Compare Linear and Exponential Growth - Algebra 2
Card 1 of 30
Find the constant difference over $4$ units if $m=-2$ for a linear function.
Find the constant difference over $4$ units if $m=-2$ for a linear function.
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$-8$. Difference over 4 units is $4 \times (-2) = -8$.
$-8$. Difference over 4 units is $4 \times (-2) = -8$.
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Identify the interval length $h$ used in $f(x+h)-f(x)$ when comparing $x=2$ to $x=7$.
Identify the interval length $h$ used in $f(x+h)-f(x)$ when comparing $x=2$ to $x=7$.
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$5$. Distance from 2 to 7 is $7-2 = 5$ units.
$5$. Distance from 2 to 7 is $7-2 = 5$ units.
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Find $m$ if $f(10)-f(6)=20$ for a linear function.
Find $m$ if $f(10)-f(6)=20$ for a linear function.
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$5$. Using $\frac{\text{difference}}{\text{interval}} = \frac{20}{4} = 5$.
$5$. Using $\frac{\text{difference}}{\text{interval}} = \frac{20}{4} = 5$.
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Identify the function type if outputs increase by $+6$ each time $x$ increases by $1$.
Identify the function type if outputs increase by $+6$ each time $x$ increases by $1$.
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Linear. Adding constant amounts indicates linear growth pattern.
Linear. Adding constant amounts indicates linear growth pattern.
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Identify the function type if outputs are multiplied by $1.25$ each time $x$ increases by $1$.
Identify the function type if outputs are multiplied by $1.25$ each time $x$ increases by $1$.
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Exponential. Multiplying by constant factors indicates exponential growth pattern.
Exponential. Multiplying by constant factors indicates exponential growth pattern.
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Find the constant factor over $2$ units if $b=3$ for an exponential function.
Find the constant factor over $2$ units if $b=3$ for an exponential function.
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$9$. Factor over 2 units is $b^2 = 3^2 = 9$.
$9$. Factor over 2 units is $b^2 = 3^2 = 9$.
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Which value is constant for linear functions: $\frac{\Delta y}{\Delta x}$ or $\frac{y_2}{y_1}$?
Which value is constant for linear functions: $\frac{\Delta y}{\Delta x}$ or $\frac{y_2}{y_1}$?
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$\frac{\Delta y}{\Delta x}$. Linear functions have constant slope (rate of change).
$\frac{\Delta y}{\Delta x}$. Linear functions have constant slope (rate of change).
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What is the general form of a linear function used to show equal differences?
What is the general form of a linear function used to show equal differences?
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$f(x)=mx+b$. Standard linear form where $m$ is the slope (constant difference).
$f(x)=mx+b$. Standard linear form where $m$ is the slope (constant difference).
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Identify the constant factor for $f(x)=5\cdot 2^x$ over a $1$-unit interval.
Identify the constant factor for $f(x)=5\cdot 2^x$ over a $1$-unit interval.
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$2$. The base $2$ is the constant factor per unit interval.
$2$. The base $2$ is the constant factor per unit interval.
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What expression gives the constant change over $1$ unit for $f(x)=mx+b$?
What expression gives the constant change over $1$ unit for $f(x)=mx+b$?
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$f(x+1)-f(x)=m$. Shows the slope $m$ as the constant difference per unit.
$f(x+1)-f(x)=m$. Shows the slope $m$ as the constant difference per unit.
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What is $f(x+1)-f(x)$ for $f(x)=mx+b$ after simplification?
What is $f(x+1)-f(x)$ for $f(x)=mx+b$ after simplification?
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$m$. Expanding $(mx+m+b)-(mx+b)$ gives $m$.
$m$. Expanding $(mx+m+b)-(mx+b)$ gives $m$.
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Identify the constant difference for $f(x)=-4x+10$ over a 1-unit interval.
Identify the constant difference for $f(x)=-4x+10$ over a 1-unit interval.
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$-4$. The coefficient $-4$ is the slope, giving constant difference $-4$.
$-4$. The coefficient $-4$ is the slope, giving constant difference $-4$.
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Which test identifies exponential growth in a table: constant $f(x+1)-f(x)$ or constant $\frac{f(x+1)}{f(x)}$?
Which test identifies exponential growth in a table: constant $f(x+1)-f(x)$ or constant $\frac{f(x+1)}{f(x)}$?
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Constant $\frac{f(x+1)}{f(x)}$. Exponential functions have constant ratios between consecutive outputs.
Constant $\frac{f(x+1)}{f(x)}$. Exponential functions have constant ratios between consecutive outputs.
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Choose the correct statement for linear functions: constant difference or constant ratio?
Choose the correct statement for linear functions: constant difference or constant ratio?
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Constant difference. Linear functions add the same amount over equal intervals.
Constant difference. Linear functions add the same amount over equal intervals.
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Find the slope $m$ if $f(x+1)-f(x)=12$ for all $x$.
Find the slope $m$ if $f(x+1)-f(x)=12$ for all $x$.
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$12$. If constant difference is 12, then slope $m = 12$.
$12$. If constant difference is 12, then slope $m = 12$.
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Find the base $b$ if $\frac{f(x+1)}{f(x)}=0.8$ for all $x$ and $f(x)\neq 0$.
Find the base $b$ if $\frac{f(x+1)}{f(x)}=0.8$ for all $x$ and $f(x)\neq 0$.
