Constructing Linear and Exponential Functions - Algebra 2
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Identify the function type if the table has a constant ratio $\frac{y_{k+1}}{y_k}$ for equal steps in $x$.
Identify the function type if the table has a constant ratio $\frac{y_{k+1}}{y_k}$ for equal steps in $x$.
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Exponential. Constant ratios indicate exponential relationship.
Exponential. Constant ratios indicate exponential relationship.
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What is the linear function through points $(0,-3)$ and $(4,5)$ in slope-intercept form?
What is the linear function through points $(0,-3)$ and $(4,5)$ in slope-intercept form?
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$y=2x-3$. Initial value $-3$, slope $\frac{5-(-3)}{4-0} = 2$.
$y=2x-3$. Initial value $-3$, slope $\frac{5-(-3)}{4-0} = 2$.
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What is the linear function through points $(1,7)$ and $(3,11)$ in slope-intercept form?
What is the linear function through points $(1,7)$ and $(3,11)$ in slope-intercept form?
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$y=2x+5$. Slope $\frac{11-7}{3-1} = 2$, $y$-intercept when $x=0$ is $5$.
$y=2x+5$. Slope $\frac{11-7}{3-1} = 2$, $y$-intercept when $x=0$ is $5$.
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What is the exponential function for “starts at $50$ and increases by $3%$ per year”?
What is the exponential function for “starts at $50$ and increases by $3%$ per year”?
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$y=50\cdot(1.03)^x$. Initial $50$, growth factor $1 + 0.03 = 1.03$.
$y=50\cdot(1.03)^x$. Initial $50$, growth factor $1 + 0.03 = 1.03$.
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What is the exponential function for “starts at $200$ and decreases by $5%$ per month”?
What is the exponential function for “starts at $200$ and decreases by $5%$ per month”?
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$y=200\cdot(0.95)^x$. Initial $200$, decay factor $1 - 0.05 = 0.95$.
$y=200\cdot(0.95)^x$. Initial $200$, decay factor $1 - 0.05 = 0.95$.
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What is the linear function for “starts at $12$ and increases by $4$ each step”?
What is the linear function for “starts at $12$ and increases by $4$ each step”?
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$y=4x+12$. Linear with slope $4$ and $y$-intercept $12$.
$y=4x+12$. Linear with slope $4$ and $y$-intercept $12$.
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What is the linear function for “starts at $30$ and decreases by $2$ each hour”?
What is the linear function for “starts at $30$ and decreases by $2$ each hour”?
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$y=-2x+30$. Linear with slope $-2$ and $y$-intercept $30$.
$y=-2x+30$. Linear with slope $-2$ and $y$-intercept $30$.
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What is the common ratio $r$ for the geometric sequence $3,12,48,192,\dots$?
What is the common ratio $r$ for the geometric sequence $3,12,48,192,\dots$?
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$r=4$. Each term multiplies by $\frac{12}{3} = 4$.
$r=4$. Each term multiplies by $\frac{12}{3} = 4$.
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What is $a_6$ for the arithmetic sequence with $a_1=2$ and $d=5$?
What is $a_6$ for the arithmetic sequence with $a_1=2$ and $d=5$?
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$a_6=27$. $a_6 = 2 + (6-1) \cdot 5 = 2 + 25 = 27$.
$a_6=27$. $a_6 = 2 + (6-1) \cdot 5 = 2 + 25 = 27$.
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What is $a_5$ for the geometric sequence with $a_1=7$ and $r=2$?
What is $a_5$ for the geometric sequence with $a_1=7$ and $r=2$?
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$a_5=112$. $a_5 = 7 \cdot 2^{5-1} = 7 \cdot 16 = 112$.
$a_5=112$. $a_5 = 7 \cdot 2^{5-1} = 7 \cdot 16 = 112$.
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What is the linear function through points $(1,7)$ and $(3,11)$ in slope-intercept form?
What is the linear function through points $(1,7)$ and $(3,11)$ in slope-intercept form?
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$y=2x+5$. Slope $\frac{11-7}{3-1} = 2$, $y$-intercept when $x=0$ is $5$.
$y=2x+5$. Slope $\frac{11-7}{3-1} = 2$, $y$-intercept when $x=0$ is $5$.
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What is the constant ratio property of a geometric sequence?
What is the constant ratio property of a geometric sequence?
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$\frac{a_{n+1}}{a_n}=r$ (constant, $a_n\neq 0$). Consecutive terms have the same ratio $r$.
$\frac{a_{n+1}}{a_n}=r$ (constant, $a_n\neq 0$). Consecutive terms have the same ratio $r$.
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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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$a_n=a_1+(n-1)d$. Start with $a_1$, add $d$ for each step to term $n$.
$a_n=a_1+(n-1)d$. Start with $a_1$, add $d$ for each step to term $n$.
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Identify the function type if the graph is a straight line with constant slope.
Identify the function type if the graph is a straight line with constant slope.
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Linear. Straight lines have constant slope (linear functions).
Linear. Straight lines have constant slope (linear functions).
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Identify the function type if the graph curves and multiplies by a constant factor each $1$ step in $x$.
Identify the function type if the graph curves and multiplies by a constant factor each $1$ step in $x$.
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Exponential. Curved growth by constant factors indicates exponential.
Exponential. Curved growth by constant factors indicates exponential.
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What is the slope of the line passing through $(0,4)$ and $(5,4)$?
