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: 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 2: 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 3: What is the decay factor $b$ if an exponential model keeps $80\%$ each time $x$ increases by $1$?

Answer: $b=0.8$. Keeping $80\%$ means multiplying by $0.8$ each step.

Flashcard 4: What is the common difference $d$ for the arithmetic sequence $5,9,13,17,\dots$?

Answer: $d=4$. Each term increases by $9-5 = 4$.

Flashcard 5: 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 6: What is the standard form of a linear equation (with integer coefficients)?

Answer: $Ax+By=C$. Linear equation with integer coefficients $A$, $B$, $C$.

Flashcard 7: 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 8: 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 $b = 2$, $a = 6$.

Flashcard 9: 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 10: What is the initial value in $y=mx+b$?

Answer: $b$ (the $y$-intercept, value when $x=0$). Where the line crosses the $y$-axis.

Flashcard 11: 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 12: What is the explicit formula for a geometric sequence with first term $a_1$ and ratio $r$?

Answer: $a_n=a_1\cdot r^{n-1}$. Start with $a_1$, multiply by $r$ raised to $(n-1)$ power.

Flashcard 13: 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 14: What is the recursive formula for an arithmetic sequence with difference $d$?

Answer: $a_n=a_{n-1}+d$ (with a given $a_1$). Each term equals the previous term plus constant $d$.

Flashcard 15: What is $b$ in $y=a\cdot b^x$ if the function passes through $(0,9)$ and $(1,3)$?

Answer: $b=\frac{1}{3}$. At $(1,3)$: $3 = 9 \cdot b$, so $b = \frac{1}{3}$.

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 the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

Answer: $y-y_1=m(x-x_1)$. Uses a known point and slope to build the equation.

Flashcard 18: 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 19: What is the recursive formula for a geometric sequence with ratio $r$?

Answer: $a_n=r\cdot a_{n-1}$ (with a given $a_1$). Each term equals the previous term times constant $r$.

Flashcard 20: What condition on $b$ gives exponential growth in $y=a\cdot b^x$ (assuming $a>0$)?

Answer: $b>1$. Base greater than 1 means increasing exponential function.

Flashcard 21: 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 22: In $y=a\cdot b^x$, what does $b$ represent?

Answer: $b$ is the growth/decay factor per $1$ unit of $x$. How much the output multiplies for each unit increase in $x$.

Flashcard 23: What is the slope of the line passing through $(2,-1)$ and $(2,6)$?

Answer: Undefined slope (vertical line $x=2$). Vertical lines have undefined slope (zero run).

Flashcard 24: What is the $y$-intercept of the line $3x+2y=8$?

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

Flashcard 25: 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 26: 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 27: 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 28: 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 29: What is the slope-intercept form of a linear function?

Answer: $y=mx+b$. Standard linear form with slope $m$ and $y$-intercept $b$.

Flashcard 30: In $y=a\cdot b^x$, what does $a$ represent?

Answer: $a$ is the initial value $y(0)$. The output value when $x = 0$.

Flashcard 31: 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 32: What condition on $b$ gives exponential decay in $y=a\cdot b^x$ (assuming $a>0$)?

Answer: $0<b<1$. Base between 0 and 1 means decreasing exponential function.

Flashcard 33: What is the linear equation of the vertical line passing through $(3,0)$?

Answer: $x=3$. Vertical lines have form $x = \text{constant}$.

Flashcard 34: 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 35: 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 36: 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 37: Identify the function type if the graph is a straight line with constant slope.

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

Flashcard 38: 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 39: 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 40: What is the growth factor $b$ if an exponential model doubles each time $x$ increases by $1$?

Answer: $b=2$. Doubling means multiplying by $2$ each step.

Flashcard 41: What is the exponential function through $(0,81)$ and $(4,9)$ in the form $y=a\cdot b^x$?

Answer: $y=81\cdot\left(\frac{1}{\sqrt{3}}\right)^x$. $a = 81$, $b^4 = \frac{9}{81} = \frac{1}{9}$, so $b = \frac{1}{\sqrt{3}}$.

Flashcard 42: What is the constant difference property of an arithmetic sequence?

Answer: $a_{n+1}-a_n=d$ (constant). Consecutive terms have the same difference $d$.

Flashcard 43: 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 44: What is the slope formula $m$ using two points $(x_1,y_1)$ and $(x_2,y_2)$?

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

Flashcard 45: 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 46: What is the general form of an exponential function showing initial value and growth factor?

Answer: $y=a\cdot b^x$. Initial value $a$ times growth factor $b$ raised to power $x$.

Flashcard 47: What is the linear equation of the horizontal line passing through $(0,-7)$?

Answer: $y=-7$. Horizontal lines have form $y = \text{constant}$.

Flashcard 48: 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 49: What is the explicit formula for the geometric sequence with $a_1=5$ and $r=\frac{1}{3}$?

Answer: $a_n=5\cdot\left(\frac{1}{3}\right)^{n-1}$. Using the explicit geometric sequence formula.

Flashcard 50: 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 51: 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 52: What is the explicit formula for the arithmetic sequence with $a_1=-3$ and $d=2$?

Answer: $a_n=-3+2(n-1)$. Using the explicit arithmetic sequence formula.

Flashcard 53: What is the common ratio $r$ for the geometric sequence $3,12,48,192,\dots$?

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