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$0.8$. If constant ratio is 0.8, then base $b = 0.8$.
$0.8$. If constant ratio is 0.8, then base $b = 0.8$.
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What expression gives the constant factor over $1$ unit for $f(x)=ab^x$?
What expression gives the constant factor over $1$ unit for $f(x)=ab^x$?
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$\frac{f(x+1)}{f(x)}=b$. Shows the base $b$ as the constant multiplicative factor per unit.
$\frac{f(x+1)}{f(x)}=b$. Shows the base $b$ as the constant multiplicative factor per unit.
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What is the key condition needed to use $\frac{f(x+h)}{f(x)}$ as a test?
What is the key condition needed to use $\frac{f(x+h)}{f(x)}$ as a test?
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$f(x)\neq 0$. Division by zero is undefined, so $f(x) \neq 0$ is required.
$f(x)\neq 0$. Division by zero is undefined, so $f(x) \neq 0$ is required.
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Find $b$ if $\frac{f(4)}{f(1)}=\frac{1}{8}$ for an exponential function.
Find $b$ if $\frac{f(4)}{f(1)}=\frac{1}{8}$ for an exponential function.
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$\frac{1}{2}$. Since $b^3 = \frac{1}{8}$ and $(\frac{1}{2})^3 = \frac{1}{8}$, base is $\frac{1}{2}$.
$\frac{1}{2}$. Since $b^3 = \frac{1}{8}$ and $(\frac{1}{2})^3 = \frac{1}{8}$, base is $\frac{1}{2}$.
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Choose the correct statement for exponential functions: constant difference or constant ratio?
Choose the correct statement for exponential functions: constant difference or constant ratio?
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Constant ratio. Exponential functions multiply by the same factor over equal intervals.
Constant ratio. Exponential functions multiply by the same factor over equal intervals.
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Identify the base $b$ if an exponential function doubles each time $x$ increases by $1$.
Identify the base $b$ if an exponential function doubles each time $x$ increases by $1$.
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$2$. Doubling means multiplying by 2, so base $b = 2$.
$2$. Doubling means multiplying by 2, so base $b = 2$.
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Identify the base $b$ if an exponential function is halved each time $x$ increases by $1$.
Identify the base $b$ if an exponential function is halved each time $x$ increases by $1$.
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$\frac{1}{2}$. Halving means multiplying by $\frac{1}{2}$, so base $b = \frac{1}{2}$.
$\frac{1}{2}$. Halving means multiplying by $\frac{1}{2}$, so base $b = \frac{1}{2}$.
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What is the common ratio $b$ using two exponential values $f(x)$ and $f(x+1)$?
What is the common ratio $b$ using two exponential values $f(x)$ and $f(x+1)$?
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$b=\frac{f(x+1)}{f(x)}$. Consecutive exponential values have ratio equal to the base.
$b=\frac{f(x+1)}{f(x)}$. Consecutive exponential values have ratio equal to the base.
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What is the constant difference over $h$ units for $f(x)=mx+b$?
What is the constant difference over $h$ units for $f(x)=mx+b$?
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$f(x+h)-f(x)=mh$. Generalizes to any interval $h$ with difference $mh$.
$f(x+h)-f(x)=mh$. Generalizes to any interval $h$ with difference $mh$.
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Identify the constant difference for $f(x)=\frac{1}{2}x+6$ over a $1$-unit interval.
Identify the constant difference for $f(x)=\frac{1}{2}x+6$ over a $1$-unit interval.
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$\frac{1}{2}$. The coefficient $\frac{1}{2}$ is the slope.
$\frac{1}{2}$. The coefficient $\frac{1}{2}$ is the slope.
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What is the general form of an exponential function used to show equal factors?
What is the general form of an exponential function used to show equal factors?
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$f(x)=ab^x$. Standard exponential form where $b$ is the base (constant factor).
$f(x)=ab^x$. Standard exponential form where $b$ is the base (constant factor).
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Find $f(x+3)-f(x)$ for $f(x)=-6x+4$.
Find $f(x+3)-f(x)$ for $f(x)=-6x+4$.
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$-18$. Difference over 3 units is $3 \times (-6) = -18$.
$-18$. Difference over 3 units is $3 \times (-6) = -18$.
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What expression gives the constant change over $1$ unit for $f(x)=mx+b$?
What expression gives the constant change over $1$ unit for $f(x)=mx+b$?
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$f(x+1)-f(x)=m$. Shows the slope $m$ as the constant difference per unit.
$f(x+1)-f(x)=m$. Shows the slope $m$ as the constant difference per unit.
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Identify the constant factor for $f(x)=3\cdot \left(\frac{1}{4}\right)^x$ over $1$ unit.
Identify the constant factor for $f(x)=3\cdot \left(\frac{1}{4}\right)^x$ over $1$ unit.
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$\frac{1}{4}$. The base $\frac{1}{4}$ is the constant factor per unit.
$\frac{1}{4}$. The base $\frac{1}{4}$ is the constant factor per unit.
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What is the general form of a linear function used to show equal differences?
What is the general form of a linear function used to show equal differences?
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$f(x)=mx+b$. Standard linear form where $m$ is the slope (constant difference).
$f(x)=mx+b$. Standard linear form where $m$ is the slope (constant difference).
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