What is the slope of the line passing through $(0,4)$ and $(5,4)$?
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$m=0$. Horizontal line has zero slope (no rise).
$m=0$. Horizontal line has zero slope (no rise).
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What is $a$ in $y=a\cdot b^x$ if the function passes through $(0,-8)$?
What is $a$ in $y=a\cdot b^x$ if the function passes through $(0,-8)$?
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$a=-8$. The initial value when $x = 0$ is $a$.
$a=-8$. The initial value when $x = 0$ is $a$.
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What is $b$ if $y=a\cdot b^x$ passes through $(0,2)$ and $(3,16)$?
What is $b$ if $y=a\cdot b^x$ passes through $(0,2)$ and $(3,16)$?
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$b=2$. $a = 2$, $16 = 2 \cdot b^3$, so $b^3 = 8$ and $b = 2$.
$b=2$. $a = 2$, $16 = 2 \cdot b^3$, so $b^3 = 8$ and $b = 2$.
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What is the linear function through $(2,0)$ and $(0,6)$ in slope-intercept form?
What is the linear function through $(2,0)$ and $(0,6)$ in slope-intercept form?
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$y=-3x+6$. Slope $\frac{0-6}{2-0} = -3$, $y$-intercept $6$.
$y=-3x+6$. Slope $\frac{0-6}{2-0} = -3$, $y$-intercept $6$.
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What is the exponential function through $(0,4)$ and $(2,36)$ in the form $y=a\cdot b^x$?
What is the exponential function through $(0,4)$ and $(2,36)$ in the form $y=a\cdot b^x$?
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$y=4\cdot 3^x$. $a = 4$, $36 = 4 \cdot b^2$, so $b^2 = 9$ and $b = 3$.
$y=4\cdot 3^x$. $a = 4$, $36 = 4 \cdot b^2$, so $b^2 = 9$ and $b = 3$.
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What is the slope of a line that rises $6$ units when $x$ increases by $2$?
What is the slope of a line that rises $6$ units when $x$ increases by $2$?
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$m=3$. Slope equals rise over run: $\frac{6}{2} = 3$.
$m=3$. Slope equals rise over run: $\frac{6}{2} = 3$.
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What is the slope of a line that falls $10$ units when $x$ increases by $5$?
What is the slope of a line that falls $10$ units when $x$ increases by $5$?
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$m=-2$. Slope equals rise over run: $\frac{-10}{5} = -2$.
$m=-2$. Slope equals rise over run: $\frac{-10}{5} = -2$.
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Identify the function type if the table has a constant first difference in $y$ for equal steps in $x$.
Identify the function type if the table has a constant first difference in $y$ for equal steps in $x$.
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Linear. Constant differences indicate linear relationship.
Linear. Constant differences indicate linear relationship.
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What is the slope of the line through $(2,5)$ and $(6,13)$?
What is the slope of the line through $(2,5)$ and $(6,13)$?
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$m=2$. $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.
$m=2$. $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.
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What is the linear function with slope $m=-4$ and passing through $(2,1)$?
What is the linear function with slope $m=-4$ and passing through $(2,1)$?
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$y=-4x+9$. Using point-slope form with $(2,1)$ and $m=-4$.
$y=-4x+9$. Using point-slope form with $(2,1)$ and $m=-4$.
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What is the $y$-intercept of the line $3x+2y=8$?
What is the $y$-intercept of the line $3x+2y=8$?
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$b=4$. Solve $3(0) + 2y = 8$ to get $y = 4$.
$b=4$. Solve $3(0) + 2y = 8$ to get $y = 4$.
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What is the slope of the line $5x-10y=20$?
What is the slope of the line $5x-10y=20$?
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$m=\frac{1}{2}$. Rewrite as $y = \frac{1}{2}x - 2$, so slope is $\frac{1}{2}$.
$m=\frac{1}{2}$. Rewrite as $y = \frac{1}{2}x - 2$, so slope is $\frac{1}{2}$.
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What is the exponential function through $(0,6)$ and $(1,15)$ in the form $y=a\cdot b^x$?
What is the exponential function through $(0,6)$ and $(1,15)$ in the form $y=a\cdot b^x$?
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$y=6\cdot\left(\frac{5}{2}\right)^x$. $a = 6$ from $(0,6)$, $b = \frac{15}{6} = \frac{5}{2}$.
$y=6\cdot\left(\frac{5}{2}\right)^x$. $a = 6$ from $(0,6)$, $b = \frac{15}{6} = \frac{5}{2}$.
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What is the exponential function through $(0,10)$ and $(2,40)$ in the form $y=a\cdot b^x$?
What is the exponential function through $(0,10)$ and $(2,40)$ in the form $y=a\cdot b^x$?
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$y=10\cdot 2^x$. $a = 10$, $b^2 = 4$ so $b = 2$.
$y=10\cdot 2^x$. $a = 10$, $b^2 = 4$ so $b = 2$.
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What is the exponential function through $(1,12)$ and $(3,48)$ in the form $y=a\cdot b^x$?
What is the exponential function through $(1,12)$ and $(3,48)$ in the form $y=a\cdot b^x$?
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$y=6\cdot 2^x$. From ratio $\frac{48}{12} = 4 = 2^2$, so $b = 2$, $a = 6$.
$y=6\cdot 2^x$. From ratio $\frac{48}{12} = 4 = 2^2$, so $b = 2$, $a = 6$.
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