PSAT Math Flashcards: Linear And Exponential Growth

What this deck covers

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

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 slope of the line $4y=-2x+8$.

Answer: $m=-\frac{1}{2}$. Rewrite as $y=-\frac{1}{2}x+2$ to identify slope.

Flashcard 2: What is the constant difference for the sequence $4,9,14,19,\dots$?

Answer: $5$. Each term increases by $5$ from the previous term.

Flashcard 3: If $y=5\cdot 2^x$, what is $y$ when $x=3$?

Answer: $40$. Calculate $5\cdot 2^3=5\cdot 8=40$.

Flashcard 4: Which model is appropriate when the amount changes by a constant ratio each step: linear or exponential?

Answer: Exponential. Constant ratio between consecutive terms indicates exponential growth.

Flashcard 5: What does the slope $m$ represent in $y=mx+b$?

Answer: Rate of change: rac{\Delta y}{\Delta x}. Slope measures vertical change per unit horizontal change.

Flashcard 6: If a quantity increases by $15\%$ each step, what is the factor $b$?

Answer: $b=1.15$. $b=1+0.15=1.15$ for 15% increase.

Flashcard 7: Find the $y$-intercept of $y=3x-7$.

Answer: $b=-7$. The constant term in slope-intercept form is the $y$-intercept.

Flashcard 8: A value starts at $80$ and is multiplied by $1.25$ each step. What is the value after $3$ steps?

Answer: $156.25$. $80\times 1.25^3 = 80\times 1.953125 = 156.25$

Flashcard 9: A quantity is multiplied by $1.08$ each year. What is the yearly percent change?

Answer: $8\%$ increase. Factor $1.08=1+0.08$ means $8\%$ growth.

Flashcard 10: Identify the initial value in $y=a\cdot b^x$ (the value when $x=0$).

Answer: $a$. When $x=0$, $b^0=1$, leaving only $a$.

Flashcard 11: What is the initial value of $y=a\cdot b^x$ at $x=0$?

Answer: $y(0)=a$. Since $b^0=1$, the function equals $a$ when $x=0$.

Flashcard 12: Which statement defines exponential growth in a table: constant difference or constant ratio?

Answer: Constant ratio. Exponential sequences have equal ratios between consecutive terms.

Flashcard 13: What is $f(3)$ for the linear model $f(n)=10+4n$?

Answer: $f(3)=22$. Substitute $n=3$: $f(3)=10+4(3)=10+12=22$.

Flashcard 14: Find the linear equation in $y=mx+b$ form with slope $3$ and $y$-intercept $-4$.

Answer: $y=3x-4$. Direct substitution into slope-intercept form.

Flashcard 15: What is the formula for slope using 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 between two points.

Flashcard 16: State the formula for a linear function with initial value $a$ and constant change $d$ per step.

Answer: $f(n)=a+dn$. Linear functions add a constant $d$ to initial value $a$ for each step $n$.

Flashcard 17: What is the recursive form for linear growth with initial value $a$ and constant change $d$?

Answer: $f(0)=a,\ f(n)=f(n-1)+d$. Each term equals the previous term plus constant $d$.

Flashcard 18: Find the next value in the exponential sequence $3,6,12,24,\ldots$.

Answer: $48$. Common ratio is $2$, so $24\times 2=48$.

Flashcard 19: What condition on $b$ in $y=a\cdot b^x$ gives exponential decay?

Answer: $0<b<1$. Values multiply by less than 1 each step.

Flashcard 20: What is the average rate of change of $f$ from $x=a$ to $x=b$?

Answer: $\frac{f(b)-f(a)}{b-a}$. Measures how much $f$ changes per unit of $x$ on average.

Flashcard 21: What does the slope $m$ represent in $y=mx+b$?

Answer: $m=\frac{\Delta y}{\Delta x}$ (constant rate of change). Change in $y$ over change in $x$ stays constant.

Flashcard 22: Which condition on $b$ in $y=a\cdot b^x$ gives exponential growth?

Answer: $b>1$. Values increase when the base exceeds 1.

Flashcard 23: Identify the constant ratio property that indicates exponential growth in a table.

Answer: Equal $y$-multipliers for equal $x$-steps. Exponential growth has constant ratios between consecutive $y$-values.

Flashcard 24: In $y=a\cdot b^x$, what condition on $b$ indicates growth?

Answer: Growth when $b>1$. Values multiply by more than 1 each step, causing increase.

Flashcard 25: What condition on $b$ in $y=a\cdot b^x$ gives exponential growth?

Answer: $b>1$. Values multiply by more than 1 each step.

Flashcard 26: 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 27: What value of $r$ indicates exponential growth in $y=a\cdot r^t$ (with $a>0$)?

