Construct and Interpret Linear Functions - 8th Grade Math
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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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$m = 2$. Use $m = rac{13-5}{6-2} = rac{8}{4} = 2$.
$m = 2$. Use $m = rac{13-5}{6-2} = rac{8}{4} = 2$.
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A gym membership costs $20$ to join and $15$ per month. What is the initial value in this model?
A gym membership costs $20$ to join and $15$ per month. What is the initial value in this model?
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$20$. The joining fee is the cost when months = 0.
$20$. The joining fee is the cost when months = 0.
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Write the equation of the line passing through $ (0,-4) $ and $ (5,1) $.
Write the equation of the line passing through $ (0,-4) $ and $ (5,1) $.
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$y=x-4$. Slope is $\frac{1-(-4)}{5-0}=1$, and $b=-4$ from $ (0,-4) $.
$y=x-4$. Slope is $\frac{1-(-4)}{5-0}=1$, and $b=-4$ from $ (0,-4) $.
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What is the slope (rate of change) formula using two points $(x_1,y_1)$ and $(x_2,y_2)$?
What is the slope (rate of change) formula using two points $(x_1,y_1)$ and $(x_2,y_2)$?
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$m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.
$m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.
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A runner’s distance is $d=6t+2$ (miles). What does the $2$ mean in context?
A runner’s distance is $d=6t+2$ (miles). What does the $2$ mean in context?
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The runner started $2$ miles from the starting point at $t=0$. The constant term represents starting position.
The runner started $2$ miles from the starting point at $t=0$. The constant term represents starting position.
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In $y=mx+b$, what does $m$ represent in a real-world situation?
In $y=mx+b$, what does $m$ represent in a real-world situation?
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$m$ is the rate of change (change in $y$ per $1$ unit of $x$). Slope tells how much $y$ changes for each unit increase in $x$.
$m$ is the rate of change (change in $y$ per $1$ unit of $x$). Slope tells how much $y$ changes for each unit increase in $x$.
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A tank has $50$ liters and drains $4$ liters per minute. What is $V(t)$ after $t$ minutes?
A tank has $50$ liters and drains $4$ liters per minute. What is $V(t)$ after $t$ minutes?
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$V(t)=50-4t$. Initial volume minus drain rate times time.
$V(t)=50-4t$. Initial volume minus drain rate times time.
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A taxi charges a $\$3$ start fee plus $$2$ per mile. What is the function for cost $C$ vs. miles $m$?
A taxi charges a $\$3$ start fee plus $$2$ per mile. What is the function for cost $C$ vs. miles $m$?
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$C=2m+3$. Fixed fee is initial value, per-mile rate is slope.
$C=2m+3$. Fixed fee is initial value, per-mile rate is slope.
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What is the slope-intercept form of a linear function?
What is the slope-intercept form of a linear function?
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$y=mx+b$. Standard form where $m$ is slope and $b$ is $y$-intercept.
$y=mx+b$. Standard form where $m$ is slope and $b$ is $y$-intercept.
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Find the function rule for the linear relationship with points $(3,-1)$ and $(7,11)$.
Find the function rule for the linear relationship with points $(3,-1)$ and $(7,11)$.
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$y=3x-10$. Slope is $\frac{11-(-1)}{7-3}=3$; use point to find $b=-10$.
$y=3x-10$. Slope is $\frac{11-(-1)}{7-3}=3$; use point to find $b=-10$.
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Find the slope of the line through $(2,5)$ and $(6,13)$.
Find 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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From the table points $(1,4)$ and $(4,13)$, what is the rate of change?
From the table points $(1,4)$ and $(4,13)$, what is the rate of change?
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$m=3$. $m=\frac{13-4}{4-1}=\frac{9}{3}=3$
$m=3$. $m=\frac{13-4}{4-1}=\frac{9}{3}=3$
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Which point on a graph gives the initial value of a linear function?
Which point on a graph gives the initial value of a linear function?
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The $y$-intercept, $(0,b)$. When $x=0$, $y=b$, which is the initial value.
The $y$-intercept, $(0,b)$. When $x=0$, $y=b$, which is the initial value.
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Identify the slope of the line given by $y=-3x+7$.
Identify the slope of the line given by $y=-3x+7$.
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$m=-3$. The coefficient of $x$ in slope-intercept form is the slope.
$m=-3$. The coefficient of $x$ in slope-intercept form is the slope.
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In $y=mx+b$, what does $b$ represent in a real-world situation?
In $y=mx+b$, what does $b$ represent in a real-world situation?
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$b$ is the initial value (the $y$-value when $x=0$). The $y$-intercept is the starting value when input is zero.
$b$ is the initial value (the $y$-value when $x=0$). The $y$-intercept is the starting value when input is zero.
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Identify the initial value for the function $y=4x-9$.
Identify the initial value for the function $y=4x-9$.
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$b=-9$. The constant term in slope-intercept form is the initial value.
$b=-9$. The constant term in slope-intercept form is the initial value.
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Write the linear function for a line with slope $m=-2$ and $y$-intercept $b=6$.
