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: 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 2: What is the difference between consecutive outputs for $f(x)=mx+b$ when $x$ increases by $h$?

Answer: $mh$. Linear functions change by slope times interval length.

Flashcard 3: What is the slope $m$ in terms of two points $(x_1,y_1)$ and $(x_2,y_2)$ on a line?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Standard slope formula using rise over run.

Flashcard 4: 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 5: What is $\frac{f(x+1)}{f(x)}$ for $f(x)=ab^x$ after simplification?

Answer: $b$. Simplifying $\frac{ab^{x+1}}{ab^x}$ gives $b$.

Flashcard 6: Find $m$ if $f(3)-f(-1)=16$ for a linear function.

Answer: $4$. Using $\frac{\text{difference}}{\text{interval}} = \frac{16}{4} = 4$.

Flashcard 7: What does "equal factors over equal intervals" mean for a function table?

Answer: For fixed $h$, $\frac{f(x+h)}{f(x)}$ is constant. The ratio between outputs stays the same for equal input gaps.

Flashcard 8: 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 9: 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 10: Find $f(x+5)-f(x)$ for $f(x)=2x-1$.

Answer: $10$. Difference over 5 units is $5 \times 2 = 10$.

Flashcard 11: Find $f(x+1)-f(x)$ for $f(x)=\left(\frac{1}{3}\right)x-2$.

Answer: $\frac{1}{3}$. The slope is the coefficient of $x$, which is $\frac{1}{3}$.

Flashcard 12: 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 13: Identify the constant difference for $f(x)=7x-3$ over a $1$-unit interval.

Answer: $7$. The coefficient $7$ is the slope, giving constant difference $7$.

Flashcard 14: 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 15: Find $f(x+1)-f(x)$ for $f(x)=3x+8$.

Answer: $3$. Linear functions have constant difference equal to the slope.

Flashcard 16: Which value is constant for exponential functions: $\frac{\Delta y}{\Delta x}$ or $\frac{y_2}{y_1}$ over equal $\Delta x$?

Answer: $\frac{y_2}{y_1}$ over equal $\Delta x$. Exponential functions have constant ratios over equal input intervals.

Flashcard 17: Find $\frac{f(x+3)}{f(x)}$ for $f(x)=2\cdot \left(\frac{1}{2}\right)^x$.

Answer: $\frac{1}{8}$. Factor over 3 units is $(\frac{1}{2})^3 = \frac{1}{8}$.

Flashcard 18: 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 19: What is the constant ratio over $h$ units for $f(x)=ab^x$?

Answer: $\frac{f(x+h)}{f(x)}=b^h$. Generalizes to any interval $h$ with factor $b^h$.

Flashcard 20: Identify the function type if $\frac{f(x+3)}{f(x)}$ is constant for all $x$ and $f(x)\neq 0$.

Answer: Exponential. Constant ratio over any interval indicates exponential growth.

Flashcard 21: 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 22: 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 23: Find $b$ if $\frac{f(5)}{f(2)}=27$ for an exponential function.

Answer: $3$. Since $b^3 = 27$ and $3^3 = 27$, base is $3$.

Flashcard 24: Find $\frac{f(x+1)}{f(x)}$ for $f(x)=9\cdot 1.1^x$.

Answer: $1.1$. Consecutive exponential values have ratio equal to base $1.1$.

Flashcard 25: 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 26: 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 27: 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 28: What is the factor between outputs for $f(x)=ab^x$ when $x$ increases by $h$?

Answer: $b^h$. Exponential functions multiply by base raised to interval length.

Flashcard 29: What does "equal differences over equal intervals" mean for a function table?

Answer: For fixed $h$, $f(x+h)-f(x)$ is constant. The difference between outputs stays the same for equal input gaps.

Flashcard 30: 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 31: Find $\frac{f(x+1)}{f(x)}$ for $f(x)=12\cdot \left(\frac{5}{2}\right)^x$.

Answer: $\frac{5}{2}$. The base is $\frac{5}{2}$ from the exponential form.

Flashcard 32: 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 33: 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 34: 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 35: 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 36: Identify the function type if $f(x+2)-f(x)$ is constant for all $x$.

Answer: Linear. Constant difference over any interval indicates linear growth.

Flashcard 37: 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 38: 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 39: Find $f(x+3)-f(x)$ for $f(x)=-6x+4$.

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

Flashcard 40: What is the slope $m$ in terms of two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Standard slope formula using rise over run.

Flashcard 41: Which test identifies linear growth in a table: constant $f(x+1)-f(x)$ or constant $\frac{f(x+1)}{f(x)}$?

Answer: Constant $f(x+1)-f(x)$. Linear functions have constant differences between consecutive outputs.

Flashcard 42: 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 43: 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 44: 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 45: 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 46: 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 47: 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 48: 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 49: 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 50: Identify the interval length $h$ used in $\frac{f(x+h)}{f(x)}$ when comparing $x=-1$ to $x=3$.

Answer: $4$. Distance from $-1$ to $3$ is $3 - (-1) = 4$ units.

Flashcard 51: 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 52: Find $\frac{f(x+2)}{f(x)}$ for $f(x)=7\cdot 3^x$.

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