Temperature during an illness. The temperature of a person...

Temperature during an illness. The temperature of a person during an intestinal illness is given by $T(t)=-0.1 t^{2}+1.2 t+98.6, \quad 0 \leq t \leq 12$ where $T$ is the temperature $\left(^{\circ} \mathrm{F}\right)$ at time $t,$ in days. Find the relative extrema and sketch a graph of the function.

Temperature during an illness. The temperature of a person during an intestinal illness is given by $T(t)=-0.1 t^{2}+1.2 t+98.6, \quad 0 \leq t \leq 12$
where $T$ is the temperature $\left(^{\circ} \mathrm{F}\right)$ at time $t,$ in days. Find the relative extrema and sketch a graph of the function.

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, generally expressed in the form ax² + bx + c. These functions produce parabolic graphs that open upward if a is positive or downward if a is negative. Understanding their structure is fundamental for analyzing their behavior, such as determining the vertex, axis of symmetry, and zeros, which are key aspects in many mathematical and applied contexts.

Relative Extrema

Relative extrema refer to the local maximum or minimum points of a function where the function's value is higher or lower than all other values in a nearby interval. For quadratic functions, the relative extremum is always at the vertex. When the parabola opens downward (a negative coefficient for the squared term), the vertex represents the relative maximum, and when it opens upward, the vertex represents the relative minimum.

Finding the Vertex of a Parabola

The vertex of a parabola is a pivotal concept in understanding quadratic functions. It can be determined either by the formula x = -b/(2a) in standard form or by completing the square. The vertex provides both the point of symmetry for the parabola and the location of the function’s relative extreme value, making it essential for optimization and graphing tasks.

Sketching Graphs of Functions

Sketching the graph of a function involves plotting key points and understanding the overall shape dictated by the function's formula. For quadratic functions, important features include the vertex, axis of symmetry, and intercepts. A clear grasp of these components, along with the behavior dictated by the leading coefficient, enables one to accurately represent the function's graph, highlight relative extrema, and include domain restrictions when necessary.

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