Interpret Points on Proportional Graphs - 7th Grade Math

1. What does a point $(x,y)$ represent on a graph of a proportional relationship in context?

   $x$ units of input correspond to $y$ units of output. Each coordinate shows the amount of one quantity for the given amount of the other.

2. Identify the graph feature that must be true for any proportional relationship.

   The graph is a straight line that passes through $(0,0)$. Proportional relationships always pass through the origin.

3. Identify the missing coordinate: if $y=\frac{3}{2}x$, what is the point when $x=1$?

   $(1,\frac{3}{2})$

4. A graph has points $(3,9)$ and $(4,15)$. Is it proportional based on $\frac{y}{x}$?

   No, because $\frac{9}{3}=3$ but $\frac{15}{4}\ne 3$

5. What is the unit rate $r$ in a proportional relationship written as $y=rx$?

   $r$ is the output per $1$ unit of input. Unit rate means the rate for one unit of input.

6. Decide if $(3,12)$ and $(5,18)$ can be on the same proportional graph through $(0,0)$.

   No; $\frac{12}{3}=4$ but $\frac{18}{5}=3.6$. Different ratios mean they can't share the same proportional line.

7. What equation form describes a proportional relationship shown on a coordinate plane?

   $y=rx$, where $r$ is the constant of proportionality. This linear equation passes through the origin with slope $r$.

8. Find the missing value $x$ if $y=2x$ and $y=18$.

   $x=9$. Divide $y$ by the unit rate: $18 \div 2 = 9$.

9. What must be true about the graph of a proportional relationship in the coordinate plane?

   It is a straight line through $(0,0)$. Proportional relationships always pass through the origin.

10. Find $r$ for a proportional relationship that includes the point $(4,10)$.

    $r=\frac{10}{4}=\frac{5}{2}$. Divide $y$ by $x$: $\frac{10}{4}=\frac{5}{2}$.

11. What equation form describes any proportional relationship between $x$ and $y$ on a graph?

    $y=rx$. The constant $r$ is the unit rate that relates $x$ and $y$ proportionally.

12. What is the unit rate $r$ in the proportional relationship $y=rx$ in terms of $x$ and $y$?

    $r=\frac{y}{x}$ for $x\ne 0$. Divide output by input to find the constant ratio.

13. What does the point $(0,0)$ mean in any proportional relationship context?

    Zero input gives zero output; there is no starting amount. Proportional relationships always start at the origin.

14. What equation form represents any proportional relationship graphed on a coordinate plane?

    $y=rx$, where $r$ is the constant of proportionality

15. A proportional graph shows $(2,8)$. What point must also be on the graph?

    $(1,4)$. Find $r=\frac{8}{2}=4$, so $(1,4)$ must be on the graph.

16. Find the output when $r=3$ and $x=7$ in the proportional relationship $y=rx$.

    $y=21$. Substitute into $y=rx$: $y=3(7)=21$.

17. What does the point $(0,0)$ mean in a proportional relationship context?

    Zero input gives zero output. In proportional relationships, no input always produces no output.

18. Which point on $y=rx$ shows the unit rate directly without dividing?

    $(1,r)$. When $x=1$, $y=r$ directly shows the unit rate.

19. A graph shows $(1,6)$ and $(0,0)$. What is the equation of the proportional relationship?

    $y=6x$. Since $(1,6)$ is on the graph, $r=6$.

20. In a proportional relationship, what does the point $(10,0)$ mean about the unit rate $r$?

    $r=0$ (the relationship is $y=0$ for all $x$). If $y=0$ when $x=10$, then $r=\frac{0}{10}=0$.

21. Which statement is always true for a proportional relationship: $\frac{y}{x}$ is constant or $y-x$ is constant?

    $\frac{y}{x}$ is constant. The ratio remains the same for all points on the line.

22. What does the point $(0,0)$ mean on a proportional relationship graph in context?

    Zero input gives zero output

23. Which point must be on every proportional relationship graph: $(0,0)$ or $(0,3)$?

    $(0,0)$

24. Identify the point that shows the unit rate if $r=\frac{3}{4}$.

    $(1,\frac{3}{4})$. When $x=1$, $y$ equals the unit rate $r$.

25. Find the unit rate $r$ if a proportional graph contains the point $(4,10)$.

    $r=\frac{10}{4}=\frac{5}{2}$. Divide $y$ by $x$ to find the constant rate.

26. A graph shows $5$ dollars for $2$ hours. What point represents this on a $(\text{hours},\text{dollars})$ graph?

    $(2,5)$. The ordered pair shows (input, output) values.

27. What is the unit rate $r$ if a proportional graph contains the point $(1,5)$?

    $r=5$

28. A proportional relationship has unit rate $r=5$. What is the point on the graph when $x=1$?

    $(1,5)$. When $x=1$, $y$ equals the unit rate.

