Graphing Proportional Relationships

When two quantities are proportional, their relationship can be written as an equation in the form $y = kx$. Here, $k$ is called the constant of proportionality. Graphing these relationships on a coordinate plane provides a clear visual way to understand how the two quantities interact.

Key Features of the Graph

Every graph of a proportional relationship shares two strict rules:

1. It is a straight line. The rate of change is constant, meaning the line does not curve or bend.
2. It passes through the origin $(0, 0)$. If $x = 0$, then $y = k(0) = 0$. This means that when one quantity is zero, the other must also be zero.

If a graph is a straight line but does not cross through $(0, 0)$, it is not a proportional relationship.

How to Graph a Proportional Relationship

Let's graph the equation $y = 2x$.

Step 1: Create a table of values. Pick a few simple values for $x$ and solve for $y$:

• If $x = 0$, $y = 2(0) = 0$. Point: $(0, 0)$
• If $x = 1$, $y = 2(1) = 2$. Point: $(1, 2)$
• If $x = 2$, $y = 2(2) = 4$. Point: $(2, 4)$

Step 2: Plot the points. Place the points $(0, 0)$, $(1, 2)$, and $(2, 4)$ on the coordinate plane.

Step 3: Draw the line. Use a straightedge to connect the points. Draw arrows on the ends of the line to show that the relationship continues infinitely.

Finding the Constant of Proportionality from a Graph

The constant of proportionality $k$ is exactly the same as the slope (or steepness) of the line.

To find $k$ from a graph:

1. Pick any clear point on the line other than the origin $(0, 0)$. For example, $(3, 9)$.
2. Use the ratio $k = \frac{y}{x}$.
3. Plug in your point's coordinates: $k = \frac{9}{3} = 3$.

The constant of proportionality is $3$, meaning the equation for the line is $y = 3x$. Every time $x$ increases by $1$, $y$ increases by $3$.