Proportional Relationships

• Study Guide: Quiz #11

Identifying Proportional Relations From Equations

• All proportional relationships follow the equational form: y = kx, where k is the constant of proportionality.
• y = 3x is proportional.
• d = 9t is proportional.
• y = x/8 is proportional.
• y = 2 - (1/x) is not proportional.
• y = 2/x is not proportional.
• y = 4x + 5 is not proportional.

Proportional Relationship From A Graph

• All must be true for a graph to show a porportional relationship:
  • There must be a straight line.
  • The line must go through the origin (0, 0).
• A straight line that goes through the origin (0, 0) indicates a proportional relationship.
• A straight line that does not go through the origin (0, 0) does not indicate a proportional relationship.

Finding The Constant of Proportionality Reading A Graph

• Ordered pairs are in the form (x, y) and these are often plotted on graphs.
• When x is 1, y is the constant (1, k).
• Example 1:
  • The points (0, 10), (1, 8), (2, 6), (3, 4), (4, 2) are plotted on a coordinate plane.
  • Relationship is not proportional, as the line does not go through the origin (0, 0), even though it is straight.
• Example 2:
  • There is a subscription service where there is a proportional relationship between the months of subscription and the total amount of money paid.
• After 6 months, $47.94 has been paid; point (6, 47.94) is on graph.
  • The constant of proportionality is given by (1, k) -> (1, 10) -> k = 10.
  • Every month of the streaming service costs $10.
  • Equation for the relationship between total cost (C) and number of months (m) is expressed as y = 10x, simplified to c = 10t.
• From the equation c = 10t:
  • At t = 2, then c = 10(2) = 20 as expressed by point (2, 20).
  • At t = 8, then c = 10(8) = 80 as expressed by point (8, 80).
  • At t = 14, then c = 10(14) = 140 as expressed by point (14, 140).
  • Point (8, 80) indicates it costs $80 for 8 months of streaming service.
• Example 3:
  • Mia and Jamal bike home from school at a steady pace.
  • Mia bikes 1.25 kilometers in 4 mins as expressed by point (4, 1.25).
  • Jamal bikes 1.75 kilometers in 7 mins as expressed by point (7, 1.75).
  • Both lines are proportional, and crosses through (0,0).
• To find each value of k, use k = y/x.
• Mia:
  • At point (4, 1.25), k = 1.25/4 = 0.31 as expressed by its point (1, 0.31).
• Jamal:
  • At point (7, 1.75), k = 1.75/7 = 0.25 as expressed by the its point (1, 0.25).
• Mia bikes faster because her constant is greater than Jamal's; 0.31 > 0.25.

Test your understanding of proportional relationships with questions on identifying equations, graphs, and constants of proportionality. Explore real-world examples involving costs, biking speeds, and more. Determine the constant of proportionality.