29. A proportional relationship has $r=\frac{3}{4}$. What point shows the unit rate?

    $(1,\frac{3}{4})$. The point $(1,r)$ directly displays the unit rate.

30. Choose the correct interpretation of $(0,0)$ for $y=\frac{3}{2}x$: output is $0$ when input is $0$ or output is $\frac{3}{2}$ when input is $0$?

    Output is $0$ when input is $0$. The origin represents zero of both quantities.

31. Identify whether the point $(3,8)$ can be on $y=\frac{8}{3}x$.

    Yes, because $8=\frac{8}{3}\cdot 3$. Check if $y=rx$: $8=\frac{8}{3} \cdot 3=8$ ✓

32. What is the $y$-intercept of any proportional relationship graph $y=rx$?

    The $y$-intercept is $0$ (the graph passes through $(0,0)$)

33. What ratio is constant for every point $(x,y)$ on a proportional graph with $x\ne 0$?

    $\frac{y}{x}=r$

34. Identify the constant of proportionality $r$ if the graph passes through $(6,2)$.

    $r=\frac{2}{6}=\frac{1}{3}$

35. How do you find the constant of proportionality from a point $(x,y)$ with $x\ne 0$?

    $r=\frac{y}{x}$. Divide output by input to find the constant ratio.

36. Identify $y$ when $r=3$ and $x=7$ for a proportional relationship $y=rx$.

    $y=21$

37. A graph has points $(2,8)$ and $(5,20)$. Is it proportional based on $\frac{y}{x}$?

    Yes, because $\frac{8}{2}=\frac{20}{5}=4$

38. What does it mean if a proportional graph includes the point $(1,0.6)$ in context?

    The unit rate is $0.6$ output per $1$ input

39. What does it mean if a graph is a line but does not pass through $(0,0)$?

    It is not proportional; it has a nonzero starting value. A $y$-intercept other than zero indicates a starting value.

40. If the constant of proportionality is $r=-2$, what does the point $(1,-2)$ show?

    The unit rate is $-2$ output per $1$ input

41. Choose the correct interpretation of $(4,12)$ for $y=3x$: $4$ items cost $12$ dollars or $12$ items cost $4$ dollars?

    $4$ items cost $12$ dollars

42. Find the unit rate $r$ for $y=rx$ if the graph includes the point $(4,10)$.

    $r=\frac{10}{4}=\frac{5}{2}$

43. Find and correct the error: A student says $(1,r)$ means $1$ unit of $y$ for $r$ units of $x$.

    Correct: $(1,r)$ means $r$ units of $y$ for $1$ unit of $x$. The point $(1,r)$ shows $1$ unit of input produces $r$ units of output.

44. Find the missing value $y$ if $y=3x$ and $x=7$.

    $y=21$. Multiply $x$ by the unit rate: $3 \times 7 = 21$.

45. A proportional relationship passes through $(0,0)$ and $(6,9)$. What is $r$?

    $r=\frac{9}{6}=\frac{3}{2}$. Use $r=\frac{y}{x}=\frac{9}{6}=\frac{3}{2}$.

46. Find $x$ when $r=\frac{1}{4}$ and $y=6$ for $y=rx$.

    $x=24$

47. What does the point $(1,r)$ represent on the graph of $y=rx$?

    The unit rate: $r$ output per $1$ input

48. What does the slope of a proportional relationship line represent in context?

    The unit rate $r$ (change in $y$ per $1$ in $x$). Slope equals the constant ratio between $y$ and $x$.

49. Which point on a proportional graph directly shows the unit rate $r$: $(0,0)$ or $(1,r)$?

    $(1,r)$. The $y$-coordinate equals $r$ when $x=1$.

50. Find the input when $r=5$ and $y=40$ in the proportional relationship $y=rx$.

    $x=8$. Solve $40=5x$ by dividing both sides by 5.

51. What does a point $(x,y)$ on a proportional relationship graph represent in context?

    $(x,y)$ means $x$ units of input correspond to $y$ units of output

52. A graph includes $(1,2.5)$. What is the unit rate $r$?

    $r=2.5$. At $(1,r)$, the $y$-value equals the unit rate.

53. What does the point $(1,r)$ mean on the graph of $y=rx$?

    The output for $1$ unit of input is $r$ (unit rate). This special point reveals the constant of proportionality directly.

54. A recipe uses $6$ cups of flour for $4$ batches. What is the unit rate $r$ in $y=rx$ with $x=\text{batches}$?

    $r=\frac{6}{4}=\frac{3}{2}$ cups per batch. Divide total cups by total batches for cups per batch.

55. What does the point $(1,r)$ mean on the graph $y=rx$?

    The output for $1$ input is $r$; $r$ is the unit rate. The $x$-coordinate of 1 makes the $y$-coordinate equal to $r$.

56. Identify the $y$-intercept of any proportional relationship graph.

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

57. Which graph feature shows a relationship is proportional: straight line through $(0,0)$ or any curve?

    A straight line through $(0,0)$

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