Recognize Constant Rate Changes - Algebra 2
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What does a constant ratio $\frac{y}{x}$ (with $x\neq 0$) indicate about $y$ versus $x$?
What does a constant ratio $\frac{y}{x}$ (with $x\neq 0$) indicate about $y$ versus $x$?
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A proportional linear relationship $y=kx$. Constant ratio means $y$ is proportional to $x$.
A proportional linear relationship $y=kx$. Constant ratio means $y$ is proportional to $x$.
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Identify whether this indicates constant rate: When $x$ increases by $2$, $y$ always increases by $10$.
Identify whether this indicates constant rate: When $x$ increases by $2$, $y$ always increases by $10$.
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Yes, constant rate ($m=5$). Fixed change of $10$ for every $2$ in $x$ means constant rate.
Yes, constant rate ($m=5$). Fixed change of $10$ for every $2$ in $x$ means constant rate.
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What is the constant rate if $y$ always increases by $10$ when $x$ increases by $2$?
What is the constant rate if $y$ always increases by $10$ when $x$ increases by $2$?
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$5$ per unit $x$. Rate per unit: $\frac{10}{2}=5$ per unit $x$.
$5$ per unit $x$. Rate per unit: $\frac{10}{2}=5$ per unit $x$.
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What does it mean for $y$ to change at a constant rate per unit of $x$?
What does it mean for $y$ to change at a constant rate per unit of $x$?
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$\frac{\Delta y}{\Delta x}$ is constant for equal $\Delta x$ intervals. The ratio of change in $y$ to change in $x$ stays the same.
$\frac{\Delta y}{\Delta x}$ is constant for equal $\Delta x$ intervals. The ratio of change in $y$ to change in $x$ stays the same.
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What must be true about slopes between any two pairs of points on a line?
What must be true about slopes between any two pairs of points on a line?
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They are equal (constant slope). All points on a line have the same slope between any pair.
They are equal (constant slope). All points on a line have the same slope between any pair.
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What phrase most strongly signals constant rate: "per" or "percent"?
What phrase most strongly signals constant rate: "per" or "percent"?
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"Per" (constant additive rate per unit). "Per" indicates additive rate; "percent" indicates multiplicative rate.
"Per" (constant additive rate per unit). "Per" indicates additive rate; "percent" indicates multiplicative rate.
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What is the constant rate of change for the proportional relationship $y=5x$?
What is the constant rate of change for the proportional relationship $y=5x$?
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$5$. In $y=kx$, the constant $k$ is the rate.
$5$. In $y=kx$, the constant $k$ is the rate.
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Find the rate of change from $(-1,4)$ to $(3,-8)$.
Find the rate of change from $(-1,4)$ to $(3,-8)$.
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$m=\frac{-8-4}{3-(-1)}=-3$. Using slope formula: $\frac{-8-4}{3-(-1)}=\frac{-12}{4}=-3$.
$m=\frac{-8-4}{3-(-1)}=-3$. Using slope formula: $\frac{-8-4}{3-(-1)}=\frac{-12}{4}=-3$.
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Identify whether the points $(0,1)$, $(1,2)$, $(2,4)$ show a constant rate of change.
Identify whether the points $(0,1)$, $(1,2)$, $(2,4)$ show a constant rate of change.
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No, slopes differ ($1$ then $2$). Slopes change from $1$ to $2$, so rate is not constant.
No, slopes differ ($1$ then $2$). Slopes change from $1$ to $2$, so rate is not constant.
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What is the constant rate of change in $y=2+4x$?
What is the constant rate of change in $y=2+4x$?
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$4$. The coefficient of $x$ gives the rate of change.
$4$. The coefficient of $x$ gives the rate of change.
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Identify the constant rate in the verbal rule: "Start at $10$ and add $2$ each day."
Identify the constant rate in the verbal rule: "Start at $10$ and add $2$ each day."
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$2$ per day. The additive change per time unit.
$2$ per day. The additive change per time unit.
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What is the constant rate of change for the line $3x-2y=8$?
What is the constant rate of change for the line $3x-2y=8$?
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$\frac{3}{2}$. Rearranging to $y=\frac{3}{2}x-4$ shows slope $\frac{3}{2}$.
$\frac{3}{2}$. Rearranging to $y=\frac{3}{2}x-4$ shows slope $\frac{3}{2}$.
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Which change pattern indicates constant rate: constant differences in $y$ or constant ratios in $y$?
Which change pattern indicates constant rate: constant differences in $y$ or constant ratios in $y$?
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Constant differences in $y$ (for equal $\Delta x$). Constant differences indicate linear; constant ratios indicate exponential.
Constant differences in $y$ (for equal $\Delta x$). Constant differences indicate linear; constant ratios indicate exponential.
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Identify whether the data show constant rate: $x:0,1,2$ and $y:5,8,11$.
Identify whether the data show constant rate: $x:0,1,2$ and $y:5,8,11$.
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Yes, constant rate ($\Delta y=3$ each step). Equal differences of $3$ between consecutive $y$-values.
Yes, constant rate ($\Delta y=3$ each step). Equal differences of $3$ between consecutive $y$-values.
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Which equation form explicitly shows constant rate and initial value: $y=mx+b$ or $y=ax^2+c$?
Which equation form explicitly shows constant rate and initial value: $y=mx+b$ or $y=ax^2+c$?
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$y=mx+b$. Linear form shows constant rate $m$ and starting value $b$.
