Algebra Flashcards: Interpret Exponential Functions And Growth Rate

What this deck covers

This deck focuses on Interpret Exponential Functions And Growth Rate, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

Flashcard 6: Identify the effective factor per 1 unit $t$ for $y=8\cdot(0.98)^{\frac{t}{2}}$.

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

Flashcard 7: What does it mean about growth or decay if $0<b<1$ in $y=a\cdot b^t$?

Answer: Decay. Base between 0 and 1 means quantity decreases.

Flashcard 8: What is the factor $b$ for a $3.5\%$ increase per period in $y=a\cdot b^t$?

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

Flashcard 9: Use exponent rules to rewrite $4^{t+3}$ as a product of two powers of $4$.

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

Flashcard 10: State the formula that converts percent rate $r$ to factor $b$ for decay per period.

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

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

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

Flashcard 12: Use exponent rules to rewrite $y=(1.2)^{\frac{t}{10}}$ in the form y=\left(<span class="fill-in-blank">&nbsp;</span>\right)^t.

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

Flashcard 13: What does it mean about growth or decay if $b>1$ in $y=a\cdot b^t$?

Answer: Growth. Base greater than 1 means quantity increases.

Flashcard 14: What is the factor $b$ for a $6\%$ decrease per period in $y=a\cdot b^t$?

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

Flashcard 15: Classify $y=10\cdot(1.03)^{4t}$ as exponential growth or decay.

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

Flashcard 16: What is the percent rate of change if the factor is $b=0.84$ per time period?

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

Flashcard 17: Use exponent rules to rewrite $y=a\cdot b^{\frac{t}{c}}$ as y=a\cdot(<span class="fill-in-blank">&nbsp;</span>)^t.

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

Flashcard 18: Identify the percent rate of change per period for $y=a(1+r)^t$ when $r=0.12$.

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

Flashcard 19: Classify $y=(1.5)^{-t}$ as exponential growth or decay.

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

Flashcard 20: What exponent property justifies rewriting $a^{kt}$ as $(a^k)^t$?

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

Flashcard 21: What is the percent rate of change if the factor is $b=1.15$ per time period?

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

Flashcard 22: Classify $y=(1.01)^{12t}$ as exponential growth or decay.

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

Flashcard 23: Identify the percent rate of change per step for $y=7(0.6)^t$.

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

Flashcard 24: What is the percent rate $r$ if $y=a(1.005)^t$ models change per period?

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

Flashcard 25: Use exponent rules to rewrite $2^{t-5}$ as a quotient of two powers of $2$.

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

Flashcard 26: Classify $y=(1.01)^{12t}$ as exponential growth or decay.

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

Flashcard 27: Use exponent rules to simplify $(5^{2t})^3$ to a single exponent on base $5$.

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

Flashcard 28: What exponent property is used to rewrite $a^{m+n}$ as $a^m\cdot a^n$?

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

Flashcard 29: What does it mean about growth or decay if $0<b<1$ in $y=a\cdot b^t$?

Answer: Decay. Base between 0 and 1 means quantity decreases.

Flashcard 30: Which value of $b$ gives no percent change in $y=a\cdot b^t$?

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

Flashcard 31: Use exponent rules to rewrite $y=(0.5)^{2t}$ in the form y=(<span class="fill-in-blank">&nbsp;</span>)^t.

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

Flashcard 32: What does it mean about growth or decay if $b>1$ in $y=a\cdot b^t$?

Answer: Growth. Base greater than 1 means quantity increases.

Flashcard 33: Use exponent rules to rewrite $y=(1.01)^{12t}$ in the form y=\left(<span class="fill-in-blank">&nbsp;</span>\right)^t.

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

Flashcard 34: Classify $y=8\cdot(0.98)^{\frac{t}{2}}$ as exponential growth or decay.

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

Flashcard 35: Identify the effective factor per 1 unit $t$ for $y=8\cdot(0.98)^{\frac{t}{2}}$.

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

Flashcard 36: What is the factor $b$ for a $3.5\%$ increase per period in $y=a\cdot b^t$?

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

Flashcard 37: What is the percent rate of change for $y=a(1)^t$?

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

Flashcard 38: Identify the percent decrease per period for $y=a(0.75)^t$.

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

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

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

Flashcard 40: What is the percent rate of change if the factor is $b=1.15$ per time period?

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

Flashcard 41: What is the initial value $a$ in $y=a\cdot b^t$ for $y=5(1.08)^t$?

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

Flashcard 42: What is the percent change per unit $t$ for $y=(0.9)^{\frac{t}{3}}$ in terms of a factor?

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

Flashcard 43: Use exponent rules to rewrite $y=(1.5)^{-t}$ with a positive exponent on the base.

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

Flashcard 44: What exponent property justifies rewriting $a^{\frac{t}{10}}$ as $(a^{\frac{1}{10}})^t$?

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

Flashcard 45: Identify the effective factor per 1 unit $t$ for $y=10\cdot(1.03)^{4t}$.

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

Flashcard 46: What is the factor $b$ for a $6\%$ decrease per period in $y=a\cdot b^t$?

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

Flashcard 47: Identify the percent rate of change per step for $y=7(0.6)^t$.

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

Flashcard 48: Use exponent rules to rewrite $4^{t+3}$ as a product of two powers of $4$.

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

Flashcard 49: Identify whether $y=(0.97)^t$ represents exponential growth or decay.

