Algebra 2 Flashcards: Compare Linear And Exponential Growth

What this deck covers

This deck focuses on Compare Linear And Exponential Growth, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

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: Find the constant difference over $4$ units if $m=-2$ for a linear function.

Answer: $-8$. Difference over 4 units is $4 \times (-2) = -8$.

Flashcard 2: Identify the interval length $h$ used in $f(x+h)-f(x)$ when comparing $x=2$ to $x=7$.

Answer: $5$. Distance from 2 to 7 is $7-2 = 5$ units.

Flashcard 3: Find $m$ if $f(10)-f(6)=20$ for a linear function.

Answer: $5$. Using $\frac{\text{difference}}{\text{interval}} = \frac{20}{4} = 5$.

Flashcard 4: Identify the function type if outputs increase by $+6$ each time $x$ increases by $1$.

Answer: Linear. Adding constant amounts indicates linear growth pattern.

Flashcard 5: Identify the function type if outputs are multiplied by $1.25$ each time $x$ increases by $1$.

Answer: Exponential. Multiplying by constant factors indicates exponential growth pattern.

Flashcard 6: Find the constant factor over $2$ units if $b=3$ for an exponential function.

Answer: $9$. Factor over 2 units is $b^2 = 3^2 = 9$.

Flashcard 7: Which value is constant for linear functions: $\frac{\Delta y}{\Delta x}$ or $\frac{y_2}{y_1}$?

Answer: $\frac{\Delta y}{\Delta x}$. Linear functions have constant slope (rate of change).

Flashcard 8: What is the general form of a linear function used to show equal differences?

Answer: $f(x)=mx+b$. Standard linear form where $m$ is the slope (constant difference).

Flashcard 9: Identify the constant factor for $f(x)=5\cdot 2^x$ over a $1$-unit interval.

Answer: $2$. The base $2$ is the constant factor per unit interval.

Flashcard 10: What expression gives the constant change over $1$ unit for $f(x)=mx+b$?

Answer: $f(x+1)-f(x)=m$. Shows the slope $m$ as the constant difference per unit.

Flashcard 11: What is $f(x+1)-f(x)$ for $f(x)=mx+b$ after simplification?

Answer: $m$. Expanding $(mx+m+b)-(mx+b)$ gives $m$.

Flashcard 12: Identify the constant difference for $f(x)=-4x+10$ over a 1-unit interval.

Answer: $-4$. The coefficient $-4$ is the slope, giving constant difference $-4$.

Flashcard 13: Which test identifies exponential growth in a table: constant $f(x+1)-f(x)$ or constant $\frac{f(x+1)}{f(x)}$?

Answer: Constant $\frac{f(x+1)}{f(x)}$. Exponential functions have constant ratios between consecutive outputs.

Flashcard 14: Choose the correct statement for linear functions: constant difference or constant ratio?

Answer: Constant difference. Linear functions add the same amount over equal intervals.

Flashcard 15: Find the slope $m$ if $f(x+1)-f(x)=12$ for all $x$.

Answer: $12$. If constant difference is 12, then slope $m = 12$.

Flashcard 16: Find the base $b$ if $\frac{f(x+1)}{f(x)}=0.8$ for all $x$ and $f(x)\neq 0$.

Answer: $0.8$. If constant ratio is 0.8, then base $b = 0.8$.

Flashcard 17: What expression gives the constant factor over $1$ unit for $f(x)=ab^x$?

Answer: $\frac{f(x+1)}{f(x)}=b$. Shows the base $b$ as the constant multiplicative factor per unit.

Flashcard 18: What is the key condition needed to use $\frac{f(x+h)}{f(x)}$ as a test?

Answer: $f(x)\neq 0$. Division by zero is undefined, so $f(x) \neq 0$ is required.

Flashcard 19: Find $b$ if $\frac{f(4)}{f(1)}=\frac{1}{8}$ for an exponential function.

Answer: $\frac{1}{2}$. Since $b^3 = \frac{1}{8}$ and $(\frac{1}{2})^3 = \frac{1}{8}$, base is $\frac{1}{2}$.

Flashcard 20: Choose the correct statement for exponential functions: constant difference or constant ratio?

Answer: Constant ratio. Exponential functions multiply by the same factor over equal intervals.

Flashcard 21: Identify the base $b$ if an exponential function doubles each time $x$ increases by $1$.

Answer: $2$. Doubling means multiplying by 2, so base $b = 2$.

Flashcard 22: Identify the base $b$ if an exponential function is halved each time $x$ increases by $1$.

Answer: $\frac{1}{2}$. Halving means multiplying by $\frac{1}{2}$, so base $b = \frac{1}{2}$.

Flashcard 23: What is the common ratio $b$ using two exponential values $f(x)$ and $f(x+1)$?

Answer: $b=\frac{f(x+1)}{f(x)}$. Consecutive exponential values have ratio equal to the base.

Flashcard 24: What is the constant difference over $h$ units for $f(x)=mx+b$?

Answer: $f(x+h)-f(x)=mh$. Generalizes to any interval $h$ with difference $mh$.

Flashcard 25: Identify the constant difference for $f(x)=\frac{1}{2}x+6$ over a $1$-unit interval.

Answer: $\frac{1}{2}$. The coefficient $\frac{1}{2}$ is the slope.

Flashcard 26: What is the general form of an exponential function used to show equal factors?

Answer: $f(x)=ab^x$. Standard exponential form where $b$ is the base (constant factor).

Flashcard 27: Find $f(x+3)-f(x)$ for $f(x)=-6x+4$.

Answer: $-18$. Difference over 3 units is $3 \times (-6) = -18$.

Flashcard 28: What expression gives the constant change over $1$ unit for $f(x)=mx+b$?

Answer: $f(x+1)-f(x)=m$. Shows the slope $m$ as the constant difference per unit.

Flashcard 29: Identify the constant factor for $f(x)=3\cdot \left(\frac{1}{4}\right)^x$ over $1$ unit.

Answer: $\frac{1}{4}$. The base $\frac{1}{4}$ is the constant factor per unit.

Flashcard 30: What is the general form of a linear function used to show equal differences?

Answer: $f(x)=mx+b$. Standard linear form where $m$ is the slope (constant difference).