$y=mx+b$. Linear form shows constant rate $m$ and starting value $b$.
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Identify the constant rate of change for $y=-\frac{3}{2}x+8$.
Identify the constant rate of change for $y=-\frac{3}{2}x+8$.
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$-\frac{3}{2}$. The coefficient of $x$ is the rate of change.
$-\frac{3}{2}$. The coefficient of $x$ is the rate of change.
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Identify whether this is constant rate: A tank loses $3$ liters every minute.
Identify whether this is constant rate: A tank loses $3$ liters every minute.
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Yes, constant rate ($-3$ liters per minute). Fixed amount lost per time unit indicates constant rate.
Yes, constant rate ($-3$ liters per minute). Fixed amount lost per time unit indicates constant rate.
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Compute the unit rate if $\Delta y=-6$ when $\Delta x=3$.
Compute the unit rate if $\Delta y=-6$ when $\Delta x=3$.
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$\frac{\Delta y}{\Delta x}=-2$. Divide change in $y$ by change in $x$: $\frac{-6}{3}=-2$.
$\frac{\Delta y}{\Delta x}=-2$. Divide change in $y$ by change in $x$: $\frac{-6}{3}=-2$.
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Identify whether this indicates constant rate: When $x$ increases by $2$, $y$ always increases by $10$.
Identify whether this indicates constant rate: When $x$ increases by $2$, $y$ always increases by $10$.
Tap to reveal answer
Yes, constant rate ($m=5$). Fixed change of $10$ for every $2$ in $x$ means constant rate.
Yes, constant rate ($m=5$). Fixed change of $10$ for every $2$ in $x$ means constant rate.
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Identify whether the sequence rule $y=2+4x$ represents constant change per step in $x$.
Identify whether the sequence rule $y=2+4x$ represents constant change per step in $x$.
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Yes, constant change of $4$ per $1$ in $x$. Linear form with coefficient $4$ as the constant rate.
Yes, constant change of $4$ per $1$ in $x$. Linear form with coefficient $4$ as the constant rate.
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Which description matches constant rate: "$y$ changes by a fixed amount" or "$y$ changes by a fixed factor"?
Which description matches constant rate: "$y$ changes by a fixed amount" or "$y$ changes by a fixed factor"?
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"$y$ changes by a fixed amount". Fixed amount indicates additive; fixed factor indicates multiplicative.
"$y$ changes by a fixed amount". Fixed amount indicates additive; fixed factor indicates multiplicative.
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What does "constant percent rate" usually indicate: linear or exponential change?
What does "constant percent rate" usually indicate: linear or exponential change?
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Exponential change. Constant percent growth creates exponential patterns.
Exponential change. Constant percent growth creates exponential patterns.
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Identify the constant rate of change for $y=7x-3$.
Identify the constant rate of change for $y=7x-3$.
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$7$. The coefficient of $x$ is the rate of change.
$7$. The coefficient of $x$ is the rate of change.
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Find the constant speed if distance is $60$ miles in $2$ hours at steady speed.
Find the constant speed if distance is $60$ miles in $2$ hours at steady speed.
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$30$ miles per hour. Rate equals distance divided by time: $\frac{60}{2}=30$.
$30$ miles per hour. Rate equals distance divided by time: $\frac{60}{2}=30$.
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Which relationship has constant rate: $y=x+3$ or $y=3^x$?
Which relationship has constant rate: $y=x+3$ or $y=3^x$?
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$y=x+3$. Linear has constant rate; exponential has constant ratio.
$y=x+3$. Linear has constant rate; exponential has constant ratio.
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What table feature confirms a constant rate when $x$ increases by $1$ each row?
What table feature confirms a constant rate when $x$ increases by $1$ each row?
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The first differences of $y$ are constant. Equal spacing in $x$ with constant $y$ differences confirms linearity.
The first differences of $y$ are constant. Equal spacing in $x$ with constant $y$ differences confirms linearity.
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Which statement indicates constant rate: "adds $4$ each hour" or "multiplies by $4$ each hour"?
Which statement indicates constant rate: "adds $4$ each hour" or "multiplies by $4$ each hour"?
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"Adds $4$ each hour". Addition indicates constant rate, multiplication indicates exponential.
"Adds $4$ each hour". Addition indicates constant rate, multiplication indicates exponential.
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Identify whether the data show constant rate: $x:0,1,2$ and $y:5,9,16$.
Identify whether the data show constant rate: $x:0,1,2$ and $y:5,9,16$.
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No, first differences are not constant. Differences are $4$ then $7$, which are not equal.
No, first differences are not constant. Differences are $4$ then $7$, which are not equal.
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Which situation represents constant rate: "$60$ miles in $2$ hours at steady speed" or "speeding up"?
Which situation represents constant rate: "$60$ miles in $2$ hours at steady speed" or "speeding up"?
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"$60$ miles in $2$ hours at steady speed". Steady speed means constant rate; speeding up means changing rate.
"$60$ miles in $2$ hours at steady speed". Steady speed means constant rate; speeding up means changing rate.
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What formula gives the constant rate of change between two points $(x_1,y_1)$ and $(x_2,y_2)$?
What formula gives the constant rate of change between two points $(x_1,y_1)$ and $(x_2,y_2)$?
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$m=\frac{y_2-y_1}{x_2-x_1}$. Standard slope formula using two points.
$m=\frac{y_2-y_1}{x_2-x_1}$. Standard slope formula using two points.
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