Algebra 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.

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 exponential function if it starts at $50$ and decreases by $10\%$ each step?

Answer: $y=50\cdot 0.9^x$. Initial value 50 with decay factor $1-0.10=0.9$.

Flashcard 2: What is $a_5$ for the geometric sequence with $a_1=3$ and $r=2$?

Answer: $a_5=48$. Use formula: $a_5=3\cdot 2^{5-1}=3\cdot 16=48$.

Flashcard 3: What is the linear function through $(2,5)$ with slope $3$?

Answer: $y=3x-1$. Use point-slope form: $y-5=3(x-2)$, then simplify.

Flashcard 4: What is the linear function with slope $4$ that passes through $(3,-2)$?

Answer: $y=4x-14$. Use point-slope form: $y-(-2)=4(x-3)$, then simplify.

Flashcard 5: What is the initial value $a$ of $y=9\left(\frac{1}{3}\right)^x$?

Answer: $9$. The coefficient $a$ is the initial value when $x=0$.

Flashcard 6: What is the factor $b$ for an exponential function with $y(0)=6$ and $y(3)=48$?

Answer: $b=2$. Solve $6b^3=48$ to get $b^3=8$, so $b=2$.

Flashcard 7: What is the exponential function with $a=3$ and factor $b=2$?

Answer: $y=3\cdot 2^x$. Direct substitution into the exponential form $y=a\cdot b^x$.

Flashcard 8: In $y=a\cdot b^x$, what does $a$ represent when $x=0$?

Answer: $a=y(0)$. When $x=0$, $b^0=1$, so $y=a\cdot 1=a$.

Flashcard 9: What is $a_5$ for the geometric sequence with $a_1=3$ and $r=2$?

Answer: $a_5=48$. Use formula: $a_5=3\cdot 2^{5-1}=3\cdot 16=48$.

Flashcard 10: What is the explicit formula for a geometric sequence with first term $a_1$ and ratio $r$?

Answer: $a_n=a_1\cdot r^{n-1}$. General term formula using first term and common ratio.

Flashcard 11: What is an explicit formula for the arithmetic sequence $7,12,17,22,\dots$?

Answer: $a_n=7+5(n-1)$. First term $a_1=7$ with common difference $d=5$.

Flashcard 12: What is the exponential function if it starts at $20$ and triples each step?

Answer: $y=20\cdot 3^x$. Initial value 20 with multiplicative factor 3 per step.

Flashcard 13: What is the slope of the line through $(1,4)$ and $(5,12)$?

Answer: $m=2$. Calculate $m=\frac{12-4}{5-1}=\frac{8}{4}=2$.

Flashcard 14: What is the factor $b$ for $12\%$ decay per step?

Answer: $b=0.88$. Calculate $b=1-0.12=0.88$ for 12% decay.

Flashcard 15: What is the slope-intercept form of a linear function?

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

Flashcard 16: What is the linear function through $(2,5)$ with slope $3$?

Answer: $y=3x-1$. Use point-slope form: $y-5=3(x-2)$, then simplify.

Flashcard 17: Identify whether $y=7-3x$ has slope $-3$ or $3$.

Answer: $-3$. Rewrite as $y=-3x+7$ to identify the slope coefficient.

Flashcard 18: What is the explicit formula for a geometric sequence with first term $a_1$ and ratio $r$?

Answer: $a_n=a_1\cdot r^{n-1}$. General term formula using first term and common ratio.

Flashcard 19: What is the exponential function through $(0,12)$ and $(2,3)$ in the form $y=a\cdot b^x$?

Answer: $y=12\cdot \left(\frac{1}{2}\right)^x$. Use $a=12$ and solve $12b^2=3$ to get $b=\frac{1}{2}$.

Flashcard 20: What is the exponential model for percent decay rate $r$ per step?

Answer: $y=a(1-r)^x$. Convert decay rate $r$ to multiplicative factor $1-r$.

Flashcard 21: Identify the slope of the linear function $y=mx+b$.

