Let's analyze the function [tex]\( b(t) = -0.32t^2 + 2.7t + 253 \)[/tex], which models the population of bobcats in northern Arizona since 2008.
1. What does [tex]\( t \)[/tex] represent?
[tex]\( t \)[/tex] represents the number of years since 2008. This means [tex]\( t = 0 \)[/tex] corresponds to the year 2008, [tex]\( t = 1 \)[/tex] corresponds to 2009, and so on.
2. What is the domain for this function?
The domain represents all possible values that [tex]\( t \)[/tex] can take. Since [tex]\( t \)[/tex] is the number of years since 2008, it must be a non-negative number. Therefore, the domain is:
[tex]\[
t \in [0, \infty)
\][/tex]
3. Which range values would not make sense for this function?
The range represents the possible values of the population of bobcats. Since the population cannot be negative, negative values do not make sense for this function. Therefore, the impossible range values are:
[tex]\[
\text{{Negative values}}
\][/tex]
4. Would the graph be continuous or discrete, and why?
A function's graph can be either continuous or discrete. A continuous graph implies that the function can take on any value within the domain, forming a smooth curve. Since the function [tex]\( b(t) \)[/tex] is a quadratic equation, it represents a smooth, continuous curve describing the change in population over time. Therefore, the graph would be:
[tex]\[
\text{{Continuous}}
\][/tex]
To summarize:
- [tex]\( t \)[/tex] represents the number of years since 2008.
- The domain for this function is [tex]\( [0, \infty) \)[/tex].
- Negative values would not make sense as range values.
- The graph would be continuous because it is a quadratic function representing population change smoothly over time.