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

Flashcard 50: Classify $y=(0.5)^{2t}$ as exponential growth or decay.

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

Flashcard 51: What is the growth/decay factor $b$ in $y=a\cdot b^t$ for $y=5(1.08)^t$?

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

Flashcard 52: What is the percent rate of change for $y=(0.97)^t$ per 1 unit of $t$?

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

Flashcard 53: State the formula that converts percent rate $r$ to factor $b$ for growth per period.

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

Flashcard 54: What is the percent rate $r$ if $y=a(1.005)^t$ models change per period?

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

Flashcard 55: Identify the effective growth factor per 1 unit of $t$ for $y=(1.2)^{\frac{t}{10}}$.

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

Flashcard 56: What is the percent rate of change for $y=a(1)^t$?

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

Flashcard 57: Use exponent rules to rewrite $2^{t-5}$ as a quotient of two powers of $2$.

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

Flashcard 58: State the formula that converts percent rate $r$ to factor $b$ for growth per period.

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

Flashcard 59: Use exponent rules to rewrite $y=(1.2)^{\frac{t}{10}}$ in the form y=\left(<span class="fill-in-blank">&nbsp;</span>\right)^t.

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

Flashcard 60: Classify $y=(0.99)^{12t}$ as exponential growth or decay.

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

Flashcard 61: Use exponent rules to rewrite $y=a\cdot b^{ct}$ as y=a\cdot(<span class="fill-in-blank">&nbsp;</span>)^t.

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

Flashcard 62: Identify the effective growth factor per 1 unit of $t$ for $y=(1.01)^{12t}$.

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

Flashcard 63: Identify whether $y=(1.02)^t$ represents exponential growth or decay.

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

Flashcard 64: What is the percent change per unit $t$ for $y=(0.9)^{\frac{t}{3}}$ in terms of a factor?

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

Flashcard 65: Use exponent rules to rewrite $y=a\cdot b^{\frac{t}{c}}$ as y=a\cdot(<span class="fill-in-blank">&nbsp;</span>)^t.

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

Flashcard 66: What is the percent rate of change per unit $t$ for $y=a(1.04)^{\frac{t}{2}}$ in factor form?

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

Flashcard 67: What is the initial value $a$ in $y=a\cdot b^t$ for $y=5(1.08)^t$?

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

Flashcard 68: Identify the effective growth factor per 1 unit of $t$ for $y=(1.01)^{12t}$.

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

Flashcard 69: Which value of $b$ gives no percent change in $y=a\cdot b^t$?

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

Flashcard 70: Use exponent rules to rewrite $y=(0.5)^{2t}$ in the form y=(<span class="fill-in-blank">&nbsp;</span>)^t.

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

Flashcard 71: Classify $y=(0.5)^{2t}$ as exponential growth or decay.

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

Flashcard 72: What is the growth/decay factor $b$ in $y=a\cdot b^t$ for $y=5(1.08)^t$?

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

Flashcard 73: Classify $y=(0.99)^{12t}$ as exponential growth or decay.

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

Flashcard 74: Use exponent rules to simplify $(5^{2t})^3$ to a single exponent on base $5$.

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

Flashcard 75: What is the percent rate of change for $y=(0.97)^t$ per 1 unit of $t$?

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

Flashcard 76: State the formula that converts percent rate $r$ to factor $b$ for decay per period.

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

Flashcard 77: Classify $y=10\cdot(1.03)^{4t}$ as exponential growth or decay.

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

Flashcard 78: Use exponent rules to rewrite $y=2\cdot(3^t)^4$ as y=2\cdot(<span class="fill-in-blank">&nbsp;</span>)^t.

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

Flashcard 79: What is the percent rate $r$ if $y=a(0.92)^t$ models change per period?

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

Flashcard 80: What is the percent rate $r$ if $y=a(0.92)^t$ models change per period?

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

Flashcard 81: What exponent property justifies rewriting $a^{kt}$ as $(a^k)^t$?

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

Flashcard 82: Use exponent rules to rewrite $y=2\cdot(3^t)^4$ as y=2\cdot(<span class="fill-in-blank">&nbsp;</span>)^t.

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

Flashcard 83: What exponent property rewrites $a^{-t}$ as $\left(\frac{1}{a}\right)^t$?

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

Flashcard 84: What exponent property justifies rewriting $a^{\frac{t}{10}}$ as $(a^{\frac{1}{10}})^t$?

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

Flashcard 85: Identify the percent increase per period for $y=a(1.2)^t$.

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

Flashcard 86: Identify whether $y=(0.97)^t$ represents exponential growth or decay.

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

Flashcard 87: Identify whether $y=(1.02)^t$ represents exponential growth or decay.

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

Flashcard 88: What is the percent rate of change for $y=(1.02)^t$ per 1 unit of $t$?

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

Flashcard 89: What exponent property is used to rewrite $a^{m-n}$ as $\frac{a^m}{a^n}$?

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

Flashcard 90: What exponent property rewrites $a^{-t}$ as $\left(\frac{1}{a}\right)^t$?

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

Flashcard 91: Identify the percent rate of change per step for $y=7(1.25)^t$.

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

Flashcard 92: Use exponent rules to rewrite $y=(1.01)^{12t}$ in the form y=\left( \text{<span class="fill-in-blank">&nbsp;</span>} \right)^t.

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