Interpreting Parameters in Linear/Exponential Models - Algebra
Card 1 of 30
Identify the meaning of $m=\frac{1}{2}$ in $y=\frac{1}{2}x+8$ in context.
Identify the meaning of $m=\frac{1}{2}$ in $y=\frac{1}{2}x+8$ in context.
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$y$ increases by $\frac{1}{2}$ per $1$ unit of $x$. Fractional slope shows $y$ increases by the fraction amount.
$y$ increases by $\frac{1}{2}$ per $1$ unit of $x$. Fractional slope shows $y$ increases by the fraction amount.
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Identify the meaning of $a=1500$ in $D(t)=1500(0.95)^t$ where $t$ is years and $D$ is value.
Identify the meaning of $a=1500$ in $D(t)=1500(0.95)^t$ where $t$ is years and $D$ is value.
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Initial value is $1500$ dollars at $t=0$. Parameter $a$ gives the starting value of the item.
Initial value is $1500$ dollars at $t=0$. Parameter $a$ gives the starting value of the item.
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Identify the yearly percent change in $D(t)=1500(0.95)^t$ where $t$ is years.
Identify the yearly percent change in $D(t)=1500(0.95)^t$ where $t$ is years.
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$5%$ decrease per year. Since $0.95 = 1 - 0.05$, the yearly decrease is $5%$.
$5%$ decrease per year. Since $0.95 = 1 - 0.05$, the yearly decrease is $5%$.
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Which parameter in $y=mx+b$ is interpreted as a fixed fee plus a variable fee?
Which parameter in $y=mx+b$ is interpreted as a fixed fee plus a variable fee?
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$b$ is the fixed fee; $m$ is the variable fee per unit of $x$. Linear models separate fixed costs from variable costs.
$b$ is the fixed fee; $m$ is the variable fee per unit of $x$. Linear models separate fixed costs from variable costs.
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Identify the meaning of $b=8$ in $y= rac{1}{2}x+8$ in context.
Identify the meaning of $b=8$ in $y= rac{1}{2}x+8$ in context.
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Initial value is $8$ at $x=0$. The intercept gives the starting value of the function.
Initial value is $8$ at $x=0$. The intercept gives the starting value of the function.
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What does $b<0$ mean in $y=mx+b$ in a context where negative values are allowed?
What does $b<0$ mean in $y=mx+b$ in a context where negative values are allowed?
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The initial value at $x=0$ is below $0$. Negative intercept means the line starts below the $x$-axis.
The initial value at $x=0$ is below $0$. Negative intercept means the line starts below the $x$-axis.
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What does $b$ represent in $N(t)=a(b)^t$ when $t$ is in days?
What does $b$ represent in $N(t)=a(b)^t$ when $t$ is in days?
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The factor multiplied each day. Base $b$ shows the daily multiplicative change factor.
The factor multiplied each day. Base $b$ shows the daily multiplicative change factor.
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Identify the decay factor if a quantity decreases by $7%$ each time period.
Identify the decay factor if a quantity decreases by $7%$ each time period.
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$0.93$. Decay factor equals $1$ minus the decimal decay rate.
$0.93$. Decay factor equals $1$ minus the decimal decay rate.
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What does the slope represent in $d=60t$ where $d$ is miles and $t$ is hours?
What does the slope represent in $d=60t$ where $d$ is miles and $t$ is hours?
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Speed is $60$ miles per hour. The coefficient of $t$ represents constant velocity.
Speed is $60$ miles per hour. The coefficient of $t$ represents constant velocity.
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Identify the weekly percent change in $N(t)=200(1.5)^t$ where $t$ is weeks.
Identify the weekly percent change in $N(t)=200(1.5)^t$ where $t$ is weeks.
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$50%$ increase per week. Since $1.5 = 1 + 0.5$, the increase is $50%$.
$50%$ increase per week. Since $1.5 = 1 + 0.5$, the increase is $50%$.
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What does the intercept represent in $d=60t$ for distance traveled from a start point?
What does the intercept represent in $d=60t$ for distance traveled from a start point?
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Initial distance is $0$ miles at $t=0$. Zero intercept means starting at the reference point.
Initial distance is $0$ miles at $t=0$. Zero intercept means starting at the reference point.
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Identify the initial value in $y=3x+5$ without any context given.
Identify the initial value in $y=3x+5$ without any context given.
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$5$ (the value when $x=0$). The constant term gives the value when $x = 0$.
$5$ (the value when $x=0$). The constant term gives the value when $x = 0$.
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What does $b$ represent in $N(t)=a(b)^t$ when $t$ is in months?
What does $b$ represent in $N(t)=a(b)^t$ when $t$ is in months?
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The factor multiplied each month. Base $b$ shows the monthly multiplicative change factor.
The factor multiplied each month. Base $b$ shows the monthly multiplicative change factor.
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What does the unit of slope $m$ equal if $x$ is in hours and $y$ is in miles?
What does the unit of slope $m$ equal if $x$ is in hours and $y$ is in miles?
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$m$ has units of miles per hour. Units of slope equal units of $y$ divided by units of $x$.
$m$ has units of miles per hour. Units of slope equal units of $y$ divided by units of $x$.
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Which option is the correct interpretation of $b$ in $y=a(b)^x$: additive change or multiplicative change?
Which option is the correct interpretation of $b$ in $y=a(b)^x$: additive change or multiplicative change?
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Multiplicative change. Exponential models involve repeated multiplication by the base.
