Interpret Exponential Functions and Growth Rate - Algebra 1

1. Identify the percent increase per period for $y=a(1.2)^t$.

   $20\%$ increase per unit $t$. Rate $r = 1.2 - 1 = 0.2 = 20\%$ increase.

2. Classify $y=8\cdot(0.98)^{\frac{t}{2}}$ as exponential growth or decay.

   Exponential decay. Base $0.98 < 1$ gives decay even with fractional exponent.

3. Identify the percent rate of change per period for $y=a(1+r)^t$ when $r=0.12$.

   $12\%$ increase per period. Factor $b = 1 + r = 1 + 0.12 = 1.12$ gives $12\%$ increase.

4. Identify the effective factor per 1 unit $t$ for $y=10\cdot(1.03)^{4t}$.

   $(1.03)^4$ per unit $t$. Rewrite as $((1.03)^4)^t$ to see factor per $t$.

5. Identify the percent rate of change per step for $y=7(1.25)^t$.

   $25\%$ increase per unit $t$. Rate $r = 1.25 - 1 = 0.25 = 25\%$ increase.

6. What is the factor $b$ for a $3.5\%$ increase per period in $y=a\cdot b^t$?

   $b=1.035$. Increase: $b = 1 + 0.035 = 1.035$.

7. Use exponent rules to rewrite $4^{t+3}$ as a product of two powers of $4$.

   $4^{t+3}=4^t\cdot 4^3$. Product rule: $a^{m+n} = a^m \cdot a^n$.

8. State the formula that converts percent rate $r$ to factor $b$ for decay per period.

   $b=1-r$. Subtract decay rate from 1 to get factor.

9. Classify $y=10\cdot(1.03)^{4t}$ as exponential growth or decay.

   Exponential growth. Base $1.03 > 1$ gives growth when compounded.

10. What is the percent rate of change if the factor is $b=0.84$ per time period?

    $16\%$ decrease. Rate $r = 1 - b = 1 - 0.84 = 0.16$.

11. Classify $y=(1.5)^{-t}$ as exponential growth or decay.

    Exponential decay. Negative exponent creates base $< 1$, indicating decay.

12. Identify the percent rate of change per step for $y=7(0.6)^t$.

    $40\%$ decrease per unit $t$. Rate $r = 1 - 0.6 = 0.4 = 40\%$ decrease.

13. Classify $y=(1.01)^{12t}$ as exponential growth or decay.

    Exponential growth. Base $1.01 > 1$ gives growth, even compounded.

14. Use exponent rules to simplify $(5^{2t})^3$ to a single exponent on base $5$.

    $5^{6t}$. Power rule: $(a^m)^n = a^{mn}$.

15. What exponent property is used to rewrite $a^{m+n}$ as $a^m\cdot a^n$?

    Product of powers: $a^{m+n}=a^m\cdot a^n$. Combines powers with same base by adding exponents.

16. What does it mean about growth or decay if $0<b<1$ in $y=a\cdot b^t$?

    Decay. Base between 0 and 1 means quantity decreases.

17. Use exponent rules to rewrite $y=(0.5)^{2t}$ in the form y=(___)^t.

    $y=\left((0.5)^2\right)^t$. Power rule: $(0.5)^{2t} = ((0.5)^2)^t$.

18. What does it mean about growth or decay if $b>1$ in $y=a\cdot b^t$?

    Growth. Base greater than 1 means quantity increases.

19. Use exponent rules to rewrite $y=(1.01)^{12t}$ in the form $y=\left(\_\_\_\_\right)^t$.

    $y=\left((1.01)^{12}\right)^t$. Power rule: $(a^m)^n = a^{mn}$ rearranged.

20. Identify the effective factor per 1 unit $t$ for $y=8\cdot(0.98)^{\frac{t}{2}}$.

    $(0.98)^{\frac{1}{2}}$ per unit $t$. Rewrite as $((0.98)^{1/2})^t$ to find factor per $t$.

21. What is the percent rate of change for $y=a(1)^t$?

    $0\%$ change. Base equals 1, so no growth or decay occurs.

22. Identify the percent decrease per period for $y=a(0.75)^t$.

    $25\%$ decrease per unit $t$. Rate $r = 1 - 0.75 = 0.25 = 25\%$ decrease.

23. What is the percent rate of change if the factor is $b=1.15$ per time period?

    $15\%$ increase. Rate $r = b - 1 = 1.15 - 1 = 0.15$.

24. What is the initial value $a$ in $y=a\cdot b^t$ for $y=5(1.08)^t$?

    $a=5$. The coefficient multiplying the exponential term.

25. Use exponent rules to rewrite $y=(1.5)^{-t}$ with a positive exponent on the base.

    $y=\left(\frac{1}{1.5}\right)^t$. Negative exponent: $a^{-t} = (1/a)^t$.

26. What is the factor $b$ for a $6\%$ decrease per period in $y=a\cdot b^t$?

    $b=0.94$. Decrease: $b = 1 - 0.06 = 0.94$.

27. What is the growth/decay factor $b$ in $y=a\cdot b^t$ for $y=5(1.08)^t$?

    $b=1.08$. The base of the exponential term.

