• Identify polynomial degrees and solutions, and graph basic odd and even polynomials
• Graph a piecewise-defined function
• Describe how algebraic functions, like polynomials, differ from transcendental functions, like sine and exponential functions
• Draw the graph of a function after it has been moved up or down, stretched or shrunk, or flipped across an axis

Quadratic Quest: Ecosystem Equilibrium

Aria, an environmental scientist, is studying the population dynamics of a species of butterfly in a local ecosystem. The population’s growth and decline over time can be modeled using quadratic functions, which will help Aria understand the factors affecting the species and predict future population changes.

 

Aria presents a quadratic function that models the butterfly population over time: [latex]P(t)=−kt^2+mt+b[/latex], where [latex]t[/latex] is time in years, [latex]P(t)[/latex] is the population, [latex]k[/latex] is the rate of population decline due to limiting factors, [latex]m[/latex] is the initial population growth rate, and [latex]b[/latex] is the initial population.

Having identified how the coefficients of our quadratic function mirror the dynamics of the butterfly population, we can broaden our perspective to understand the full scope of the population’s potential over time. This requires us to examine the domain and range of our function.

Now that we’ve established the theoretical limits of the butterfly population, it’s time to visualize these dynamics. A graph will not only illustrate the population’s trajectory over time but also highlight key moments and values that are particularly relevant for the conservation board’s strategy planning.

After graphing the population trends and understanding the key features of our quadratic model, let’s challenge ourselves further. Can we predict when the butterfly population will return to its initial size? This insight is crucial for Aria to evaluate the long-term impact of environmental changes on the species.

Having determined when the population will return back to its starting number, let’s consider how environmental factors might alter this course. Suppose the rate of population decline is reduced by half due to successful conservation measures. What would our new population model look like under these improved conditions?

Finally, with a new model reflecting a slower decline in the butterfly population, let’s delve into the effects of a successful conservation effort on the population’s growth rate and maximum size. By transforming our original function, we can uncover the new face of the population graph and interpret the implications of these positive changes for the ecosystem.

By engaging with these tasks, Aria can effectively communicate the population dynamics to the conservation board, facilitating the development of protective measures. This activity illustrates the practical application of quadratic functions in environmental science, underscoring their importance in ecological analysis and prediction. Excellent work today!