Sketch and label the graph of a function that describes the...

Sketch and label the graph of a function that describes the given situation.
An island warthog population initially grows quickly, but as space and food become sparse on the island, the population growth slows down. Eventually the population of the island levels off at 512 warthogs.

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Key Concepts

Logistic Growth Model

This is a mathematical model used to describe populations that experience rapid initial growth but slow down as limiting factors such as space, food, or other resources come into play. The growth pattern typically starts off exponentially but then gradually tapers off as the population nears its maximum sustainable size.

Carrying Capacity

The carrying capacity is the maximum population size that an environment can sustain over the long term given the available resources. In graphical terms, it is represented as the horizontal asymptote, where the growth of the population levels off.

Horizontal Asymptote

A horizontal asymptote is the line that a graph approaches as the independent variable (usually time) tends to infinity. In the context of logistic growth, it represents the carrying capacity of an environment, indicating the level at which the population stabilizes.

S-shaped Curve

An S-shaped curve, or sigmoid curve, is characteristic of the logistic growth model. It visually represents the phases of growth: initial exponential increase, followed by a slowdown as resources become limited, and finally leveling off when the population reaches its carrying capacity.

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