Answer: $r>1$. Growth factor greater than 1 means the quantity increases.

Flashcard 28: If $f(n)=8\cdot(1.5)^n$, what is the growth factor and percent increase per step?

Answer: $r=1.5$, $50\%$ increase. Growth factor 1.5 means $(1.5-1)\times 100\%=50\%$ increase.

Flashcard 29: What is the slope of $3x+2y=10$?

Answer: $-\frac{3}{2}$. Rewrite as $y=-\frac{3}{2}x+5$; slope is coefficient of $x$.

Flashcard 30: If a quantity grows by $20\%$ each step, what is the growth factor $b$?

Answer: $b=1.2$. $b=1+0.20=1.2$ for 20% growth.

Flashcard 31: What is the explicit form for linear growth with initial value $a$ and constant change $d$?

Answer: $f(n)=a+dn$. Direct formula: start at $a$, add $d$ for each step.

Flashcard 32: What is the meaning of slope $m$ in $y=mx+b$?

Answer: $m=\frac{\Delta y}{\Delta x}$ (rate of change per $1$ in $x$). Slope measures vertical change per unit horizontal change.

Flashcard 33: In $y=a\cdot b^x$, what condition on $b$ indicates decay?

Answer: Decay when $0<b<1$. Values multiply by less than 1 each step, causing decrease.

Flashcard 34: A population is $200$ and grows by $5\%$ per year. What is the expression after $t$ years?

Answer: $200(1.05)^t$. $5\%$ growth means multiply by $1.05$ each year.

Flashcard 35: State the exponential model for initial value $a$ with growth rate $r$ per step.

Answer: $y=a(1+r)^x$. Growth rate $r$ means multiply by $(1+r)$ each step.

Flashcard 36: What is the relationship between percent rate $p$ and growth factor $r$ for growth?

Answer: $r=1+\frac{p}{100}$. Convert percent increase to decimal and add 1.

Flashcard 37: What condition on $r$ gives exponential decay in $f(n)=a\cdot r^n$?

Answer: $0<r<1$. Values decrease when multiplying by factor between 0 and 1.

Flashcard 38: Convert a $35\%$ decrease per step into an exponential multiplier $b$.

Answer: $b=0.65$. Subtract the decay rate from 1: $b=1-0.35=0.65$.

Flashcard 39: What condition on $r$ gives exponential growth in $f(n)=a\cdot r^n$?

Answer: $r>1$. Values increase when multiplying by factor greater than 1.

Flashcard 40: Find the value of $y$ for $y=10\cdot (2)^x$ when $x=3$.

Answer: $80$. $y=10\cdot 2^3=10\cdot 8=80$.

Flashcard 41: Find $b$ in $y=mx+b$ for the line with $m=2$ passing through $(3,7)$.

Answer: $b=1$. Substitute $(3,7)$: $7 = 2(3) + b$, so $b = 1$.

Flashcard 42: What factor $b$ corresponds to a decrease of $p\%$ per step?

Answer: $b=1-\frac{p}{100}$. Subtract the percent as a decimal from 1.

Flashcard 43: What is the percent rate $r$ if an exponential model uses factor $b$ each step?

Answer: $r=(b-1)\times 100\%$. Convert growth factor to percent: subtract 1, multiply by 100.

Flashcard 44: What is the meaning of $b$ in $y=mx+b$ when $x=0$?

Answer: The $y$-intercept (initial value). When $x=0$, $y=m(0)+b=b$, so $b$ is where the line crosses the $y$-axis.

Flashcard 45: State the percent-change form of an exponential model with initial value $a$ and rate $r$.

Answer: $f(t)=a(1+r)^t$. Growth factor $(1+r)$ represents $100\%$ plus the growth rate $r$.

Flashcard 46: Identify whether $f(n)=12+3n$ is linear or exponential.

Answer: Linear. Has form $a+dn$ with constant change per step.

Flashcard 47: What is the growth factor $b$ if a quantity decreases by $p\%$ each step?

Answer: b=1-rac{p}{100}. A $p\%$ decrease means multiplying by $(100-p)/100$.

Flashcard 48: What is the meaning of slope $m$ in $y=mx+b$?

Answer: $m=\frac{\Delta y}{\Delta x}$ (constant rate of change). Slope measures vertical change per horizontal unit.

Flashcard 49: What is the slope of the line $3x-2y=8$?

Answer: $\frac{3}{2}$. Rewrite as $y=\frac{3}{2}x-4$ to find slope.

Flashcard 50: What is the growth factor $r$ if the percent rate is $p\%$?

Answer: $r=1+\frac{p}{100}$. Add percent rate (as decimal) to 1 for growth factor.

Flashcard 51: What is the meaning of slope $m$ in $y=mx+b$?

