Proportional Relationships and Graphs
Identify proportional relationships from tables, graphs, and equations. Recognize the constant of proportionality k in y = kx and interpret it as a unit rate in context.
A lesson of Pre-Algebra, in Ratios, Rates & Proportional Reasoning, about 10 minutes.
Proportional Relationships
By the end of this lesson, you will be able to identify proportional relationships from tables, graphs, and equations, and find the constant of proportionality.
Proportional Relationships
Two quantities are proportional when they have a constant ratio — one is always the same multiple of the other.
The equation of proportionality: $y = kx$, where $k$ is the constant of proportionality (also the unit rate).
Identifying Proportionality in a Table
| Hours worked ($x$) | Earnings ($y$) | Ratio $y/x$ |
|---|---|---|
| 2 | 20 | 10 |
| 5 | 50 | 10 |
| 8 | 80 | 10 |
Every ratio equals 10 → proportional. The constant of proportionality $k = 10$ (earn $10 per hour).
Non-proportional: if the ratios differ, or if the table includes (0, non-zero), the relationship is not proportional.
Graphing Proportional Relationships
The graph of $y = kx$ is always:
- A straight line
- Through the origin (0, 0)
If the graph is a straight line but does NOT pass through the origin → not proportional (it's still linear, but not proportional).
Finding k from a Table or Graph
From a table: pick any row and compute $k = y/x$.
From a graph: pick any point $(x, y)$ on the line and compute $k = y/x$.
Example: A graph passes through (3, 12). $k = 12/3 = 4$. Equation: $y = 4x$.
Summary
Proportional: $y = kx$; constant ratio $y/x = k$; graph is a line through the origin.
Constant of proportionality $k$: the unit rate; the slope of the line.
Check: compute $y/x$ for multiple points — if all equal, proportional.