Quadratic functions can be used to describe the rate at which an object will fall and predict when it will land or how high it may go. Create a quadratic function using the function y=−16t^2 to help predict when an object will return to the ground from being kicked, dropped, or thrown. In 3 sentences, describe the scenario in which you will use the function. Write the complete function and explain the meaning of the variable t and the constant value. Simplify your answer to shorten it.

Answer 1

Imagine a soccer ball is kicked from ground level with an initial upward velocity. To model its height above the ground over time, we can use the quadratic function \( y = -16t^2 + vt + h \), where \(y\) represents the height of the ball (in feet) at any time \(t\) (in seconds), \(v\) is the initial upward velocity in feet per second, and \(h\) is the initial height (here, \(h = 0\) since the ball starts from the ground). For a soccer ball kicked with an initial velocity of 32 feet per second, the complete function becomes \( y = -16t^2 + 32t \).

In this function, \(t\) represents the time in seconds after the ball has been kicked, with the constant -16 indicating the acceleration due to gravity (in feet per second squared), which causes the ball to descend over time. Through this model, we can predict when the ball will return to the ground by solving for \(t\) when \(y = 0\).