Graphs of Proportionality

Understand proportional relationships and learn how to find the constant of proportionality.

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Graphs of Proportional Relationships

In elementary school, we dealt with graphs of proportional relationships where the domain of $x$ was non-negative. In middle school, we extend our consideration to include the region where the domain of $x$ is negative.

The table below shows the values of $x$ and $y$ in the proportional relationship $y=2x$ .

Try plotting the $x$ and $y$ values from the above table as coordinates on the graph below. Press the button with the range of $x$ written on it to plot the corresponding coordinate points.

When we plot the values of $x$ and $y$ in the proportional relationship $y=ax$ as coordinates, the points line up on a straight line. If we take an infinite number of $x$ values, the set of corresponding points forms a straight line. This is called the graph of a proportional relationship.

When the Constant of Proportionality is Negative

Next, let's look at the graph when the constant of proportionality is negative.

The table below shows the corresponding values of $x$ and $y$ in the proportional relationship $y=-2x$ .

When we plot the $x$ and $y$ values from the above table as coordinates, we get the graph below.

When the constant of proportionality is negative, the slope of the graph is downward to the right.

The Magnitude of the Constant of Proportionality and the Slope of the Graph

We've seen that when the constant of proportionality is positive, the graph slopes upward to the right, and when it's negative, the graph slopes downward to the right. So, how does the slope of the graph change as the value of the constant of proportionality changes?

Move the slider on the graph below to change the value of the constant of proportionality and see how the graph changes.

Drawing Graphs

Drawing a Graph from the Origin and One Other Point

When drawing a graph, first confirm that it passes through the origin $(0,0)$ , then find one more point and draw a line. For example, in the case of $y=2x$ , when $x=1$ , $y=2$ , so we connect these two points $(0,0)$ and $(1,2)$ .

Drawing a Graph from an Equation

Similarly, when drawing a graph directly from an equation, we find the origin and one more point and draw a line. In the equation $y=ax$ , substitute an appropriate value for $x$ to find the corresponding $y$ value, and plot that point. This completes the straight line graph.