Shh! Please Be Discreet (Discrete)!

Learning Focus

Decide whether a situation is discrete or continuous, and understand how that affects domain and graph.

Big idea

• Discrete: inputs can only be separate, countable values (whole tickets, people, days). Graph = isolated points.
• Continuous: inputs can be any value in an interval (time, distance, volume). Graph = unbroken line or curve.

Ask: "Does a value like $2.5$ make sense as an input?"

The equation may look the same in both cases; the domain is what changes.

Worked example

Tickets cost \8$each:$C(n)=8n$.

• $n$ is a count of tickets, so the domain is $\{0,1,2,3,\dots\}$.
• The graph is a set of points on a line, not a solid line.

A water tank filling at $8$ gallons per minute: $V(t)=8t$.

• $t$ can be $2.5$ minutes, so the domain is $t\ge0$ (continuous).
• The graph is a solid ray.

Takeaways

• Counting things $\Rightarrow$ usually discrete; measuring things $\Rightarrow$ usually continuous.
• Discrete graphs are separate points; continuous graphs are connected.
• Same formula can model both—the domain differs.