Write the linear function for a line with slope $m=-2$ and $y$-intercept $b=6$.
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$y=-2x+6$. Substitute slope and $y$-intercept into $y=mx+b$.
$y=-2x+6$. Substitute slope and $y$-intercept into $y=mx+b$.
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What is the $y$-intercept of a linear function in terms of a point on its graph?
What is the $y$-intercept of a linear function in terms of a point on its graph?
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The point where $x=0$, written as $(0,b)$. The $y$-intercept occurs where the line crosses the $y$-axis.
The point where $x=0$, written as $(0,b)$. The $y$-intercept occurs where the line crosses the $y$-axis.
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Find $b$ for a line with slope $m=3$ that passes through $(2,11)$.
Find $b$ for a line with slope $m=3$ that passes through $(2,11)$.
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$b=5$. Use $11=3(2)+b$, so $11=6+b$, thus $b=5$.
$b=5$. Use $11=3(2)+b$, so $11=6+b$, thus $b=5$.
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What is the meaning of a negative slope for a linear relationship?
What is the meaning of a negative slope for a linear relationship?
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As $x$ increases, $y$ decreases at a constant rate. Negative slope means the line goes down from left to right.
As $x$ increases, $y$ decreases at a constant rate. Negative slope means the line goes down from left to right.
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What does $m$ represent in $y=mx+b$ in a real-world situation?
What does $m$ represent in $y=mx+b$ in a real-world situation?
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rate of change per $1$ unit increase in $x$. Slope tells how much $y$ changes for each unit increase in $x$.
rate of change per $1$ unit increase in $x$. Slope tells how much $y$ changes for each unit increase in $x$.
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Find $b$ for the line with $m=3$ that passes through $(2,7)$.
Find $b$ for the line with $m=3$ that passes through $(2,7)$.
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$b=1$. Substitute into $7=3(2)+b$ to get $b=1$.
$b=1$. Substitute into $7=3(2)+b$ to get $b=1$.
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A tank has $50$ liters and drains $4$ liters per minute. What is the function $V(t)$?
A tank has $50$ liters and drains $4$ liters per minute. What is the function $V(t)$?
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$V(t)=-4t+50$. Starting volume minus drain rate times minutes.
$V(t)=-4t+50$. Starting volume minus drain rate times minutes.
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Interpret $m=-3$ in $y=-3x+10$ for a real situation where $x$ is hours and $y$ is dollars.
Interpret $m=-3$ in $y=-3x+10$ for a real situation where $x$ is hours and $y$ is dollars.
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dollars decrease by $3$ each hour. Negative slope means decreasing relationship.
dollars decrease by $3$ each hour. Negative slope means decreasing relationship.
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Interpret $b=8$ in $y=1.5x+8$ when $x$ is minutes and $y$ is pages read.
Interpret $b=8$ in $y=1.5x+8$ when $x$ is minutes and $y$ is pages read.
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at $x=0$, $y=8$ pages already read. The $y$-intercept shows the starting point before timing begins.
at $x=0$, $y=8$ pages already read. The $y$-intercept shows the starting point before timing begins.
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On a graph, a line crosses the $y$-axis at $(0,5)$ and rises $2$ for every run of $1$. What is $y=$?
On a graph, a line crosses the $y$-axis at $(0,5)$ and rises $2$ for every run of $1$. What is $y=$?
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$y=2x+5$. Rise of $2$ over run of $1$ gives slope $m=2$; crosses at $(0,5)$ so $b=5$.
$y=2x+5$. Rise of $2$ over run of $1$ gives slope $m=2$; crosses at $(0,5)$ so $b=5$.
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A table shows $x: 0,2,4$ and $y: 3,7,11$. What is the rate of change?
A table shows $x: 0,2,4$ and $y: 3,7,11$. What is the rate of change?
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$m=2$. $y$ increases by $4$ when $x$ increases by $2$, so $m=2$.
$m=2$. $y$ increases by $4$ when $x$ increases by $2$, so $m=2$.
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What does $b$ represent in $y=mx+b$ in a real-world situation?
What does $b$ represent in $y=mx+b$ in a real-world situation?
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initial value (starting amount) when $x=0$. The $y$-intercept represents the starting value before any change occurs.
initial value (starting amount) when $x=0$. The $y$-intercept represents the starting value before any change occurs.
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Using the table $x: 0,2,4$ and $y: 3,7,11$, what is the initial value?
Using the table $x: 0,2,4$ and $y: 3,7,11$, what is the initial value?
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$b=3$. When $x=0$, $y=3$ is the initial value.
$b=3$. When $x=0$, $y=3$ is the initial value.
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A line passes through $(0,-2)$ and $(4,6)$. What is the linear function $y=$?
A line passes through $(0,-2)$ and $(4,6)$. What is the linear function $y=$?
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$y=2x-2$. $m=\frac{6-(-2)}{4-0}=2$; $y$-intercept is $-2$.
$y=2x-2$. $m=\frac{6-(-2)}{4-0}=2$; $y$-intercept is $-2$.
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