “A population doubles every $5$ years.”
Identify the model type: “A population doubles every $5$ years.”
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Exponential growth. Doubling represents multiplication by 2 each interval, which is exponential growth.
Exponential growth. Doubling represents multiplication by 2 each interval, which is exponential growth.
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What feature distinguishes linear change from exponential change in a table?
What feature distinguishes linear change from exponential change in a table?
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Constant difference (not constant ratio). Each y-value minus the previous equals the same number (constant difference).
Constant difference (not constant ratio). Each y-value minus the previous equals the same number (constant difference).
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What feature distinguishes exponential change from linear change in a table?
What feature distinguishes exponential change from linear change in a table?
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Constant ratio (not constant difference). Each y-value divided by the previous equals the same number (constant ratio).
Constant ratio (not constant difference). Each y-value divided by the previous equals the same number (constant ratio).
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Which option represents exponential decay: $P(t)=200(0.9)^t$ or $P(t)=200-0.9t$?
Which option represents exponential decay: $P(t)=200(0.9)^t$ or $P(t)=200-0.9t$?
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$P(t)=200(0.9)^t$. The exponential form with base $0.9 < 1$ indicates decay, not linear decrease.
$P(t)=200(0.9)^t$. The exponential form with base $0.9 < 1$ indicates decay, not linear decrease.
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Which option represents exponential growth: $A(t)=50+3t$ or $A(t)=50(1.03)^t$?
Which option represents exponential growth: $A(t)=50+3t$ or $A(t)=50(1.03)^t$?
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$A(t)=50(1.03)^t$. The exponential form with base $1.03 > 1$ indicates growth, not linear increase.
$A(t)=50(1.03)^t$. The exponential form with base $1.03 > 1$ indicates growth, not linear increase.
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What is the decay factor (multiplier) for $15%$ decay per interval?
What is the decay factor (multiplier) for $15%$ decay per interval?
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$0.85$. Decay factor is $1 - 0.15 = 0.85$.
$0.85$. Decay factor is $1 - 0.15 = 0.85$.
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What is the growth factor (multiplier) for $8%$ growth per interval?
What is the growth factor (multiplier) for $8%$ growth per interval?
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$1.08$. Growth factor is $1 + 0.08 = 1.08$.
$1.08$. Growth factor is $1 + 0.08 = 1.08$.
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What decimal is equivalent to a $7%$ decay rate?
What decimal is equivalent to a $7%$ decay rate?
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$r=0.07$. Convert $7%$ to decimal: $7 ÷ 100 = 0.07$.
$r=0.07$. Convert $7%$ to decimal: $7 ÷ 100 = 0.07$.
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What decimal is equivalent to a $12%$ growth rate?
What decimal is equivalent to a $12%$ growth rate?
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$r=0.12$. Convert $12%$ to decimal: $12 ÷ 100 = 0.12$.
$r=0.12$. Convert $12%$ to decimal: $12 ÷ 100 = 0.12$.
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What is the multiplier for a $r%$ decrease per interval?
What is the multiplier for a $r%$ decrease per interval?
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$1-r$ (with $r$ as a decimal). Subtract the decimal rate from 1 to get the decay multiplier.
$1-r$ (with $r$ as a decimal). Subtract the decimal rate from 1 to get the decay multiplier.
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What is the multiplier for a $r%$ increase per interval?
What is the multiplier for a $r%$ increase per interval?
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$1+r$ (with $r$ as a decimal). Add the decimal rate to 1 to get the growth multiplier.
$1+r$ (with $r$ as a decimal). Add the decimal rate to 1 to get the growth multiplier.
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What is the general exponential form for constant percent change over $t$ intervals?
What is the general exponential form for constant percent change over $t$ intervals?
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$A(t)=A_0(1+r)^t$. Standard exponential function where $A_0$ is initial value and $(1+r)$ is the growth factor.
$A(t)=A_0(1+r)^t$. Standard exponential function where $A_0$ is initial value and $(1+r)$ is the growth factor.
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Find the percent rate per interval for $A(t)=90(0.93)^t$.
Find the percent rate per interval for $A(t)=90(0.93)^t$.
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$7%$ decrease. Base $0.93 = 1 - 0.07$, so the rate is $7%$ decrease.
$7%$ decrease. Base $0.93 = 1 - 0.07$, so the rate is $7%$ decrease.
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Which situation uses constant percent change: “earns $5%$ interest yearly” or “earns $50$ yearly”?
Which situation uses constant percent change: “earns $5%$ interest yearly” or “earns $50$ yearly”?
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“Earns $5%$ interest yearly”. Percentage interest creates exponential growth, not constant addition.
“Earns $5%$ interest yearly”. Percentage interest creates exponential growth, not constant addition.
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Identify whether the change is constant percent: values $100,110,121$ at equal intervals.
Identify whether the change is constant percent: values $100,110,121$ at equal intervals.
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Yes; ratio is constant at $1.1$. Ratios: $\frac{110}{100} = 1.1$ and $\frac{121}{110} = 1.1$ are constant.
Yes; ratio is constant at $1.1$. Ratios: $\frac{110}{100} = 1.1$ and $\frac{121}{110} = 1.1$ are constant.
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What model matches “starts at $1200$ and doubles every interval”?
What model matches “starts at $1200$ and doubles every interval”?
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$A(t)=1200(2)^t$. Initial value $1200$, doubling means multiplier is $2$.
$A(t)=1200(2)^t$. Initial value $1200$, doubling means multiplier is $2$.
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Identify whether the change is constant percent: values $100,120,140$ at equal intervals.
Identify whether the change is constant percent: values $100,120,140$ at equal intervals.
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No; differences are constant, not ratios. Differences of $20$ are constant, indicating linear (not exponential) change.
No; differences are constant, not ratios. Differences of $20$ are constant, indicating linear (not exponential) change.
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What is the multiplier for “increases by $30%$ each interval”?
What is the multiplier for “increases by $30%$ each interval”?
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$1.30$. For $30%$ increase: $1 + 0.30 = 1.30$.
$1.30$. For $30%$ increase: $1 + 0.30 = 1.30$.
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What is the value of $b$ in $A(t)=A_0b^t$ if the quantity decreases by $18%$ each interval?
What is the value of $b$ in $A(t)=A_0b^t$ if the quantity decreases by $18%$ each interval?
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$b=0.82$. Decrease by $18%$ means $b = 1 - 0.18 = 0.82$.
$b=0.82$. Decrease by $18%$ means $b = 1 - 0.18 = 0.82$.
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What is the common ratio for values $50,60,72$ at equal intervals?
What is the common ratio for values $50,60,72$ at equal intervals?
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$1.2$. Each consecutive ratio: $\frac{60}{50} = 1.2$ and $\frac{72}{60} = 1.2$.
$1.2$. Each consecutive ratio: $\frac{60}{50} = 1.2$ and $\frac{72}{60} = 1.2$.
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