Answer: $m$. The coefficient of $x$ represents the rate of change.

Flashcard 22: What is the equation of the line through $(0,-3)$ and $(4,5)$?

Answer: $y=2x-3$. First find slope $m=2$, then use point $(0,-3)$ for y-intercept.

Flashcard 23: What condition on $b$ makes $y=a\cdot b^x$ exponential growth (assuming $a>0$)?

Answer: $b>1$. Base greater than 1 causes exponential increase.

Flashcard 24: What is the exponential function with $a=3$ and factor $b=2$?

Answer: $y=3\cdot 2^x$. Direct substitution into the exponential form $y=a\cdot b^x$.

Flashcard 25: What is the common difference $d$ of $-2,1,4,7,\dots$?

Answer: $d=3$. Calculate $d=1-(-2)=3$ from consecutive terms.

Flashcard 26: In $y=a\cdot b^x$, what does $b$ represent?

Answer: Growth/decay factor per $1$ unit of $x$. The multiplier that determines how $y$ changes per unit increase in $x$.

Flashcard 27: Identify whether the table pattern is linear or exponential: $y$ multiplies by $3$ each time $x$ increases by $1$.

Answer: Exponential. Constant multiplicative factor indicates exponential relationship.

Flashcard 28: What is the slope formula for two points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula for calculating slope between two points.

Flashcard 29: What is the linear function passing through $(5,1)$ and $(5,9)$?

Answer: $x=5$. Vertical line since both points have the same x-coordinate.

Flashcard 30: What is the general form of an exponential function with initial value $a$ and factor $b$?

Answer: $y=a\cdot b^x$. Standard exponential form with initial value $a$ and base $b$.

Flashcard 31: What is the common ratio definition for a geometric sequence?

Answer: Constant ratio: $\frac{a_{n+1}}{a_n}=r$. Each term is multiplied by the same constant ratio $r$.

Flashcard 32: What is the linear function rule if a table shows $(0,2)$ and $(3,11)$?

Answer: $y=3x+2$. Calculate slope $m=\frac{11-2}{3-0}=3$ and use y-intercept $b=2$.

Flashcard 33: What is the equation of the line with slope $-\frac{1}{2}$ and $y$-intercept $6$?

Answer: $y=-\frac{1}{2}x+6$. Direct substitution into slope-intercept form $y=mx+b$.

Flashcard 34: What is the slope formula for two points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula for calculating slope between two points.

Flashcard 35: What is an explicit formula for the geometric sequence $10,5,2.5,1.25,\dots$?

Answer: $a_n=10\left(\frac{1}{2}\right)^{n-1}$. First term $a_1=10$ with common ratio $r=\frac{1}{2}$.

Flashcard 36: What is the exponential function rule if a table shows $(0,6)$ and $(2,24)$?

Answer: $y=6\cdot 2^x$. Use $a=6$ and solve $6b^2=24$ to get $b=2$.

Flashcard 37: What is the linear function passing through $(5,1)$ and $(5,9)$?

Answer: $x=5$. Vertical line since both points have the same x-coordinate.

Flashcard 38: What is the factor $b$ for $25\%$ growth per step?

Answer: $b=1.25$. Calculate $b=1+0.25=1.25$ for 25% growth.

Flashcard 39: What is the linear function with slope $4$ that passes through $(3,-2)$?

Answer: $y=4x-14$. Use point-slope form: $y-(-2)=4(x-3)$, then simplify.

Flashcard 40: Identify whether the sequence $81,27,9,3,\dots$ is arithmetic or geometric.

Answer: Geometric. Constant ratio of $\frac{1}{3}$ between consecutive terms.

Flashcard 41: What is the standard form of a linear equation?

Answer: $Ax+By=C$. General linear equation form where $A$, $B$, $C$ are constants.

Flashcard 42: Identify whether the sequence $5,9,13,17,\dots$ is arithmetic or geometric.

Answer: Arithmetic. Constant difference of 4 between consecutive terms.