28. What is the percent rate of change for $y=(0.97)^t$ per 1 unit of $t$?

    $3\%$ decrease per unit $t$. Base $0.97 = 1 - 0.03$, so decay rate is $3\%$.

29. State the formula that converts percent rate $r$ to factor $b$ for growth per period.

    $b=1+r$. Add growth rate to 1 to get factor.

30. What is the percent rate $r$ if $y=a(1.005)^t$ models change per period?

    $0.5\%$ increase. Rate $r = 1.005 - 1 = 0.005 = 0.5\%$ growth.

31. Identify the effective growth factor per 1 unit of $t$ for $y=(1.2)^{\frac{t}{10}}$.

    $(1.2)^{\frac{1}{10}}$ per unit $t$. Rewrite as $((1.2)^{1/10})^t$ to see factor per $t$.

32. Use exponent rules to rewrite $2^{t-5}$ as a quotient of two powers of $2$.

    $2^{t-5}=\frac{2^t}{2^5}$. Quotient rule: $a^{m-n} = \frac{a^m}{a^n}$.

33. Use exponent rules to rewrite $y=(1.2)^{\frac{t}{10}}$ in the form $y=\left(\_\_\_\_\right)^t$.

    $y=\left((1.2)^{\frac{1}{10}}\right)^t$. Power rule: $a^{m/n} = (a^{1/n})^m$ rearranged.

34. Use exponent rules to rewrite $y=a\cdot b^{ct}$ as $y=a\cdot(\_\_\_)^t$.

    $y=a\cdot(b^c)^t$. Power rule applied to exponential functions.

35. Identify the effective growth factor per 1 unit of $t$ for $y=(1.01)^{12t}$.

    $(1.01)^{12}$ per unit $t$. Rewrite as $((1.01)^{12})^t$ to see factor per $t$.

36. Identify whether $y=(1.02)^t$ represents exponential growth or decay.

    Exponential growth. Base $1.02 > 1$, indicating growth.

37. What is the percent change per unit $t$ for $y=(0.9)^{\frac{t}{3}}$ in terms of a factor?

    Factor per $t$ is $(0.9)^{\frac{1}{3}}$. Rewrite as $((0.9)^{1/3})^t$ to find factor per $t$.

38. Use exponent rules to rewrite $y=a\cdot b^{\frac{t}{c}}$ as $y=a\cdot(\_\_\_)^t$.

    $y=a\cdot\left(b^{\frac{1}{c}}\right)^t$. Fractional exponent moved inside parentheses as power.

39. What is the percent rate of change per unit $t$ for $y=a(1.04)^{\frac{t}{2}}$ in factor form?

    Factor per $t$ is $(1.04)^{\frac{1}{2}}$. Rewrite as $((1.04)^{1/2})^t$ to find factor per $t$.

40. Which value of $b$ gives no percent change in $y=a\cdot b^t$?

    $b=1$. Factor of 1 means no change from period to period.

41. Classify $y=(0.5)^{2t}$ as exponential growth or decay.

    Exponential decay. Base $0.5 < 1$, so represents decay.

42. Classify $y=(0.99)^{12t}$ as exponential growth or decay.

    Exponential decay. Base $0.99 < 1$ gives decay, even compounded.

43. Use exponent rules to rewrite $y=2\cdot(3^t)^4$ as $y=2\cdot(\_\_\_)^t$.

    $y=2\cdot(3^4)^t$. Power rule: $(a^t)^4 = a^{4t} = (a^4)^t$.

44. What is the percent rate $r$ if $y=a(0.92)^t$ models change per period?

    $8\%$ decrease. Rate $r = 1 - 0.92 = 0.08 = 8\%$ decay.

45. What exponent property justifies rewriting $a^{kt}$ as $(a^k)^t$?

    Power of a power: $(a^m)^n=a^{mn}$. Allows converting between forms $a^{kt}$ and $(a^k)^t$.

46. What exponent property justifies rewriting $a^{\frac{t}{10}}$ as $(a^{\frac{1}{10}})^t$?

    $a^{mn}=(a^m)^n$ with $m=\frac{1}{10}$. Fractional exponent can be moved inside parentheses.

47. Identify whether $y=(0.97)^t$ represents exponential growth or decay.

    Exponential decay. Base $0.97 < 1$, indicating decay.

48. What is the percent rate of change for $y=(1.02)^t$ per 1 unit of $t$?

    $2\%$ increase per unit $t$. Base $1.02 = 1 + 0.02$, so growth rate is $2\%$.

49. What exponent property is used to rewrite $a^{m-n}$ as $\frac{a^m}{a^n}$?

    Quotient of powers: $a^{m-n}=\frac{a^m}{a^n}$. Divides powers with same base by subtracting exponents.

50. What exponent property rewrites $a^{-t}$ as $\left(\frac{1}{a}\right)^t$?

    Negative exponent: $a^{-n}=\frac{1}{a^n}$. Converts negative exponent to reciprocal base.

51. Use exponent rules to rewrite $y=(1.01)^{12t}$ in the form y=\left( \text{____} \right)^t.

    $y=\left((1.01)^{12}\right)^t$. Power rule: $(a^m)^n = a^{mn}$ rearranged.

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