Interpreting Parameters in Linear/Exponential Models - Algebra 2
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What does the $y$-intercept represent on a graph of a contextual linear function?
What does the $y$-intercept represent on a graph of a contextual linear function?
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It represents the starting amount at $x = 0$. This is where the line crosses the y-axis at $x = 0$.
It represents the starting amount at $x = 0$. This is where the line crosses the y-axis at $x = 0$.
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What condition on $b$ indicates exponential decay in $y = a , b^x$?
What condition on $b$ indicates exponential decay in $y = a , b^x$?
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Decay occurs when $0 < b < 1$. Values between 0 and 1 multiply to decrease the quantity.
Decay occurs when $0 < b < 1$. Values between 0 and 1 multiply to decrease the quantity.
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What does the parameter $b$ represent in an exponential model $y = a , b^x$ in context?
What does the parameter $b$ represent in an exponential model $y = a , b^x$ in context?
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$b$ is the growth factor per $1$ unit increase in $x$. Each unit increase in $x$ multiplies the output by this factor.
$b$ is the growth factor per $1$ unit increase in $x$. Each unit increase in $x$ multiplies the output by this factor.
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What does the parameter $b$ represent in a linear model $y = mx + b$ in context?
What does the parameter $b$ represent in a linear model $y = mx + b$ in context?
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$b$ is the initial value, the output when $x = 0$. The y-intercept shows the starting value before any input changes.
$b$ is the initial value, the output when $x = 0$. The y-intercept shows the starting value before any input changes.
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What does the parameter $m$ represent in a linear model $y = mx + b$ in context?
What does the parameter $m$ represent in a linear model $y = mx + b$ in context?
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$m$ is the rate of change (slope), in $y$-units per $x$-unit. The slope quantifies how much the output changes per unit input change.
$m$ is the rate of change (slope), in $y$-units per $x$-unit. The slope quantifies how much the output changes per unit input change.
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Identify the meaning of $a = 2000$ in $P(t) = 2000(1.05)^t$ for a population model.
Identify the meaning of $a = 2000$ in $P(t) = 2000(1.05)^t$ for a population model.
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Initial population is $2000$ at $t = 0$. The coefficient represents the starting population count.
Initial population is $2000$ at $t = 0$. The coefficient represents the starting population count.
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Identify the meaning of $0.6$ in $V(t) = 80(0.6)^t$ for a value-depreciation model.
Identify the meaning of $0.6$ in $V(t) = 80(0.6)^t$ for a value-depreciation model.
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Value keeps $60%$ each time unit (a $40%$ decrease each unit). The base shows the remaining fraction after each depreciation period.
Value keeps $60%$ each time unit (a $40%$ decrease each unit). The base shows the remaining fraction after each depreciation period.
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Identify the growth/decay factor for $y = 120(0.85)^t$ in context.
Identify the growth/decay factor for $y = 120(0.85)^t$ in context.
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Factor is $0.85$, meaning multiply by $0.85$ each time unit. The base shows how the quantity changes each time period.
Factor is $0.85$, meaning multiply by $0.85$ each time unit. The base shows how the quantity changes each time period.
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What is the percent change per unit for $y = 500(1.03)^t$?
What is the percent change per unit for $y = 500(1.03)^t$?
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$3%$ growth per time unit. Convert to percent: $(1.03 - 1) \times 100% = 3%$ growth.
$3%$ growth per time unit. Convert to percent: $(1.03 - 1) \times 100% = 3%$ growth.
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What is the percent change per unit for $y = 500(0.97)^t$?
What is the percent change per unit for $y = 500(0.97)^t$?
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$3%$ decay per time unit. Convert to percent: $(1 - 0.97) \times 100% = 3%$ decay.
$3%$ decay per time unit. Convert to percent: $(1 - 0.97) \times 100% = 3%$ decay.
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Identify the meaning of $b = 120$ in $y = 65t + 120$ if $t$ is hours and $y$ is miles.
Identify the meaning of $b = 120$ in $y = 65t + 120$ if $t$ is hours and $y$ is miles.
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The starting distance is $120$ miles at $t = 0$. The y-intercept represents the initial distance position.
The starting distance is $120$ miles at $t = 0$. The y-intercept represents the initial distance position.
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What does the parameter $k$ represent in $y = a , b^{x-k}$ in context?
What does the parameter $k$ represent in $y = a , b^{x-k}$ in context?
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$k$ shifts the input: the reference point occurs at $x = k$. The horizontal shift changes when the reference value occurs.
$k$ shifts the input: the reference point occurs at $x = k$. The horizontal shift changes when the reference value occurs.
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What does the parameter $(1+r)$ represent in $y = a(1+r)^t$?
What does the parameter $(1+r)$ represent in $y = a(1+r)^t$?
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$(1+r)$ is the multiplier applied each time unit. This factor determines how the quantity changes per time unit.
$(1+r)$ is the multiplier applied each time unit. This factor determines how the quantity changes per time unit.
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What does the parameter $a$ represent in $y = a(1+r)^t$ when $t$ is time?
What does the parameter $a$ represent in $y = a(1+r)^t$ when $t$ is time?
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$a$ is the amount at time $t = 0$. This represents the initial quantity before any time passes.
$a$ is the amount at time $t = 0$. This represents the initial quantity before any time passes.
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What does the parameter $c$ represent in $y = mx + b + c$ in context?
What does the parameter $c$ represent in $y = mx + b + c$ in context?
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$c$ is a vertical shift, adding $c$ to every output value. Vertical shifts move all output values up or down by $c$.
