Recognize Constant Rate Changes - Algebra 1

1. Find the unit rate if $y=-12$ when $x=3$.

   $-4$. Dividing change: $\frac{-12}{3} = -4$ units per unit.

2. Find the constant rate if $y$ changes from $30$ to $18$ while $x$ changes from $2$ to $8$.

   $-2$. Rate equals $\frac{18-30}{8-2} = \frac{-12}{6} = -2$.

3. Which expression represents a constant rate of change: $y=4x+9$ or $y=4^x+9$?

   $y=4x+9$. Linear has constant rate; exponential has variable rate.

4. Identify the constant rate if a taxi charges $3 plus $2 per mile.

   $2$ dollars per mile. The coefficient of miles gives the per-mile rate.

5. Identify the constant rate of change from the points $(-2,7)$ and $(4,-5)$.

   $-2$. Using slope formula: $\frac{-5-7}{4-(-2)} = \frac{-12}{6} = -2$.

6. Find the unit rate (slope) if $y=18$ when $x=6$.

   $3$. Dividing total change: $\frac{18}{6} = 3$ units per unit.

7. A table has equal $x$ steps of $1$ and $y$ values $1,4,9,16$. Is the rate constant?

   No; the changes are $3,5,7$. These are perfect squares; differences increase: $3, 5, 7$.

8. Identify the slope (rate) of the line through $(2,-1)$ and $(2,6)$.

   Undefined (division by $0$). Both points have same $x$-value, so $\Delta x = 0$.

9. Choose the word that signals constant rate: "per" or "percent"?

   "Per" (as in "per unit"). "Per" indicates rate per unit; "percent" often suggests proportional change.

10. Which model matches constant rate: $y=y_0+rt$ or $y=y_0(1+r)^t$?

    $y=y_0+rt$. Linear form $y_0 + rt$ has constant rate; exponential has variable rate.

11. Identify the slope between $ (2,5) $ and $ (6,13) $ to test if the rate is constant.

    $2$. Using slope formula: $\frac{13-5}{6-2} = \frac{8}{4} = 2$.

12. Which situation shows constant rate: "$y$ increases by $8$ every $2$ minutes" or "$y$ increases by $8$ every minute"?

    Both (each states a constant per-time change). Both describe fixed amounts per time interval (constant rates).

13. Identify the constant rate of change from the points $(3,2)$ and $(9,14)$.

    $2$. Using slope formula: $\frac{14-2}{9-3} = \frac{12}{6} = 2$.

14. What does a vertical line mean about a rate per unit of $x$?

    Undefined rate; it is not a function of $x$. Vertical lines have zero $\Delta x$, making slope undefined.

15. A table has equal $x$ steps of $1$ and $y$ values $2,6,10,14$. Is the rate constant?

    Yes; the change is $+4$ each step. Each $y$-value increases by exactly $4$ for each unit $x$ increase.

16. Identify the slope (rate) of the line through $(0,4)$ and $(5,4)$.

    $0$. Both points have same $y$-value, so $\Delta y = 0$.

17. Which phrase indicates constant rate: "$+12$ each year" or "multiplied by $1.12$ each year"?

    "$+12$ each year.". Addition shows constant rate; multiplication shows exponential growth.

18. A plant grows $15$ cm in $3$ weeks at a steady pace. What is the constant rate per week?

    $5$ cm per week. Divide total growth by time: $\frac{15}{3} = 5$ cm per week.

19. If $x$ increases by $4$ and $y$ decreases by $12$, what is the constant rate of change?

    $-3$. Rate equals $\frac{-12}{4} = -3$ (negative because $y$ decreases).

20. Identify the initial value in $y=mx+b$ that pairs with constant rate $m$.

    $b$ (the value of $y$ when $x=0$). The $y$-intercept $b$ is the starting value when $x = 0$.

21. In $y=-5x+12$, what is the constant rate and what does it mean per $1$ unit of $x$?

    Rate $-5$; $y$ decreases by $5$ for each $+1$ in $x$. The coefficient $-5$ means $y$ drops $5$ units per $x$ unit.

22. Find the unit rate if $y=-12$ when $x=3$.

    $-4$. Dividing change: $\frac{-12}{3} = -4$ units per unit.

23. A car travels $180$ miles in $3$ hours at constant speed. What is the constant rate?

    $60$ miles per hour. Divide distance by time: $\frac{180}{3} = 60$ miles per hour.

24. Which equation form shows a constant rate of change: $y=mx+b$ or $y=ax^2+bx+c$?

    $y=mx+b$. Linear form has constant slope $m$; quadratic has variable rate.

25. What does a horizontal line on a graph mean about the constant rate of change?

    Rate is $0$ (no change in $y$ as $x$ changes). Horizontal lines have zero slope since $y$ never changes.

26. A table has $x$ steps of $2$ and $y$ values $1,7,13,19$. What is the constant rate?

    $3$. Changes are $6, 6, 6$ for steps of $2$, so rate is $\frac{6}{2} = 3$.

27. What is the name of the constant ratio $y/x$ for a linear relationship?

    Slope (constant rate of change). In linear equations, the coefficient of $x$ represents the constant rate.