Multiplicative change. Exponential models involve repeated multiplication by the base.
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What does $b=1$ mean in an exponential model $y = a(b)^x$?
What does $b=1$ mean in an exponential model $y = a(b)^x$?
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No change; $y=a$ for all $x$. Base of $1$ means no multiplicative change occurs.
No change; $y=a$ for all $x$. Base of $1$ means no multiplicative change occurs.
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What does $a<0$ mean in an exponential model $y = a(b)^x$ if allowed by context?
What does $a<0$ mean in an exponential model $y = a(b)^x$ if allowed by context?
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All outputs are negative; the model starts below $0$ at $x=0$. Negative coefficient reflects all outputs across the $x$-axis.
All outputs are negative; the model starts below $0$ at $x=0$. Negative coefficient reflects all outputs across the $x$-axis.
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Identify the meaning of $b=0.6$ in $N(t)=200(0.6)^t$ where $t$ is weeks.
Identify the meaning of $b=0.6$ in $N(t)=200(0.6)^t$ where $t$ is weeks.
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The quantity is multiplied by $0.6$ each week. Base $0.6$ means the quantity becomes $0.6$ times its size.
The quantity is multiplied by $0.6$ each week. Base $0.6$ means the quantity becomes $0.6$ times its size.
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What does an intercept $b=0$ mean in a linear model $y = mx + b$?
What does an intercept $b=0$ mean in a linear model $y = mx + b$?
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The relationship is proportional; $y=0$ when $x=0$. The line passes through the origin with direct variation.
The relationship is proportional; $y=0$ when $x=0$. The line passes through the origin with direct variation.
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Identify the meaning of $b=1.5$ in $N(t)=200(1.5)^t$ where $t$ is weeks.
Identify the meaning of $b=1.5$ in $N(t)=200(1.5)^t$ where $t$ is weeks.
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The quantity is multiplied by $1.5$ each week. Base $1.5$ means the quantity becomes $1.5$ times larger.
The quantity is multiplied by $1.5$ each week. Base $1.5$ means the quantity becomes $1.5$ times larger.
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What does a positive slope $m>0$ mean in $y=mx+b$ in context?
What does a positive slope $m>0$ mean in $y=mx+b$ in context?
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$y$ increases by $m$ for each $1$ unit increase in $x$. Positive slope creates an upward-sloping line.
$y$ increases by $m$ for each $1$ unit increase in $x$. Positive slope creates an upward-sloping line.
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Identify the meaning of intercept $72$ in $T(h)=-4h+72$ where $h$ is hours.
Identify the meaning of intercept $72$ in $T(h)=-4h+72$ where $h$ is hours.
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Initial temperature is $72$ degrees at $h=0$. The intercept gives the starting temperature value.
Initial temperature is $72$ degrees at $h=0$. The intercept gives the starting temperature value.
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Which parameter is the starting amount in $A(t)=a(1+r)^t$?
Which parameter is the starting amount in $A(t)=a(1+r)^t$?
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$a$. Parameter $a$ is the coefficient, representing initial quantity.
$a$. Parameter $a$ is the coefficient, representing initial quantity.
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Which parameter is the constant percent rate in $A(t)=a(1+r)^t$?
Which parameter is the constant percent rate in $A(t)=a(1+r)^t$?
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$r$. Parameter $r$ appears in the base as the growth rate.
$r$. Parameter $r$ appears in the base as the growth rate.
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What does $r=0.12$ mean in $A(t)=a(1+r)^t$ when $t$ is years?
What does $r=0.12$ mean in $A(t)=a(1+r)^t$ when $t$ is years?
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$12%$ increase per year. In growth form, $r$ represents percent increase as decimal.
$12%$ increase per year. In growth form, $r$ represents percent increase as decimal.
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What does a zero slope $m=0$ mean for $y = mx + b$ in context?
What does a zero slope $m=0$ mean for $y = mx + b$ in context?
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$y$ stays constant as $x$ changes. Zero slope creates a horizontal line with no change.
$y$ stays constant as $x$ changes. Zero slope creates a horizontal line with no change.
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What does the unit of $b$ equal in $y=mx+b$ if $y$ measures dollars?
What does the unit of $b$ equal in $y=mx+b$ if $y$ measures dollars?
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$b$ is measured in dollars. The $y$-intercept has the same units as the dependent variable.
$b$ is measured in dollars. The $y$-intercept has the same units as the dependent variable.
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Identify the growth factor if a quantity increases by $7%$ each time period.
Identify the growth factor if a quantity increases by $7%$ each time period.
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$1.07$. Growth factor equals $1$ plus the decimal growth rate.
$1.07$. Growth factor equals $1$ plus the decimal growth rate.
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Identify the weekly percent change in $N(t)=200(0.6)^t$ where $t$ is weeks.
Identify the weekly percent change in $N(t)=200(0.6)^t$ where $t$ is weeks.
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$40%$ decrease per week. Since $0.6 = 1 - 0.4$, the decrease is $40%$.
$40%$ decrease per week. Since $0.6 = 1 - 0.4$, the decrease is $40%$.
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What does a negative slope $m<0$ mean for $y = mx + b$ in context?
What does a negative slope $m<0$ mean for $y = mx + b$ in context?
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$y$ decreases by $|m|$ for each $1$ unit increase in $x$. Negative slope creates a downward-sloping line.
$y$ decreases by $|m|$ for each $1$ unit increase in $x$. Negative slope creates a downward-sloping line.
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