Constructing Linear and Exponential Functions - Algebra
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Identify the $y$-intercept of the linear function $y=mx+b$.
Identify the $y$-intercept of the linear function $y=mx+b$.
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$b$. The constant term represents where the line crosses the y-axis.
$b$. The constant term represents where the line crosses the y-axis.
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What is the exponential model for percent decay rate $r$ per step?
What is the exponential model for percent decay rate $r$ per step?
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$y=a(1-r)^x$. Convert decay rate $r$ to multiplicative factor $1-r$.
$y=a(1-r)^x$. Convert decay rate $r$ to multiplicative factor $1-r$.
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In $y=a\cdot b^x$, what does $a$ represent when $x=0$?
In $y=a\cdot b^x$, what does $a$ represent when $x=0$?
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$a=y(0)$. When $x=0$, $b^0=1$, so $y=a\cdot 1=a$.
$a=y(0)$. When $x=0$, $b^0=1$, so $y=a\cdot 1=a$.
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Identify whether the table pattern is linear or exponential: $y$ increases by $-4$ each time $x$ increases by $1$.
Identify whether the table pattern is linear or exponential: $y$ increases by $-4$ each time $x$ increases by $1$.
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Linear. Constant additive change indicates linear relationship.
Linear. Constant additive change indicates linear relationship.
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Identify whether the table pattern is linear or exponential: $y$ multiplies by $3$ each time $x$ increases by $1$.
Identify whether the table pattern is linear or exponential: $y$ multiplies by $3$ each time $x$ increases by $1$.
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Exponential. Constant multiplicative factor indicates exponential relationship.
Exponential. Constant multiplicative factor indicates exponential relationship.
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What is the exponential function rule if a table shows $(0,6)$ and $(2,24)$?
What is the exponential function rule if a table shows $(0,6)$ and $(2,24)$?
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$y=6\cdot 2^x$. Use $a=6$ and solve $6b^2=24$ to get $b=2$.
$y=6\cdot 2^x$. Use $a=6$ and solve $6b^2=24$ to get $b=2$.
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What is the linear function rule if a table shows $(0,2)$ and $(3,11)$?
What is the linear function rule if a table shows $(0,2)$ and $(3,11)$?
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$y=3x+2$. Calculate slope $m=\frac{11-2}{3-0}=3$ and use y-intercept $b=2$.
$y=3x+2$. Calculate slope $m=\frac{11-2}{3-0}=3$ and use y-intercept $b=2$.
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What is an explicit formula for the geometric sequence $10,5,2.5,1.25,\dots$?
What is an explicit formula for the geometric sequence $10,5,2.5,1.25,\dots$?
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$a_n=10\left(\frac{1}{2}\right)^{n-1}$. First term $a_1=10$ with common ratio $r=\frac{1}{2}$.
$a_n=10\left(\frac{1}{2}\right)^{n-1}$. First term $a_1=10$ with common ratio $r=\frac{1}{2}$.
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What is an explicit formula for the arithmetic sequence $7,12,17,22,\dots$?
What is an explicit formula for the arithmetic sequence $7,12,17,22,\dots$?
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$a_n=7+5(n-1)$. First term $a_1=7$ with common difference $d=5$.
$a_n=7+5(n-1)$. First term $a_1=7$ with common difference $d=5$.
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What is $a_5$ for the geometric sequence with $a_1=3$ and $r=2$?
What is $a_5$ for the geometric sequence with $a_1=3$ and $r=2$?
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$a_5=48$. Use formula: $a_5=3\cdot 2^{5-1}=3\cdot 16=48$.
$a_5=48$. Use formula: $a_5=3\cdot 2^{5-1}=3\cdot 16=48$.
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What is $a_6$ for the arithmetic sequence with $a_1=4$ and $d=-3$?
What is $a_6$ for the arithmetic sequence with $a_1=4$ and $d=-3$?
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$a_6=-11$. Use formula: $a_6=4+(6-1)(-3)=4-15=-11$.
$a_6=-11$. Use formula: $a_6=4+(6-1)(-3)=4-15=-11$.
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What is the common ratio $r$ of $2,6,18,54,\dots$?
What is the common ratio $r$ of $2,6,18,54,\dots$?
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$r=3$. Calculate $r=\frac{6}{2}=3$ from consecutive terms.
$r=3$. Calculate $r=\frac{6}{2}=3$ from consecutive terms.
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What is the common difference $d$ of $-2,1,4,7,\dots$?
What is the common difference $d$ of $-2,1,4,7,\dots$?
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$d=3$. Calculate $d=1-(-2)=3$ from consecutive terms.
$d=3$. Calculate $d=1-(-2)=3$ from consecutive terms.
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Identify whether the sequence $81,27,9,3,\dots$ is arithmetic or geometric.
Identify whether the sequence $81,27,9,3,\dots$ is arithmetic or geometric.
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Geometric. Constant ratio of $\frac{1}{3}$ between consecutive terms.
Geometric. Constant ratio of $\frac{1}{3}$ between consecutive terms.
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Identify whether the sequence $5,9,13,17,\dots$ is arithmetic or geometric.
Identify whether the sequence $5,9,13,17,\dots$ is arithmetic or geometric.
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Arithmetic. Constant difference of 4 between consecutive terms.
