Recognize Constant Rate Changes - Algebra 2

1. Find the constant speed if distance is $60$ miles in $2$ hours at steady speed.

   $30$ miles per hour. Rate equals distance divided by time: $\frac{60}{2}=30$.

2. What table feature confirms a constant rate when $x$ increases by $1$ each row?

   The first differences of $y$ are constant. Equal spacing in $x$ with constant $y$ differences confirms linearity.

3. What phrase most strongly signals constant rate: "per" or "percent"?

   "Per" (constant additive rate per unit). "Per" indicates additive rate; "percent" indicates multiplicative rate.

4. Identify whether the data show constant rate: $x:0,1,2$ and $y:5,9,16$.

   No, first differences are not constant. Differences are $4$ then $7$, which are not equal.

5. What does constant rate imply about $\Delta y$ when $\Delta x$ doubles?

   $\Delta y$ doubles as well. Constant rate means proportional relationship between $\Delta x$ and $\Delta y$.

6. Identify the constant rate of change for $y=7x-3$.

   $7$. The coefficient of $x$ is the rate of change.

7. Identify whether the situation is constant rate: $y$ doubles whenever $x$ increases by $1$.

   No, that is exponential (not constant difference). Doubling is multiplicative change, not constant additive change.

8. What does a negative constant rate of change mean about $y$ as $x$ increases?

   $y$ decreases by a constant amount per unit increase in $x$. Negative slope means $y$ decreases as $x$ increases.

9. Which description matches constant rate: "$y$ changes by a fixed amount" or "$y$ changes by a fixed factor"?

   "$y$ changes by a fixed amount". Fixed amount indicates additive; fixed factor indicates multiplicative.

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

    They are equal (constant slope). All points on a line have the same slope between any pair.

11. What is the constant rate of change in $y=2+4x$?

    $4$. The coefficient of $x$ gives the rate of change.

12. What is the name of the constant rate of change in a linear function $y=mx+b$?

    The slope $m$. The coefficient of $x$ represents the constant rate.

13. What is the constant rate of change for the proportional relationship $y=5x$?

    $5$. In $y=kx$, the constant $k$ is the rate.

14. Identify the constant rate in the verbal rule: "Start at $10$ and add $2$ each day."

    $2$ per day. The additive change per time unit.

15. What is the constant rate of change for the line $3x-2y=8$?

    $\frac{3}{2}$. Rearranging to $y=\frac{3}{2}x-4$ shows slope $\frac{3}{2}$.

16. Find the rate of change from $(2,5)$ to $(6,13)$.

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

17. What does $m=0$ mean about the rate of change in $y=mx+b$?

    No change; $y$ is constant. Zero slope means $y$ stays the same regardless of $x$.

18. Identify whether $y=x^2+1$ has a constant rate of change.

    No, the rate of change is not constant. Quadratic functions have changing rates of change.

19. Identify whether the situation is constant rate: $y$ increases by $3$ whenever $x$ increases by $1$.

    Yes, constant rate ($m=3$). Fixed increase of $3$ for each unit of $x$ means constant rate.

20. What does "constant percent rate" usually indicate: linear or exponential change?

    Exponential change. Constant percent growth creates exponential patterns.

21. Identify whether the sequence rule $y=2+4x$ represents constant change per step in $x$.

    Yes, constant change of $4$ per $1$ in $x$. Linear form with coefficient $4$ as the constant rate.

22. What graph shape indicates a constant rate of change between $x$ and $y$?

    A straight line (linear relationship). Linear functions have constant slope, creating straight lines.

23. Identify the initial value in "Start at $10$ and add $2$ each day" for $y$ at day $0$.

    $10$. The starting value when the independent variable equals zero.

24. Choose the model type for constant rate: $y=mx+b$ or $y=a\cdot b^x$.

    $y=mx+b$. Linear models have constant rates; exponential models have constant ratios.

25. Find the slope of the line through $(1,9)$ and $(5,1)$.

    $m=\frac{1-9}{5-1}=-2$. Using slope formula: $\frac{1-9}{5-1}=\frac{-8}{4}=-2$.

26. What does the phrase "per unit" refer to in a constant rate of change?

    The change in one variable for $\Delta x=1$. Rate of change when $x$ increases by exactly $1$.

27. What is the constant rate if $y$ always increases by $10$ when $x$ increases by $2$?

    $5$ per unit $x$. Rate per unit: $\frac{10}{2}=5$ per unit $x$.

