Recognize Constant Rate Changes - Algebra
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Which table pattern indicates constant rate: constant first differences in $y$ or constant second differences?
Which table pattern indicates constant rate: constant first differences in $y$ or constant second differences?
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Constant first differences in $y$ (for equal $x$ steps). Linear relationships show equal $y$-differences for equal $x$-steps.
Constant first differences in $y$ (for equal $x$ steps). Linear relationships show equal $y$-differences for equal $x$-steps.
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Find the unit rate if $y=-12$ when $x=3$.
Find the unit rate if $y=-12$ when $x=3$.
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$-4$. Dividing change: $\frac{-12}{3} = -4$ units per unit.
$-4$. Dividing change: $\frac{-12}{3} = -4$ units per unit.
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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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The ratio $\Delta y / \Delta x$ is constant for all intervals. This defines linear relationships where change in $y$ per unit $x$ stays the same.
The ratio $\Delta y / \Delta x$ is constant for all intervals. This defines linear relationships where change in $y$ per unit $x$ stays the same.
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Identify if the rate is constant: $(0,2)$, $(2,5)$, $(5,11)$.
Identify if the rate is constant: $(0,2)$, $(2,5)$, $(5,11)$.
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No; slopes are $\frac{3}{2}$ and $2$. Slopes differ: $\frac{5-2}{2-0} = 1.5$, $\frac{11-5}{5-2} = 2$.
No; slopes are $\frac{3}{2}$ and $2$. Slopes differ: $\frac{5-2}{2-0} = 1.5$, $\frac{11-5}{5-2} = 2$.
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What is the name of the constant ratio $y/x$ for a linear relationship?
What is the name of the constant ratio $y/x$ for a linear relationship?
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Slope (constant rate of change). In linear equations, the coefficient of $x$ represents the constant rate.
Slope (constant rate of change). In linear equations, the coefficient of $x$ represents the constant rate.
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What is the constant rate of change in the equation $y=-x+10$?
What is the constant rate of change in the equation $y=-x+10$?
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$-$. The coefficient of $x$ gives the constant rate of change.
$-$. The coefficient of $x$ gives the constant rate of change.
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Which description indicates a constant rate: "adds $5$ each hour" or "doubles each hour"?
Which description indicates a constant rate: "adds $5$ each hour" or "doubles each hour"?
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"Adds $5$ each hour.". Fixed addition indicates constant rate; doubling shows exponential growth.
"Adds $5$ each hour.". Fixed addition indicates constant rate; doubling shows exponential growth.
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Which situation is constant rate: $y$ increases by $3$ when $x$ increases by $1$, always; or increases by $3,6,12$?
Which situation is constant rate: $y$ increases by $3$ when $x$ increases by $1$, always; or increases by $3,6,12$?
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The first situation (always $+3$ per $+1$). Consistent change shows constant rate; varying increases indicate non-linear.
The first situation (always $+3$ per $+1$). Consistent change shows constant rate; varying increases indicate non-linear.
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Find the unit rate (slope) if $y=18$ when $x=6$.
Find the unit rate (slope) if $y=18$ when $x=6$.
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$3$. Dividing total change: $\frac{18}{6} = 3$ units per unit.
$3$. Dividing total change: $\frac{18}{6} = 3$ units per unit.
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Identify whether the relationship is constant rate: points $(0,2)$, $(1,5)$, $(2,8)$.
Identify whether the relationship is constant rate: points $(0,2)$, $(1,5)$, $(2,8)$.
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Yes; slope is $3$ each step. Each consecutive slope equals $3$: $(5-2)/(1-0) = 3$, $(8-5)/(2-1) = 3$.
Yes; slope is $3$ each step. Each consecutive slope equals $3$: $(5-2)/(1-0) = 3$, $(8-5)/(2-1) = 3$.
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Identify whether the relationship is constant rate: points $(0,1)$, $(1,2)$, $(2,4)$.
Identify whether the relationship is constant rate: points $(0,1)$, $(1,2)$, $(2,4)$.
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No; slopes change ($1$ then $2$). Slopes differ: $(2-1)/(1-0) = 1$, $(4-2)/(2-1) = 2$.
No; slopes change ($1$ then $2$). Slopes differ: $(2-1)/(1-0) = 1$, $(4-2)/(2-1) = 2$.
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Which graph shape indicates a constant rate of change: straight line or curved line?
Which graph shape indicates a constant rate of change: straight line or curved line?
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Straight line. Constant slope produces straight lines; variable slope creates curves.
Straight line. Constant slope produces straight lines; variable slope creates curves.
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Identify the slope (rate) of the line through $(0,4)$ and $(5,4)$.
Identify the slope (rate) of the line through $(0,4)$ and $(5,4)$.
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$0$. Both points have same $y$-value, so $\Delta y = 0$.
$0$. Both points have same $y$-value, so $\Delta y = 0$.
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Identify the slope (rate) of the line through $(2,-1)$ and $(2,6)$.
Identify the slope (rate) of the line through $(2,-1)$ and $(2,6)$.
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Undefined (division by $0$). Both points have same $x$-value, so $\Delta x = 0$.
Undefined (division by $0$). Both points have same $x$-value, so $\Delta x = 0$.
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What is the constant rate of change in the equation $y=-x+10$?
What is the constant rate of change in the equation $y=-x+10$?
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$-$. The coefficient of $x$ gives the constant rate of change.
$-$. The coefficient of $x$ gives the constant rate of change.