Arithmetic. Constant difference of 4 between consecutive terms.
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What is the recursive formula for a geometric sequence with ratio $r$?
What is the recursive formula for a geometric sequence with ratio $r$?
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$a_n=a_{n-1}\cdot r$. Each term equals the previous term times the common ratio.
$a_n=a_{n-1}\cdot r$. Each term equals the previous term times the common ratio.
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What is the explicit formula for a geometric sequence with first term $a_1$ and ratio $r$?
What is the explicit formula for a geometric sequence with first term $a_1$ and ratio $r$?
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$a_n=a_1\cdot r^{n-1}$. General term formula using first term and common ratio.
$a_n=a_1\cdot r^{n-1}$. General term formula using first term and common ratio.
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What is the common ratio definition for a geometric sequence?
What is the common ratio definition for a geometric sequence?
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Constant ratio: $\frac{a_{n+1}}{a_n}=r$. Each term is multiplied by the same constant ratio $r$.
Constant ratio: $\frac{a_{n+1}}{a_n}=r$. Each term is multiplied by the same constant ratio $r$.
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What is the recursive formula for an arithmetic sequence with difference $d$?
What is the recursive formula for an arithmetic sequence with difference $d$?
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$a_n=a_{n-1}+d$. Each term equals the previous term plus the common difference.
$a_n=a_{n-1}+d$. Each term equals the previous term plus the common difference.
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What is the explicit formula for an arithmetic sequence with first term $a_1$ and difference $d$?
What is the explicit formula for an arithmetic sequence with first term $a_1$ and difference $d$?
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$a_n=a_1+(n-1)d$. General term formula using first term and common difference.
$a_n=a_1+(n-1)d$. General term formula using first term and common difference.
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What is the common difference definition for an arithmetic sequence?
What is the common difference definition for an arithmetic sequence?
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Constant difference: $a_{n+1}-a_n=d$. Each term increases by the same constant amount $d$.
Constant difference: $a_{n+1}-a_n=d$. Each term increases by the same constant amount $d$.
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What is the exponential function if it starts at $50$ and decreases by $10%$ each step?
What is the exponential function if it starts at $50$ and decreases by $10%$ each step?
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$y=50\cdot 0.9^x$. Initial value 50 with decay factor $1-0.10=0.9$.
$y=50\cdot 0.9^x$. Initial value 50 with decay factor $1-0.10=0.9$.
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What is the exponential function if it starts at $20$ and triples each step?
What is the exponential function if it starts at $20$ and triples each step?
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$y=20\cdot 3^x$. Initial value 20 with multiplicative factor 3 per step.
$y=20\cdot 3^x$. Initial value 20 with multiplicative factor 3 per step.
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What is the factor $b$ for an exponential function with $y(0)=6$ and $y(3)=48$?
What is the factor $b$ for an exponential function with $y(0)=6$ and $y(3)=48$?
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$b=2$. Solve $6b^3=48$ to get $b^3=8$, so $b=2$.
$b=2$. Solve $6b^3=48$ to get $b^3=8$, so $b=2$.
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What is the exponential function through $(0,12)$ and $(2,3)$ in the form $y=a\cdot b^x$?
What is the exponential function through $(0,12)$ and $(2,3)$ in the form $y=a\cdot b^x$?
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$y=12\cdot \left(\frac{1}{2}\right)^x$. Use $a=12$ and solve $12b^2=3$ to get $b=\frac{1}{2}$.
$y=12\cdot \left(\frac{1}{2}\right)^x$. Use $a=12$ and solve $12b^2=3$ to get $b=\frac{1}{2}$.
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What is the exponential function through $(0,4)$ and $(1,10)$ in the form $y=a\cdot b^x$?
What is the exponential function through $(0,4)$ and $(1,10)$ in the form $y=a\cdot b^x$?
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$y=4\cdot 2.5^x$. Use $a=4$ and calculate $b=\frac{10}{4}=2.5$.
$y=4\cdot 2.5^x$. Use $a=4$ and calculate $b=\frac{10}{4}=2.5$.
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What is the factor $b$ of $y=5\cdot 0.2^x$?
What is the factor $b$ of $y=5\cdot 0.2^x$?
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$0.2$. The base $b$ is the multiplicative factor per unit increase.
$0.2$. The base $b$ is the multiplicative factor per unit increase.
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What is the initial value $a$ of $y=9\left(\frac{1}{3}\right)^x$?
What is the initial value $a$ of $y=9\left(\frac{1}{3}\right)^x$?
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$9$. The coefficient $a$ is the initial value when $x=0$.
$9$. The coefficient $a$ is the initial value when $x=0$.
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What is the exponential function with $a=3$ and factor $b=2$?
What is the exponential function with $a=3$ and factor $b=2$?
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$y=3\cdot 2^x$. Direct substitution into the exponential form $y=a\cdot b^x$.
$y=3\cdot 2^x$. Direct substitution into the exponential form $y=a\cdot b^x$.
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What is the factor $b$ for $12%$ decay per step?
What is the factor $b$ for $12%$ decay per step?
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$b=0.88$. Calculate $b=1-0.12=0.88$ for 12% decay.
$b=0.88$. Calculate $b=1-0.12=0.88$ for 12% decay.
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