Recognize Percent Growth or Decay - Algebra 1

1. Find the percent rate per interval for $A(t)=90(1.07)^t$.

   $7\%$ increase. Base $1.07 = 1 + 0.07$, so the rate is $7\%$ increase.

2. What decimal is equivalent to a $12\%$ growth rate?

   $r=0.12$. Convert $12\%$ to decimal: $12 ÷ 100 = 0.12$.

3. What is the value of $b$ in $A(t)=A_0b^t$ if the quantity decreases by $18\%$ each interval?

   $b=0.82$. Decrease by $18\%$ means $b = 1 - 0.18 = 0.82$.

4. What feature distinguishes linear change from exponential change in a table?

   Constant difference (not constant ratio). Each y-value minus the previous equals the same number (constant difference).

5. What is the growth factor for $3\%$ growth per interval?

   $1.03$. Growth factor is $1 + 0.03 = 1.03$.

6. Identify the model type: "A population doubles every $5$ years."

   Exponential growth. Doubling represents multiplication by 2 each interval, which is exponential growth.

7. What is the common ratio for values $50,60,72$ at equal intervals?

   $1.2$. Each consecutive ratio: $\frac{60}{50} = 1.2$ and $\frac{72}{60} = 1.2$.

8. What model matches "starts at $64$ and halves each interval"?

   $A(t)=64(0.5)^t$. Initial value $64$, halving means multiplier is $0.5$.

9. Identify the model type: "A car loses $20\%$ of its value each year."

   Exponential decay. Losing a constant percentage each period creates exponential decay.

10. What is the growth factor (multiplier) for $8\%$ growth per interval?

    $1.08$. Growth factor is $1 + 0.08 = 1.08$.

11. What is the percent rate for a common ratio of $1.2$ per interval?

    $20\%$ increase. Ratio $1.2 = 1 + 0.2$, so the increase is $20\%$.

12. What model matches "starts at $1200$ and doubles every interval"?

    $A(t)=1200(2)^t$. Initial value $1200$, doubling means multiplier is $2$.

13. What model matches "starts at $500$ and increases by $6\%$ each year"?

    $A(t)=500(1.06)^t$. Initial value $500$, growth factor $1 + 0.06 = 1.06$, time variable $t$.

14. What key phrase signals constant percent growth or decay each time interval?

    "Increases/decreases by $r\%$ each (time) interval". This phrase directly indicates a constant percentage change each time period.

15. What is the per-interval multiplier for "increases by $r\%$" written with $r$ as a percent?

    $1+\frac{r}{100}$. When $r$ is a percent value, divide by $100$ then add to $1$.

16. Identify the type: "The amount is multiplied by $1.15$ each week."

    Exponential growth. Multiplier $1.15 > 1$ indicates the quantity increases each period.

17. What percent change corresponds to a multiplier of $1.25$ per interval?

    $25\%$ increase. Multiplier $1.25 = 1 + 0.25$, so the increase is $25\%$.

18. What is the multiplier for "triples each interval"?

    $3$. Tripling means multiplying by 3 each interval.

19. Identify whether the change is constant percent: values $100,110,121$ at equal intervals.

    Yes; ratio is constant at $1.1$. Ratios: $\frac{110}{100} = 1.1$ and $\frac{121}{110} = 1.1$ are constant.

20. Which equation has a constant percent rate: $y=300(0.8)^t$ or $y=300-0.8t$?

    $y=300(0.8)^t$. The exponential form has a constant base raised to variable power.

21. What is the general exponential form for constant percent change over $t$ intervals?

    $A(t)=A_0(1+r)^t$. Standard exponential function where $A_0$ is initial value and $(1+r)$ is the growth factor.

22. Which description matches $A(t)=A_0b^t$ when $0<b<1$?

    Exponential decay. When $0 < b < 1$, each multiplication makes the quantity smaller.

23. Which situation uses constant percent change: "earns $5\%$ interest yearly" or "earns $50$ yearly"?

    "Earns $5\%$ interest yearly". Percentage interest creates exponential growth, not constant addition.

24. What is the decay factor for $3\%$ decay per interval?

    $0.97$. Decay factor is $1 - 0.03 = 0.97$.

25. Which option represents exponential growth: $A(t)=50+3t$ or $A(t)=50(1.03)^t$?

    $A(t)=50(1.03)^t$. The exponential form with base $1.03 > 1$ indicates growth, not linear increase.

26. What is the per-interval multiplier for "decreases by $r\%$" written with $r$ as a percent?

    $1-\frac{r}{100}$. When $r$ is a percent value, divide by $100$ then subtract from $1$.

