Algebra 2 Flashcards: Constructing Linear And Exponential Functions

What this deck covers

This deck focuses on Constructing Linear And Exponential Functions, 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: Identify the function type if the table has a constant ratio $\frac{y_{k+1}}{y_k}$ for equal steps in $x$.

Answer: Exponential. Constant ratios indicate exponential relationship.

Flashcard 2: What is the linear function through points $(0,-3)$ and $(4,5)$ in slope-intercept form?

Answer: $y=2x-3$. Initial value $-3$, slope $\frac{5-(-3)}{4-0} = 2$.

Flashcard 3: What is the linear function through points $(1,7)$ and $(3,11)$ in slope-intercept form?

Answer: $y=2x+5$. Slope $\frac{11-7}{3-1} = 2$, $y$-intercept when $x=0$ is $5$.

Flashcard 4: What is the exponential function for “starts at $50$ and increases by $3\%$ per year”?

Answer: $y=50\cdot(1.03)^x$. Initial $50$, growth factor $1 + 0.03 = 1.03$.

Flashcard 5: What is the exponential function for “starts at $200$ and decreases by $5\%$ per month”?

Answer: $y=200\cdot(0.95)^x$. Initial $200$, decay factor $1 - 0.05 = 0.95$.

Flashcard 6: What is the linear function for “starts at $12$ and increases by $4$ each step”?

Answer: $y=4x+12$. Linear with slope $4$ and $y$-intercept $12$.

Flashcard 7: What is the linear function for “starts at $30$ and decreases by $2$ each hour”?

Answer: $y=-2x+30$. Linear with slope $-2$ and $y$-intercept $30$.

Flashcard 8: What is the common ratio $r$ for the geometric sequence $3,12,48,192,\dots$?

Answer: $r=4$. Each term multiplies by $\frac{12}{3} = 4$.

Flashcard 9: What is $a_6$ for the arithmetic sequence with $a_1=2$ and $d=5$?

Answer: $a_6=27$. $a_6 = 2 + (6-1) \cdot 5 = 2 + 25 = 27$.

Flashcard 10: What is $a_5$ for the geometric sequence with $a_1=7$ and $r=2$?

Answer: $a_5=112$. $a_5 = 7 \cdot 2^{5-1} = 7 \cdot 16 = 112$.

Flashcard 11: What is the linear function through points $(1,7)$ and $(3,11)$ in slope-intercept form?

Answer: $y=2x+5$. Slope $\frac{11-7}{3-1} = 2$, $y$-intercept when $x=0$ is $5$.

Flashcard 12: What is the constant ratio property of a geometric sequence?

Answer: $\frac{a_{n+1}}{a_n}=r$ (constant, $a_n\neq 0$). Consecutive terms have the same ratio $r$.

Flashcard 13: What is the explicit formula for an arithmetic sequence with first term $a_1$ and difference $d$?

Answer: $a_n=a_1+(n-1)d$. Start with $a_1$, add $d$ for each step to term $n$.

Flashcard 14: Identify the function type if the graph is a straight line with constant slope.

Answer: Linear. Straight lines have constant slope (linear functions).

Flashcard 15: Identify the function type if the graph curves and multiplies by a constant factor each $1$ step in $x$.

Answer: Exponential. Curved growth by constant factors indicates exponential.

Flashcard 16: What is the slope of the line passing through $(0,4)$ and $(5,4)$?

Answer: $m=0$. Horizontal line has zero slope (no rise).

Flashcard 17: What is $a$ in $y=a\cdot b^x$ if the function passes through $(0,-8)$?

Answer: $a=-8$. The initial value when $x = 0$ is $a$.

Flashcard 18: What is $b$ if $y=a\cdot b^x$ passes through $(0,2)$ and $(3,16)$?

Answer: $b=2$. $a = 2$, $16 = 2 \cdot b^3$, so $b^3 = 8$ and $b = 2$.

Flashcard 19: What is the linear function through $(2,0)$ and $(0,6)$ in slope-intercept form?

Answer: $y=-3x+6$. Slope $\frac{0-6}{2-0} = -3$, $y$-intercept $6$.

Flashcard 20: What is the exponential function through $(0,4)$ and $(2,36)$ in the form $y=a\cdot b^x$?

Answer: $y=4\cdot 3^x$. $a = 4$, $36 = 4 \cdot b^2$, so $b^2 = 9$ and $b = 3$.

Flashcard 21: What is the slope of a line that rises $6$ units when $x$ increases by $2$?

Answer: $m=3$. Slope equals rise over run: $\frac{6}{2} = 3$.

Flashcard 22: What is the slope of a line that falls $10$ units when $x$ increases by $5$?

Answer: $m=-2$. Slope equals rise over run: $\frac{-10}{5} = -2$.

Flashcard 23: Identify the function type if the table has a constant first difference in $y$ for equal steps in $x$.

Answer: Linear. Constant differences indicate linear relationship.

Flashcard 24: What is the slope of the line through $(2,5)$ and $(6,13)$?

Answer: $m=2$. $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.

Flashcard 25: What is the linear function with slope $m=-4$ and passing through $(2,1)$?

Answer: $y=-4x+9$. Using point-slope form with $(2,1)$ and $m=-4$.

Flashcard 26: What is the $y$-intercept of the line $3x+2y=8$?

Answer: $b=4$. Solve $3(0) + 2y = 8$ to get $y = 4$.

Flashcard 27: What is the slope of the line $5x-10y=20$?

Answer: $m=\frac{1}{2}$. Rewrite as $y = \frac{1}{2}x - 2$, so slope is $\frac{1}{2}$.

Flashcard 28: What is the exponential function through $(0,6)$ and $(1,15)$ in the form $y=a\cdot b^x$?

Answer: $y=6\cdot\left(\frac{5}{2}\right)^x$. $a = 6$ from $(0,6)$, $b = \frac{15}{6} = \frac{5}{2}$.

Flashcard 29: What is the exponential function through $(0,10)$ and $(2,40)$ in the form $y=a\cdot b^x$?

Answer: $y=10\cdot 2^x$. $a = 10$, $b^2 = 4$ so $b = 2$.

Flashcard 30: What is the exponential function through $(1,12)$ and $(3,48)$ in the form $y=a\cdot b^x$?

Answer: $y=6\cdot 2^x$. From ratio $\frac{48}{12} = 4 = 2^2$, so $b = 2$, $a = 6$.