A ball is thrown directly upward from a height of 6 ft with an initial velocity of 20 ft / sec .

step 1 Here we are given a quadratic function that is ST equal to minus 16 t square plus 20 t plus 6. T

A ball is thrown directly upward from a height of 6 ft with...

A ball is thrown directly upward from a height of $6 \mathrm{ft}$ with an initial velocity of $20 \mathrm{ft} / \mathrm{sec} .$ The function $s(t)=-16 t^{2}+20 t+6$ gives the height of the ball, in feet, $t$ seconds after it has been thrown. Determine the time at which the ball reaches its maximum height and find the maximum height.

A ball is thrown directly upward from a height of $6 \mathrm{ft}$ with an initial velocity of $20 \mathrm{ft} / \mathrm{sec} .$ The function $s(t)=-16 t^{2}+20 t+6$
gives the height of the ball, in feet, $t$ seconds after it has been thrown. Determine the time at which the ball reaches its maximum height and find the maximum height.

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Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two that are typically expressed in the form ax^2 + bx + c. They describe parabolic curves that can open upward or downward depending on the coefficient 'a'. In the context of projectile motion, quadratic functions are used to model the trajectory of objects under constant acceleration due to gravity.

Vertex of a Parabola

The vertex of a parabola is the point where the graph reaches its maximum or minimum value. For quadratic functions in the form ax^2 + bx + c, the vertex can be found using the formula -b/(2a) for the x-coordinate (or time in projectile motion) and then substituting that back into the function to find the y-coordinate (or height). This concept is key to determining the highest point reached by a projectile.

Projectile Motion

Projectile motion describes the motion of an object thrown or projected into the air, subject to acceleration due to gravity. The trajectory often follows a parabolic path, and understanding this motion involves analyzing the vertical component to determine factors such as maximum height, time of flight, and the point of impact.

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