28. What is the constant rate if $x:0,1,2$ and $y:5,8,11$?

    $3$ per unit $x$. Change in $y$ divided by change in $x$: $\frac{3}{1}=3$.

29. Identify whether this indicates constant rate: When $x$ increases by $2$, $y$ always increases by $10$.

    Yes, constant rate ($m=5$). Fixed change of $10$ for every $2$ in $x$ means constant rate.

30. Identify the constant rate if $y$ increases by $15$ when $x$ increases by $5$ (same every time).

    $3$ per unit $x$. Rate equals total change divided by input change: $\frac{15}{5}=3$.

31. Which situation represents constant rate: "$60$ miles in $2$ hours at steady speed" or "speeding up"?

    "$60$ miles in $2$ hours at steady speed". Steady speed means constant rate; speeding up means changing rate.

32. What formula gives the constant rate of change between two points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Standard slope formula using two points.

33. Identify whether the points $(0,2)$, $(2,6)$, $(4,10)$ show a constant rate of change.

    Yes, constant rate ($m=2$). All consecutive slopes equal $2$, confirming constant rate.

34. Which is constant rate: $y$ changes by $-6$ when $x$ changes by $3$, or by $-7$ when $x$ changes by $3$?

    The $-6$ for $\Delta x=3$ case (rate $-2$ per $1$). Rate of $-2$ per unit gives $-6$ change for $\Delta x=3$.

35. Which change pattern indicates constant rate: constant differences in $y$ or constant ratios in $y$?

    Constant differences in $y$ (for equal $\Delta x$). Constant differences indicate linear; constant ratios indicate exponential.

36. Which slope indicates constant decrease of $5$ per $1$ in $x$: $m=-5$ or $m=5$?

    $m=-5$. Negative slope means decrease of $5$ per unit increase in $x$.

37. Identify whether $y=4x+1$ has a constant rate of change.

    Yes, constant rate ($m=4$). Linear functions have constant slope everywhere.

38. Identify whether $3x-2y=8$ represents a constant rate of change relationship.

    Yes, it is linear (constant slope). Linear equations represent constant rate relationships.

39. Find the rate of change from $(-1,4)$ to $(3,-8)$.

    $m=\frac{-8-4}{3-(-1)}=-3$. Using slope formula: $\frac{-8-4}{3-(-1)}=\frac{-12}{4}=-3$.

40. Identify whether the data show constant rate: $x:0,1,2$ and $y:5,8,11$.

    Yes, constant rate ($\Delta y=3$ each step). Equal differences of $3$ between consecutive $y$-values.

41. Identify whether the points $(0,1)$, $(1,2)$, $(2,4)$ show a constant rate of change.

    No, slopes differ ($1$ then $2$). Slopes change from $1$ to $2$, so rate is not constant.

42. What does a constant ratio $\frac{y}{x}$ (with $x\neq 0$) indicate about $y$ versus $x$?

    A proportional linear relationship $y=kx$. Constant ratio means $y$ is proportional to $x$.

43. Which equation form explicitly shows constant rate and initial value: $y=mx+b$ or $y=ax^2+c$?

    $y=mx+b$. Linear form shows constant rate $m$ and starting value $b$.

44. Compute the unit rate if $\Delta y=-6$ when $\Delta x=3$.

    $\frac{\Delta y}{\Delta x}=-2$. Divide change in $y$ by change in $x$: $\frac{-6}{3}=-2$.

45. Identify the constant rate of change for $y=-\frac{3}{2}x+8$.

    $-\frac{3}{2}$. The coefficient of $x$ is the rate of change.

46. Which statement indicates constant rate: "adds $4$ each hour" or "multiplies by $4$ each hour"?

    "Adds $4$ each hour". Addition indicates constant rate, multiplication indicates exponential.

47. Identify whether this is constant rate: A population increases by $5\%$ each year.

    No, that is exponential (constant percent change). Percent change indicates exponential, not linear growth.

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

    $\frac{\Delta y}{\Delta x}$ is constant for equal $\Delta x$ intervals. The ratio of change in $y$ to change in $x$ stays the same.

49. Identify whether this is constant rate: A tank loses $3$ liters every minute.

    Yes, constant rate ($-3$ liters per minute). Fixed amount lost per time unit indicates constant rate.

50. Which relationship has constant rate: $y=x+3$ or $y=3^x$?

    $y=x+3$. Linear has constant rate; exponential has constant ratio.

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