Flashcard 43: What is the explicit formula for an arithmetic sequence with first term $a_1$ and difference $d$?

Answer: $a_n=a_1+(n-1)d$. General term formula using first term and common difference.

Flashcard 44: What is the equation of the line with slope $-\frac{1}{2}$ and $y$-intercept $6$?

Answer: $y=-\frac{1}{2}x+6$. Direct substitution into slope-intercept form $y=mx+b$.

Flashcard 45: What is the slope of a line that rises $9$ units while running $-3$ units?

Answer: $m=-3$. Slope equals rise over run: $m=\frac{9}{-3}=-3$.

Flashcard 46: What is the common difference definition for an arithmetic sequence?

Answer: Constant difference: $a_{n+1}-a_n=d$. Each term increases by the same constant amount $d$.

Flashcard 47: What is the average rate of change from $x=2$ to $x=6$ for points $(2,1)$ and $(6,9)$?

Answer: $2$. Average rate of change equals $\frac{9-1}{6-2}=\frac{8}{4}=2$.

Flashcard 48: What is the common difference $d$ of $-2,1,4,7,\dots$?

Answer: $d=3$. Calculate $d=1-(-2)=3$ from consecutive terms.

Flashcard 49: What is the point-slope form of a linear function through $(x_1,y_1)$ with slope $m$?

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

Flashcard 50: What is the exponential function through $(0,4)$ and $(1,10)$ in the form $y=a\cdot b^x$?

Answer: $y=4\cdot 2.5^x$. Use $a=4$ and calculate $b=\frac{10}{4}=2.5$.

Flashcard 51: What is the point-slope form of a linear function through $(x_1,y_1)$ with slope $m$?

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

Flashcard 52: What condition on $b$ makes $y=a\cdot b^x$ exponential growth (assuming $a>0$)?

Answer: $b>1$. Base greater than 1 causes exponential increase.

Flashcard 53: What is the average rate of change from $x=2$ to $x=6$ for points $(2,1)$ and $(6,9)$?

Answer: $2$. Average rate of change equals $\frac{9-1}{6-2}=\frac{8}{4}=2$.

Flashcard 54: What is the slope of the line through $(1,4)$ and $(5,12)$?

Answer: $m=2$. Calculate $m=\frac{12-4}{5-1}=\frac{8}{4}=2$.

Flashcard 55: Identify the slope of the linear function $y=mx+b$.

Answer: $m$. The coefficient of $x$ represents the rate of change.

Flashcard 56: Identify whether $y=7-3x$ has slope $-3$ or $3$.

Answer: $-3$. Rewrite as $y=-3x+7$ to identify the slope coefficient.

Flashcard 57: What is $a_6$ for the arithmetic sequence with $a_1=4$ and $d=-3$?

Answer: $a_6=-11$. Use formula: $a_6=4+(6-1)(-3)=4-15=-11$.

Flashcard 58: What is the common ratio $r$ of $2,6,18,54,\dots$?

Answer: $r=3$. Calculate $r=\frac{6}{2}=3$ from consecutive terms.

Flashcard 59: What is the common ratio definition for a geometric sequence?

Answer: Constant ratio: $\frac{a_{n+1}}{a_n}=r$. Each term is multiplied by the same constant ratio $r$.

Flashcard 60: What is the linear function passing through $(2,7)$ and $(6,7)$?

Answer: $y=7$. Horizontal line since both points have the same y-coordinate.

Flashcard 61: What is the linear function passing through $(2,7)$ and $(6,7)$?

Answer: $y=7$. Horizontal line since both points have the same y-coordinate.

Flashcard 62: What is the factor $b$ of $y=5\cdot 0.2^x$?

Answer: $0.2$. The base $b$ is the multiplicative factor per unit increase.

Flashcard 63: What is the exponential model for percent growth rate $r$ per step?

Answer: $y=a(1+r)^x$. Convert growth rate $r$ to multiplicative factor $1+r$.

