Graph Proportional Relationships - 8th Grade Math

1. Identify the constant of proportionality for the table rule $y = 5x$.

   $k = 5$. The coefficient of $x$ in $y = 5x$ is the constant.

2. What point must be on the graph of any proportional relationship $y = kx$?

   $(0,0)$. All proportional relationships pass through the origin.

3. What is the slope of the line representing $y = kx$ on a coordinate plane?

   $k$. In $y = kx$, the coefficient $k$ is the slope.

4. Identify the unit rate (slope) for the proportional relationship $y = 6x$.

   $6$. The coefficient of $x$ is the unit rate.

5. Identify the slope for the proportional graph that passes through $(0,0)$ and $(2,-6)$.

   $\frac{-6}{2} = -3$. Calculate slope using two points: $\frac{-6-0}{2-0}$.

6. What is the equation of a proportional relationship with unit rate $k=9$?

   $y = 9x$. Substitute $k$ into the form $y = kx$.

7. What is the slope of the line $y = -\frac{1}{2}x$ and what does its sign indicate?

   $-\frac{1}{2}$; $y$ decreases as $x$ increases. Negative slope means inverse relationship.

8. Find the unit rate from the table: $(t,d)=(1,7),(2,14),(3,21)$.

   $7$. Distance increases by $7$ for each unit of time.

9. A proportional graph includes $(1,7)$. What is the equation in the form $y = kx$?

   $y = 7x$. Since $(1,7)$ is on the line, $k = \frac{7}{1} = 7$.

10. What is the slope of a proportional relationship if $y$ increases by $9$ when $x$ increases by $3$?

    $\frac{9}{3} = 3$. Slope equals change in $y$ over change in $x$.

11. What is the $y$-intercept of any proportional relationship graphed as a line on a coordinate plane?

    $0$. All proportional relationships pass through $(0,0)$.

12. What is the slope (unit rate) of a proportional relationship written as $y = kx$?

    $k$. In $y = kx$, the coefficient $k$ represents the slope.

13. Find $k$ for a proportional relationship where $y = 18$ when $x = 9$.

    $k = 2$. Solve $k = \frac{y}{x} = \frac{18}{9} = 2$.

14. Choose the greater unit rate: $y = 2.5x$ or $y = \frac{11}{4}x$.

    $y = \frac{11}{4}x$. $\frac{11}{4} = 2.75 > 2.5$, so greater unit rate.

15. What is the definition of a proportional relationship in terms of $y$ and $x$?

    $y = kx$ for a constant $k$. Direct variation where $y$ is always $k$ times $x$.

16. Which equation represents a proportional relationship: $y = 3x + 2$ or $y = 3x$?

    $y = 3x$. Only $y = 3x$ passes through $(0,0)$.

17. What is the slope of the line $y = -\frac{3}{2}x$?

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

18. Identify the slope of the line through $(0,0)$ and $(4,-10)$.

    $-\frac{5}{2}$. Use $m = \frac{-10-0}{4-0} = \frac{-10}{4} = -\frac{5}{2}$

19. Which equation represents a proportional relationship: $y = 3x$ or $y = 3x + 2$?

    $y = 3x$. Only $y = 3x$ passes through $(0,0)$.

20. Which representation has the greater unit rate: $y=2.8x$ or a graph with slope $3.1$?

    The graph with slope $3.1$. Compare unit rates: $3.1 > 2.8$.

21. What is the unit rate (slope) for the table pair $x=4$, $y=14$ in a proportional relationship?

    $k = \frac{14}{4} = 3.5$. Divide $y$ by $x$ to find the constant of proportionality.

22. Which object is faster: A has $d=5t$, B has $d=4t$ (same units)?

    A. Compare coefficients: $5 > 4$.

23. Find and correct the error: "For $y=\frac{3}{4}x$, the slope is $\frac{4}{3}$." What is the correct slope?

    Correct slope: $\frac{3}{4}$. The coefficient $\frac{3}{4}$ is the slope, not its reciprocal.

24. Identify whether the relationship is proportional if the graph line crosses the $y$-axis at $2$.

    Not proportional. Must pass through $(0,0)$ to be proportional.

25. Find the constant of proportionality $k$ if $y$ is proportional to $x$ and $(x,y)=(8,-20)$.

    $k = \frac{-20}{8} = -\frac{5}{2}$. Use $k = \frac{y}{x}$ with the given point.

26. What is the speed (unit rate) for the distance-time equation $d = 12t$?

    $12$ (distance units per time unit). Coefficient of $t$ gives speed in distance per time.

27. Which object has greater speed: A passes through $(2,14)$ on a $d$ vs. $t$ graph, B has $d = 6t$?

    Object A. A has slope $\frac{14}{2} = 7 > 6$, so A is faster.

28. What is the unit rate for the ratio $\frac{24}{6}$ (as $y$ per $x$)?

    $4$. Divide $y$ by $x$: $\frac{24}{6} = 4$.

29. What is the constant of proportionality for a proportional relationship shown by the ratio $\frac{y}{x}$?

    $k = \frac{y}{x}$. The constant ratio of $y$ to $x$ equals $k$.

30. What is the slope formula between two points $ ($x_1$,$y_1$) $ and $ ($x_2$,$y_2$) $?

    $m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run gives the rate of change between points.

31. Which object is faster: A has slope $3$, B has slope $\frac{7}{2}$ (same units)?

    B. Compare slopes: $\frac{7}{2} = 3.5 > 3$.

32. Find the slope of the proportional relationship through $(0,0)$ and $(3,12)$.

    $m = 4$. Use $m = \frac{12-0}{3-0} = \frac{12}{3} = 4$.

33. A car travels $180$ miles in $3$ hours. What is the speed (unit rate) in $\frac{\text{miles}}{\text{hour}}$?

    $60$. Speed = $\frac{\text{distance}}{\text{time}} = \frac{180}{3}$.

34. What is the unit rate in the equation $y = kx$?

    $k$ (the value of $y$ when $x = 1$). The constant $k$ represents the rate of change per unit of $x$.

35. Compare: A has slope $3$ on a $d$ vs. $t$ graph; B has equation $d = 2.8t$. Which is faster?

    Object A. Slope $3 > 2.8$, so A is faster.

36. Compare speeds: Object A $d = 45t$ and Object B $d = 40t$. Which is faster?

    Object A. Compare coefficients: $45 > 40$.

37. What point must the graph of any proportional relationship include on the coordinate plane?

    $(0,0)$. Proportional relationships always pass through the origin.

38. Choose the correct slope for a distance-time graph that goes through $(0,0)$ and $(4,30)$.

    $\frac{30}{4} = 7.5$. Use slope formula: $\frac{30-0}{4-0}$.

39. Find the slope of the line through $(0,0)$ and $(5,20)$, and interpret it as a unit rate.

    $\frac{20}{5} = 4$. Slope = rise over run = $\frac{20-0}{5-0}$.

40. What is the slope of a distance-time graph if distance increases $150$ meters in $5$ seconds?

    $30$. Slope = $\frac{Delta d}{Delta t} = \frac{150}{5}$.

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