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A plant grows $15$ cm in $3$ weeks at a steady pace. What is the constant rate per week?
A plant grows $15$ cm in $3$ weeks at a steady pace. What is the constant rate per week?
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$5$ cm per week. Divide total growth by time: $\frac{15}{3} = 5$ cm per week.
$5$ cm per week. Divide total growth by time: $\frac{15}{3} = 5$ cm per week.
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What is the slope formula for the rate of change from $(x_1,y_1)$ to $(x_2,y_2)$?
What is the slope formula for the rate of change from $(x_1,y_1)$ to $(x_2,y_2)$?
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$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$. This formula calculates the constant rate between two points.
$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$. This formula calculates the constant rate between two points.
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What does a horizontal line on a graph mean about the constant rate of change?
What does a horizontal line on a graph mean about the constant rate of change?
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Rate is $0$ (no change in $y$ as $x$ changes). Horizontal lines have zero slope since $y$ never changes.
Rate is $0$ (no change in $y$ as $x$ changes). Horizontal lines have zero slope since $y$ never changes.
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What does a vertical line mean about a rate per unit of $x$?
What does a vertical line mean about a rate per unit of $x$?
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Undefined rate; it is not a function of $x$. Vertical lines have zero $\Delta x$, making slope undefined.
Undefined rate; it is not a function of $x$. Vertical lines have zero $\Delta x$, making slope undefined.
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A table has equal $x$ steps of $1$ and $y$ values $2,6,10,14$. Is the rate constant?
A table has equal $x$ steps of $1$ and $y$ values $2,6,10,14$. Is the rate constant?
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Yes; the change is $+4$ each step. Each $y$-value increases by exactly $4$ for each unit $x$ increase.
Yes; the change is $+4$ each step. Each $y$-value increases by exactly $4$ for each unit $x$ increase.
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A table has equal $x$ steps of $1$ and $y$ values $1,4,9,16$. Is the rate constant?
A table has equal $x$ steps of $1$ and $y$ values $1,4,9,16$. Is the rate constant?
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No; the changes are $3,5,7$. These are perfect squares; differences increase: $3, 5, 7$.
No; the changes are $3,5,7$. These are perfect squares; differences increase: $3, 5, 7$.
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If $x$ increases by $2$ each step and $y$ increases by $10$ each step, what is the constant rate $y/x$?
If $x$ increases by $2$ each step and $y$ increases by $10$ each step, what is the constant rate $y/x$?
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$5$. Rate equals $\frac{\Delta y}{\Delta x} = \frac{10}{2} = 5$.
$5$. Rate equals $\frac{\Delta y}{\Delta x} = \frac{10}{2} = 5$.
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If $x$ increases by $4$ and $y$ decreases by $12$, what is the constant rate of change?
If $x$ increases by $4$ and $y$ decreases by $12$, what is the constant rate of change?
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$-3$. Rate equals $\frac{-12}{4} = -3$ (negative because $y$ decreases).
$-3$. Rate equals $\frac{-12}{4} = -3$ (negative because $y$ decreases).
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What does a negative constant rate of change indicate about the relationship between $x$ and $y$?
What does a negative constant rate of change indicate about the relationship between $x$ and $y$?
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As $x$ increases, $y$ decreases at a constant rate. Negative slope means $y$ decreases as $x$ increases.
As $x$ increases, $y$ decreases at a constant rate. Negative slope means $y$ decreases as $x$ increases.
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What does a positive constant rate of change indicate about the relationship between $x$ and $y$?
What does a positive constant rate of change indicate about the relationship between $x$ and $y$?
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As $x$ increases, $y$ increases at a constant rate. Positive slope means $y$ increases as $x$ increases.
As $x$ increases, $y$ increases at a constant rate. Positive slope means $y$ increases as $x$ increases.
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Identify the constant rate if a taxi charges $\$3$ plus $$2$ per mile.
Identify the constant rate if a taxi charges $\$3$ plus $$2$ per mile.
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$2$ dollars per mile. The coefficient of miles gives the per-mile rate.
$2$ dollars per mile. The coefficient of miles gives the per-mile rate.
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Identify the initial value in $y=mx+b$ that pairs with constant rate $m$.
Identify the initial value in $y=mx+b$ that pairs with constant rate $m$.
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$b$ (the value of $y$ when $x=0$). The $y$-intercept $b$ is the starting value when $x = 0$.
$b$ (the value of $y$ when $x=0$). The $y$-intercept $b$ is the starting value when $x = 0$.
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Identify whether the statement implies constant rate: "The balance decreases by $25$ dollars each month."
Identify whether the statement implies constant rate: "The balance decreases by $25$ dollars each month."
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Yes; it is a constant decrease of $25$ per month. Fixed decrease per month indicates linear (constant rate) change.
Yes; it is a constant decrease of $25$ per month. Fixed decrease per month indicates linear (constant rate) change.
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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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The ratio $y/x$ is constant for all intervals. This defines linear relationships where change in $y$ per unit $x$ stays the same.
The ratio $y/x$ is constant for all intervals. This defines linear relationships where change in $y$ per unit $x$ stays the same.
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Which equation form shows a constant rate of change: $y=mx+b$ or $y=ax^2+bx+c$?
Which equation form shows a constant rate of change: $y=mx+b$ or $y=ax^2+bx+c$?
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$y=mx+b$. Linear form has constant slope $m$; quadratic has variable rate.
$y=mx+b$. Linear form has constant slope $m$; quadratic has variable rate.
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