Question

Analyzing a Function
The population of bobcats in northern Arizona since 2008 can be modeled using the function b(t) = –0.32t2 + 2.7t + 253.

What does t represent?

What is the domain for this function?

Which range values would not make sense for this function?

Would the graph be continuous or discrete, and why?

Answer 1

- What does t represent?
- t = the number of years after 2008.

- What is the domain for this function?
- Mathematically the polynomial is defined for all real t, but in the modeling context t ≥ 0 (years since 2008).
- If you restrict to when the model predicts a nonnegative population, 0 ≤ t ≤ about 32.64 (≈ year 2040.6), since the model predicts b(t)=0 at t ≈ 32.64 and negative values after that.

- Which range values would not make sense for this function?
- Negative population values do not make sense. (So realistically b(t) ≥ 0.)
- Also extremely large values above the model’s maximum (the vertex value) are not possible for this model; the model’s maximum population is b(4.21875) ≈ 258.7 bobcats.
- Note: in real life population counts are whole animals, so non-integer fractional values are not literal counts though they are acceptable in a continuous model.

- Would the graph be continuous or discrete, and why?
- Mathematically the function is continuous (a polynomial).
- Practically, actual bobcat counts are discrete (whole animals) and are often recorded at discrete times (e.g., yearly), so real data would appear as discrete points even though the model curve is continuous.