Answer: $m=\frac{\Delta y}{\Delta x}$ (constant rate of change). Slope measures rise over run between any two points.

Flashcard 52: What is the formula for slope between $(x_1,y_1)$ and $(x_2,y_2)$?

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

Flashcard 53: If $y=5(1.2)^x$, what is the percent increase for each increase of $1$ in $x$?

Answer: $20\%$ increase. Factor $1.2$ means $20\%$ increase since $(1.2-1) imes 100\%=20\%$.

Flashcard 54: What is the meaning of $b$ in $y=mx+b$?

Answer: $b$ is the $y$-intercept (value when $x=0$). The y-intercept is where the line crosses the y-axis.

Flashcard 55: Identify the constant difference property that indicates linear growth in a table.

Answer: Equal $y$-differences for equal $x$-steps. Linear growth adds the same amount for equal $x$-intervals.

Flashcard 56: Find the percent change per step for $y=8\cdot 0.93^x$.

Answer: $-7\%$. $0.93 = 1 - 0.07$, so 7% decrease.

Flashcard 57: What is the recursive form for a geometric sequence with first term $a_1$ and ratio $r$?

Answer: $a_n=ra_{n-1}$ with $a_1$ given. Each term equals previous term times constant ratio.

Flashcard 58: What is the growth factor $b$ if a quantity increases by $p\%$ each step?

Answer: b=1+rac{p}{100}. A $p\%$ increase means multiplying by $(100+p)/100$.

Flashcard 59: What condition on $b$ gives exponential growth in $y=a(b)^x$ (with $a>0$)?

Answer: $b>1$. Values multiply by more than 1, causing growth.

Flashcard 60: A population decreases by $15\%$ each year. What is the exponential factor $b$?

Answer: $0.85$. Decreasing by $15\%$ means multiplying by $1-0.15=0.85$.

Flashcard 61: A quantity decays $15\%$ per step. What is the decay factor $r$?

Answer: $0.85$. Decay factor $=1-\frac{15}{100}=0.85$.

Flashcard 62: What is the general form of an exponential growth/decay function?

Answer: $y=a\cdot b^x$ with $b>0$ and $b\ne 1$. $a$ is initial value; $b$ is the growth/decay factor.

Flashcard 63: State the formula for an exponential function with initial value $a$ and growth factor $r$ per step.

Answer: $f(n)=a\cdot r^n$. Exponential functions multiply initial value $a$ by factor $r$ for each step $n$.

Flashcard 64: Which model fits: value increases by $5\%$ each year starting at $200$?

Answer: $V(t)=200\cdot 1.05^t$. Exponential model multiplies by $1.05$ (5% increase) each year.

Flashcard 65: Which model is linear and which is exponential: $y=3x+1$ and $y=3\cdot 1.1^x$?

Answer: $y=3x+1$ is linear; $y=3\cdot 1.1^x$ is exponential. Linear has $x$; exponential has $x$ in exponent.

Flashcard 66: What does the base $b$ represent in $y=a(b)^x$?

Answer: Multiplicative factor per $1$ increase in $x$. Each unit increase in $x$ multiplies $y$ by factor $b$.

Flashcard 67: If a quantity decays by $20\%$ each step, what is the decay factor $r$?

Answer: $r=0.8$. Decay by 20% means multiply by $1-0.2=0.8$.

Flashcard 68: What is the recursive form for exponential growth with initial value $a$ and growth factor $r$?

Answer: $f(0)=a,\ f(n)=r\,f(n-1)$. Each term equals the previous term times factor $r$.

Flashcard 69: State the formula for an exponential function with initial value $a$ and growth factor $b$ per step $t$.

Answer: $f(t)=ab^t$. Exponential functions multiply by factor $b$ for each unit increase in $t$.

Flashcard 70: What is the meaning of slope $m$ in $y=mx+b$?

Answer: $m=\frac{\Delta y}{\Delta x}$ (rate of change per $1$ in $x$). Slope measures vertical change per unit horizontal change.

Flashcard 71: Identify the model type: a quantity increases by $+7$ each step. Is it linear or exponential?

Answer: Linear. Constant addition indicates linear growth.

Flashcard 72: Find $f(5)$ for the linear function $f(x)=12-3x$.

Answer: $-3$. Substitute: $f(5)=12-3(5)=12-15=-3$.

Flashcard 73: Identify whether $y=2\cdot 5^x$ is linear or exponential.

Answer: Exponential. Has form $y = a \cdot b^x$ with variable raised to power.

Flashcard 74: What is the common ratio for the sequence $3,6,12,24,\dots$?

Answer: $2$. Each term is $2$ times the previous term.

Flashcard 75: What is the meaning of slope $m$ in $y=mx+b$?