28. Identify if the rate is constant: $(0,2)$, $(2,5)$, $(5,11)$.

    No; slopes are $\frac{3}{2}$ and $2$. Slopes differ: $\frac{5-2}{2-0} = 1.5$, $\frac{11-5}{5-2} = 2$.

29. Which situation is constant rate: $y$ increases by $3$ when $x$ increases by $1$, always; or increases by $3,6,12$?

    The first situation (always $+3$ per $+1$). Consistent change shows constant rate; varying increases indicate non-linear.

30. Convert "increases by $8$ every $2$ minutes" to a per-$1$-minute constant rate.

    $4$ per minute. Divide the change by time: $\frac{8}{2} = 4$ per minute.

31. Identify the constant rate if the line passes through $(0,-3)$ and $(4,5)$.

    $2$. Using slope formula: $\frac{5-(-3)}{4-0} = \frac{8}{4} = 2$.

32. What is the constant rate of change in the equation $C=2.5t+40$?

    $2.5$. The coefficient $2.5$ represents the rate per unit of $t$.

33. What does a positive constant rate of change indicate about the relationship between $x$ and $y$?

    As $x$ increases, $y$ increases at a constant rate. Positive slope means $y$ increases as $x$ increases.

34. If $x$ increases by $2$ each step and $y$ increases by $10$ each step, what is the constant rate $\Delta y / \Delta x$?

    $5$. Rate equals $\frac{\Delta y}{\Delta x} = \frac{10}{2} = 5$.

35. Identify the slope between $(2,5)$ and $(6,13)$ to test if the rate is constant.

    $2$. Using slope formula: $\frac{13-5}{6-2} = \frac{8}{4} = 2$.

36. Identify if the rate is constant: $(0,0)$, $(2,6)$, $(5,15)$.

    Yes; each slope is $3$. All slopes equal $3$: $\frac{6-0}{2-0} = 3$, $\frac{15-6}{5-2} = 3$.

37. What must be true about slopes between any two pairs of points on a line?

    All computed slopes are equal (constant). Linear relationships maintain identical slope between any point pairs.

38. If $x$ increases by $2$ each step and $y$ increases by $10$ each step, what is the constant rate $y/x$?

    $5$. Rate equals $\frac{\Delta y}{\Delta x} = \frac{10}{2} = 5$.

39. Find the unit rate (slope) if $y=18$ when $x=6$.

    $3$. Dividing total change: $\frac{18}{6} = 3$ units per unit.

40. Which graph shape indicates a constant rate of change: straight line or curved line?

    Straight line. Constant slope produces straight lines; variable slope creates curves.

41. What is the constant rate of change in the equation $y=-x+10$?

    $-$. The coefficient of $x$ gives the constant rate of change.

42. What does a negative constant rate of change indicate about the relationship between $x$ and $y$?

    As $x$ increases, $y$ decreases at a constant rate. Negative slope means $y$ decreases as $x$ increases.

43. Which description indicates a constant rate: "adds $5$ each hour" or "doubles each hour"?

    "Adds $5$ each hour.". Fixed addition indicates constant rate; doubling shows exponential growth.

44. Identify the slope between $(1,9)$ and $(4,3)$.

    $-2$. Using slope formula: $\frac{3-9}{4-1} = \frac{-6}{3} = -2$.

45. Identify whether the relationship is constant rate: points $(0,1)$, $(1,2)$, $(2,4)$.

    No; slopes change ($1$ then $2$). Slopes differ: $(2-1)/(1-0) = 1$, $(4-2)/(2-1) = 2$.

46. What is the constant rate of change in the equation $y=7x-3$?

    $7$. The coefficient of $x$ is the slope (constant rate of change).

47. A quantity follows $y=50+6t$. What is the constant rate and in what units?

    $6$ units of $y$ per $1$ unit of $t$. The coefficient $6$ gives the rate: $6$ $y$-units per $t$-unit.

48. What does it mean for $y$ to change at a constant rate per unit of $x$?

    The ratio $y/x$ is constant for all intervals. This defines linear relationships where change in $y$ per unit $x$ stays the same.

49. Identify whether the statement implies constant rate: "The balance decreases by $25$ dollars each month."

    Yes; it is a constant decrease of $25$ per month. Fixed decrease per month indicates linear (constant rate) change.

50. Identify whether the relationship is constant rate: points $(0,2)$, $(1,5)$, $(2,8)$.

    Yes; slope is $3$ each step. Each consecutive slope equals $3$: $(5-2)/(1-0) = 3$, $(8-5)/(2-1) = 3$.

51. Which table pattern indicates constant rate: constant first differences in $y$ or constant second differences?

    Constant first differences in $y$ (for equal $x$ steps). Linear relationships show equal $y$-differences for equal $x$-steps.

52. Identify whether the statement implies constant rate: "The balance decreases by $10\%$ each month."

    No; percent decrease suggests exponential change, not constant difference. Percentage decreases create exponential, not linear relationships.

53. What is the slope formula for the rate of change from $(x_1,y_1)$ to $(x_2,y_2)$?

    $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$. This formula calculates the constant rate between two points.

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