Getting Ready for a Pool Party

Learning Focus

Interpret function features in context: intercepts, rate of change, and domain restrictions.

Big idea

In a real-world function, each feature has meaning:

• $y$-intercept: starting amount ($t=0$).
• Slope / rate of change: change in output per unit input, with units.
• $x$-intercept: when the output reaches $0$.
• Domain: values that make sense (e.g. $t$ can't be negative; time stops when the pool is empty).

Always include units and say what the numbers mean.

Worked example

A pool has $2000$ gallons and drains at $50$ gallons per minute: $V(t)=2000-50t$.

• $y$-intercept $2000$: starting volume
• Slope $-50$: loses 50 gal each minute
• $x$-intercept: $2000-50t=0\Rightarrow t=40$ min, when it is empty
• Domain: $0\le t\le40$
• $V(10)=2000-500=1500$ gal

Takeaways

• Say what each feature means in the story, with units.
• Restrict the domain to values that make sense.
• Negative slope means decreasing quantity.