Answer: $m=\frac{\Delta y}{\Delta x}$ (constant rate of change). Slope measures vertical change per unit horizontal change.

Flashcard 76: Which model fits the table $x:0,1,2$ and $y:3,6,12$?

Answer: Exponential. Values double each step ($\frac{6}{3}=\frac{12}{6}=2$), indicating exponential.

Flashcard 77: Identify the model type: $y=7x+3$ versus $y=7\cdot 3^x$.

Answer: $y=7x+3$ is linear; $y=7\cdot 3^x$ is exponential. First has constant difference; second has constant ratio.

Flashcard 78: Which condition on $b$ gives exponential growth in $y=a\,b^x$?

Answer: $b>1$. Values multiply by more than 1 each step.

Flashcard 79: What does the base $b$ indicate in $y=a\cdot b^x$?

Answer: $b>1$ growth; $0<b<1$ decay. Base greater than 1 multiplies; less than 1 divides.

Flashcard 80: Find $g(3)$ for the exponential function $g(n)=5\cdot 2^n$.

Answer: $40$. Substitute: $g(3)=5\cdot 2^3=5\cdot 8=40$.

Flashcard 81: What is the growth factor $r$ if the percent change each step is $p\%$?

Answer: $r=1+\frac{p}{100}$. Convert percent increase to decimal and add 1.

Flashcard 82: What is the general exponential growth/decay form with initial value $a$ and factor $b$?

Answer: $y=a\,b^x$. Standard exponential form with base $b$.

Flashcard 83: What is the meaning of slope $m$ in $y=mx+b$ for linear growth?

Answer: Constant rate of change per $1$ unit of $x$. Slope tells how much $y$ changes when $x$ increases by $1$.

Flashcard 84: What is the initial value of $y=7\cdot (0.8)^x$?

Answer: $7$. The coefficient $a=7$ is the initial value.

Flashcard 85: What is the meaning of $b$ in $y=mx+b$?

Answer: $b$ is the $y$-intercept (value when $x=0$). The y-intercept is where the line crosses the y-axis.

Flashcard 86: Which model is linear or exponential if values go $3,6,12,24$ for $t=0,1,2,3$?

Answer: Exponential. Each value doubles the previous, showing constant ratio of $2$.

Flashcard 87: What is the $y$-intercept $b$ of the line with slope $m=4$ passing through $(2,11)$?

Answer: $b=3$. Use $y=mx+b$: $11=4(2)+b$, so $b=11-8=3$.

Flashcard 88: Which function has the greater value at $x=5$: $y=3x+2$ or $y=2^x$?

Answer: $y=2^x$. At $x=5$: linear gives $17$, exponential gives $32$.

Flashcard 89: Which model is exponential: $y=7-2x$ or $y=7\cdot (0.8)^x$?

Answer: $y=7\cdot (0.8)^x$. Has form $y=a\cdot b^x$ with multiplicative change.

Flashcard 90: Find 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 91: Find $f(2)$ for the exponential function $f(t)=6\cdot 2^t$.

Answer: $f(2)=24$. Substitute $t=2$: $f(2)=6\cdot 2^2=6\cdot 4=24$.

Flashcard 92: What is the percent change per step if the exponential factor is $b$?

Answer: $(b-1) imes 100\%$. Convert factor to percent: subtract 1 and multiply by 100.

Flashcard 93: What is the general form of an exponential function showing growth or decay?

Answer: $y=a\cdot b^x$. Where $a$ is initial value and $b$ is growth/decay factor.

Flashcard 94: Which expression represents exponential growth with initial value $a$ and factor $r$?

Answer: $f(n)=a\cdot r^n$. Multiplies by factor $r$ for each unit increase in $n$.

Flashcard 95: Find $f(3)$ for the linear function $f(t)=12-4t$.

Answer: $f(3)=0$. Substitute $t=3$: $f(3)=12-4(3)=12-12=0$.

Flashcard 96: In $y=80\left(0.9\right)^t$, what is the initial value at $t=0$?

Answer: $80$. When $t=0$, any number to the zero power equals 1.

Flashcard 97: What is the formula for slope using 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 between two points.

Flashcard 98: What is the constant difference for the linear sequence $5,9,13,17,\dots$?

Answer: $d=4$. Each term increases by 4: $9-5=4$, $13-9=4$, etc.

Flashcard 99: If a quantity decreases by $15\%$ per step, what is the exponential factor $b$?

Answer: $0.85$. Decreasing by $15\%$ means keeping $85\%$, so $b=0.85$.

Flashcard 100: An amount is $200$ and increases by $5$ each day. What is the linear model after $d$ days?

Answer: $y=200+5d$. Start at 200, add 5 for each day (constant change).