27. Identify whether the change is constant percent: values $100,120,140$ at equal intervals.

    No; differences are constant, not ratios. Differences of $20$ are constant, indicating linear (not exponential) change.

28. What is the multiplier for a $r\%$ increase per interval?

    $1+r$ (with $r$ as a decimal). Add the decimal rate to 1 to get the growth multiplier.

29. What percent change corresponds to a multiplier of $0.6$ per interval?

    $40\%$ decrease. Multiplier $0.6 = 1 - 0.4$, so the decrease is $40\%$.

30. What is the multiplier for a $r\%$ decrease per interval?

    $1-r$ (with $r$ as a decimal). Subtract the decimal rate from 1 to get the decay multiplier.

31. Identify the type: "The amount is multiplied by $0.92$ each day."

    Exponential decay. Multiplier $0.92 < 1$ indicates the quantity decreases each period.

32. What is the multiplier for "increases by $30\%$ each interval"?

    $1.30$. For $30\%$ increase: $1 + 0.30 = 1.30$.

33. What is the decay factor (multiplier) for $15\%$ decay per interval?

    $0.85$. Decay factor is $1 - 0.15 = 0.85$.

34. What decimal is equivalent to a $7\%$ decay rate?

    $r=0.07$. Convert $7\%$ to decimal: $7 ÷ 100 = 0.07$.

35. What is the multiplier for "halves each interval"?

    $0.5$. Halving means multiplying by $\frac{1}{2} = 0.5$ each interval.

36. Identify the unit interval in "grows by $5\%$ per month."

    Month. The phrase 'per month' identifies month as the time interval unit.

37. Identify the unit interval in "decays by $2\%$ per year."

    Year. The phrase 'per year' identifies year as the time interval unit.

38. What is the starting value $A_0$ in the model $A(t)=A_0(1+r)^t$?

    The value when $t=0$. $A_0$ represents the initial amount before any time has passed.

39. What percent change corresponds to a multiplier of $0.98$ per interval?

    $2\%$ decrease. Multiplier $0.98 = 1 - 0.02$, so the decrease is $2\%$.

40. What feature distinguishes exponential change from linear change in a table?

    Constant ratio (not constant difference). Each y-value divided by the previous equals the same number (constant ratio).

41. Which option represents exponential decay: $P(t)=200(0.9)^t$ or $P(t)=200-0.9t$?

    $P(t)=200(0.9)^t$. The exponential form with base $0.9 < 1$ indicates decay, not linear decrease.

42. Which phrase indicates exponential decay rather than growth: "grows by" or "decreases by"?

    "Decreases by" (a constant percent each interval). 'Decreases by' indicates the quantity is getting smaller over time.

43. Find the percent rate per interval for $A(t)=90(0.93)^t$.

    $7\%$ decrease. Base $0.93 = 1 - 0.07$, so the rate is $7\%$ decrease.

44. Identify the constant percent rate for values $200,160,128$ at equal intervals.

    $20\%$ decrease (ratio $0.8$). Consecutive ratios: $\frac{160}{200} = 0.8$ and $\frac{128}{160} = 0.8$.

45. Identify the constant percent rate for values $30,33,36.3$ at equal intervals.

    $10\%$ increase (ratio $1.1$). Consecutive ratios: $\frac{33}{30} = 1.1$ and $\frac{36.3}{33} = 1.1$.

46. What model matches "starts at $80$ and decreases by $12\%$ each hour"?

    $A(t)=80(0.88)^t$. Initial value $80$, decay factor $1 - 0.12 = 0.88$, time variable $t$.

47. What percent change corresponds to a multiplier of $1.04$ per interval?

    $4\%$ increase. Multiplier $1.04 = 1 + 0.04$, so the increase is $4\%$.

48. Which description matches $A(t)=A_0b^t$ when $b>1$?

    Exponential growth. When $b > 1$, each multiplication makes the quantity larger.

49. Which option shows constant percent change: differences $+5,+5,+5$ or ratios $\times 1.05,\times 1.05$?

    Ratios $\times 1.05$ (constant ratio). Constant ratios indicate exponential change; constant differences indicate linear.

50. What is the multiplier for "decreases by $30\%$ each interval"?

    $0.70$. For $30\%$ decrease: $1 - 0.30 = 0.70$.

51. Identify the model type: "A tank drains $3$ gallons per minute."

    Linear change. A constant rate of change (gallons per minute) creates linear change.

52. What is the percent rate for a common ratio of $0.75$ per interval?

    $25\%$ decrease. Ratio $0.75 = 1 - 0.25$, so the decrease is $25\%$.

53. Which option is a constant percent rate: "adds $10$ each week" or "increases by $10\%$ each week"?

    "Increases by $10\%$ each week". Percentage increase indicates exponential change, not constant addition.

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