$c$ is a vertical shift, adding $c$ to every output value. Vertical shifts move all output values up or down by $c$.
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What does the parameter $h$ represent in $y = m(x-h) + b$ in context?
What does the parameter $h$ represent in $y = m(x-h) + b$ in context?
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$h$ shifts the input: the value $b$ occurs at $x = h$. The horizontal shift changes when the y-intercept value occurs.
$h$ shifts the input: the value $b$ occurs at $x = h$. The horizontal shift changes when the y-intercept value occurs.
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What does the difference $y(x+1) - y(x)$ equal for $y = mx + b$?
What does the difference $y(x+1) - y(x)$ equal for $y = mx + b$?
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The difference is constant and equals $m$. Consecutive outputs have a constant additive relationship.
The difference is constant and equals $m$. Consecutive outputs have a constant additive relationship.
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What does the $x$-intercept represent on a graph of a contextual linear function?
What does the $x$-intercept represent on a graph of a contextual linear function?
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It is when the output equals $0$ (the quantity reaches $0$). This is where the line crosses the x-axis at $y = 0$.
It is when the output equals $0$ (the quantity reaches $0$). This is where the line crosses the x-axis at $y = 0$.
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What does the $y$-intercept represent on a graph of a contextual linear function?
What does the $y$-intercept represent on a graph of a contextual linear function?
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It represents the starting amount at $x = 0$. This is where the line crosses the y-axis at $x = 0$.
It represents the starting amount at $x = 0$. This is where the line crosses the y-axis at $x = 0$.
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What does a positive slope $m>0$ mean in a contextual linear model $y = mx + b$?
What does a positive slope $m>0$ mean in a contextual linear model $y = mx + b$?
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The quantity increases by $m$ per $1$ unit increase in $x$. Positive slopes indicate an increase in the dependent variable.
The quantity increases by $m$ per $1$ unit increase in $x$. Positive slopes indicate an increase in the dependent variable.
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What does a negative slope $m<0$ mean in a contextual linear model $y = mx + b$?
What does a negative slope $m<0$ mean in a contextual linear model $y = mx + b$?
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The quantity decreases by $|m|$ per $1$ unit increase in $x$. Negative slopes indicate a decrease in the dependent variable.
The quantity decreases by $|m|$ per $1$ unit increase in $x$. Negative slopes indicate a decrease in the dependent variable.
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What condition on $b$ indicates exponential growth in $y = a , b^x$?
What condition on $b$ indicates exponential growth in $y = a , b^x$?
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Growth occurs when $b > 1$. Values greater than 1 multiply to increase the quantity.
Growth occurs when $b > 1$. Values greater than 1 multiply to increase the quantity.
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What does $r$ represent in $y = a(1-r)^x$ when $0<r<1$?
What does $r$ represent in $y = a(1-r)^x$ when $0<r<1$?
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$r$ is the percent decay rate per $x$-unit, written as a decimal. Decay rate as a decimal: $0.93$ means $7%$ decay.
$r$ is the percent decay rate per $x$-unit, written as a decimal. Decay rate as a decimal: $0.93$ means $7%$ decay.
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What does $r$ represent in $y = a(1+r)^x$ when $r$ is given as a decimal?
What does $r$ represent in $y = a(1+r)^x$ when $r$ is given as a decimal?
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$r$ is the percent rate per $x$-unit, written as a decimal. Growth rate as a decimal: $1.08$ means $8%$ growth.
$r$ is the percent rate per $x$-unit, written as a decimal. Growth rate as a decimal: $1.08$ means $8%$ growth.
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What does the parameter $a$ represent in an exponential model $y = a , b^x$ in context?
What does the parameter $a$ represent in an exponential model $y = a , b^x$ in context?
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$a$ is the initial value, the output when $x = 0$. This is the y-intercept, representing the starting amount.
$a$ is the initial value, the output when $x = 0$. This is the y-intercept, representing the starting amount.
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Identify the meaning of $m = 65$ in $y = 65t + 120$ if $t$ is hours and $y$ is miles.
Identify the meaning of $m = 65$ in $y = 65t + 120$ if $t$ is hours and $y$ is miles.
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The speed is $65$ miles per hour. The slope represents the rate of change in distance per time.
The speed is $65$ miles per hour. The slope represents the rate of change in distance per time.
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Identify the growth/decay factor for $y = 120(0.85)^t$ in context.
Identify the growth/decay factor for $y = 120(0.85)^t$ in context.
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Factor is $0.85$, meaning multiply by $0.85$ each time unit. The base shows how the quantity changes each time period.
Factor is $0.85$, meaning multiply by $0.85$ each time unit. The base shows how the quantity changes each time period.
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Identify the initial value for $y = 120(0.85)^t$ in context.
Identify the initial value for $y = 120(0.85)^t$ in context.
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Initial value is $120$ at $t = 0$. The coefficient of the exponential term at $t = 0$.
Initial value is $120$ at $t = 0$. The coefficient of the exponential term at $t = 0$.
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Identify the rate of change for $y = -2x + 9$ in context.
Identify the rate of change for $y = -2x + 9$ in context.
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Rate of change is $-2$ per $1$ unit of $x$. The coefficient of $x$ shows the change per unit input.
Rate of change is $-2$ per $1$ unit of $x$. The coefficient of $x$ shows the change per unit input.
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Identify the initial value for $y = 3x + 50$ in context.
Identify the initial value for $y = 3x + 50$ in context.
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Initial value is $50$ (the value when $x = 0$). The constant term is the value when the input is zero.
Initial value is $50$ (the value when $x = 0$). The constant term is the value when the input is zero.
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