• Identify quadratic functions in both general and standard form
• Determine the domain and range of a quadratic function by recognizing whether the vertex represents a maximum or minimum point
• Recognize key features of a parabola’s graph: vertex, axis of symmetry, y-intercept, and minimum or maximum value
• Create graphs of quadratic functions using tables and transformations

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, let’s now turn our attention to the timing of these changes. Specifically, we’ll explore when this population reaches its peak, which is a critical piece of information for Aria’s conservation efforts.

With the peak population time pinpointed, 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.