Use this information: At a time t seconds after an object is...

Use this information: At a time $t$ seconds after an object is tossed vertically upward, it reaches a height s in feet given by the equation $s=80 t-16 t^{2}$.
What is the maximum height reached by the object?

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Key Concepts

Projectile Motion

Projectile motion involves the movement of an object under the influence of gravity, typically modeled by quadratic functions for its vertical component. The quadratic equation representing the height as a function of time allows us to determine important characteristics, like the maximum height achieved, by analyzing the vertex of the parabola. This concept bridges physics and mathematics by applying algebraic methods to real-world motion.

Vertex of a Parabola

The vertex is the point on the parabola where the function reaches its maximum (if the parabola opens downward) or minimum (if it opens upward). It can be determined using the formula -b/(2a) for the x-coordinate, with the corresponding y-coordinate found by substituting this value back into the function. This method is key in optimization problems, including those involving projectile motion.

Quadratic Functions

A quadratic function is a second-degree polynomial of the form ax² + bx + c. Its graph is a parabola, and depending on the sign of the coefficient a, the parabola opens upward or downward. This concept is fundamental in analyzing and solving problems that involve finding the maximum or minimum values of a function.

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