Flashcard 64: What is the recursive formula for a geometric sequence with ratio $r$?

Answer: $a_n=a_{n-1}\cdot r$. Each term equals the previous term times the common ratio.

Flashcard 65: What is the general form of an exponential function with initial value $a$ and factor $b$?

Answer: $y=a\cdot b^x$. Standard exponential form with initial value $a$ and base $b$.

Flashcard 66: In $y=a\cdot b^x$, what does $a$ represent when $x=0$?

Answer: $a=y(0)$. When $x=0$, $b^0=1$, so $y=a\cdot 1=a$.

Flashcard 67: In $y=a\cdot b^x$, what does $b$ represent?

Answer: Growth/decay factor per $1$ unit of $x$. The multiplier that determines how $y$ changes per unit increase in $x$.

Flashcard 68: What is the standard form of a linear equation?

Answer: $Ax+By=C$. General linear equation form where $A$, $B$, $C$ are constants.

Flashcard 69: What is the slope of a line that rises $9$ units while running $-3$ units?

Answer: $m=-3$. Slope equals rise over run: $m=\frac{9}{-3}=-3$.

Flashcard 70: What is the factor $b$ of $y=5\cdot 0.2^x$?

Answer: $0.2$. The base $b$ is the multiplicative factor per unit increase.

Flashcard 71: What is the exponential function if it starts at $50$ and decreases by $10\%$ each step?

Answer: $y=50\cdot 0.9^x$. Initial value 50 with decay factor $1-0.10=0.9$.

Flashcard 72: What is the equation of the line through $(0,-3)$ and $(4,5)$?

Answer: $y=2x-3$. First find slope $m=2$, then use point $(0,-3)$ for y-intercept.

Flashcard 73: What condition on $b$ makes $y=a\cdot b^x$ exponential decay (assuming $a>0$)?

Answer: $0<b<1$. Base between 0 and 1 causes exponential decrease.

Flashcard 74: What is $a_6$ for the arithmetic sequence with $a_1=4$ and $d=-3$?

Answer: $a_6=-11$. Use formula: $a_6=4+(6-1)(-3)=4-15=-11$.

Flashcard 75: What is the common ratio $r$ of $2,6,18,54,\dots$?

Answer: $r=3$. Calculate $r=\frac{6}{2}=3$ from consecutive terms.

Flashcard 76: Identify the $y$-intercept of the linear function $y=mx+b$.

Answer: $b$. The constant term represents where the line crosses the y-axis.

Flashcard 77: Identify whether the table pattern is linear or exponential: $y$ increases by $-4$ each time $x$ increases by $1$.

Answer: Linear. Constant additive change indicates linear relationship.

Flashcard 78: What is the exponential model for percent growth rate $r$ per step?

Answer: $y=a(1+r)^x$. Convert growth rate $r$ to multiplicative factor $1+r$.

Flashcard 79: Identify whether the table pattern is linear or exponential: $y$ multiplies by $3$ each time $x$ increases by $1$.

Answer: Exponential. Constant multiplicative factor indicates exponential relationship.

Flashcard 80: What is the recursive formula for an arithmetic sequence with difference $d$?

Answer: $a_n=a_{n-1}+d$. Each term equals the previous term plus the common difference.

Flashcard 81: What is the factor $b$ for an exponential function with $y(0)=6$ and $y(3)=48$?

Answer: $b=2$. Solve $6b^3=48$ to get $b^3=8$, so $b=2$.

Flashcard 82: What condition on $b$ makes $y=a\cdot b^x$ exponential decay (assuming $a>0$)?

Answer: $0<b<1$. Base between 0 and 1 causes exponential decrease.

Flashcard 83: Identify the $y$-intercept of the linear function $y=mx+b$.

Answer: $b$. The constant term represents where the line crosses the y-axis.

Flashcard 84: What is the exponential function rule if a table shows $(0,6)$ and $(2,24)$?

Answer: $y=6\cdot 2^x$. Use $a=6$ and solve $6b^2=24$ to get $b=2$.