ball is thrown up into the air from a height of 21.76 feet above the ground. After 1.5 seconds, the ball reaches a maximum height of 57.76 feet. It then begins fall and hits the ground 3.4 seconds after it is thrown. quadratic function \( f \). \( f(x)= \) . dee feet) of the ball \( x \) seconds after it is thrown. Then, the function \( f \) is quadratic. (Its graph is a parabola.) Write an equation for the

To find the quadratic function \( f(x) \) that describes the height of the ball over time, we can use the vertex form of a quadratic equation, which is \( f(x) = a(x - h)^2 + k \). In this case, the maximum height (the vertex) is at point \( (1.5, 57.76) \). The height starts at 21.76 when \( x = 0 \) and falls to 0 after 3.4 seconds. Next, we determine \( a \) by using the point \( (0, 21.76) \): \[ f(0) = a(0 - 1.5)^2 + 57.76 = 21.76 \] This leads us to find \( a \) and form the complete function. The function is \( f(x) = -8(x - 1.5)^2 + 57.76 \), which shows how the height changes over time. The steepness of the parabola reflects the ball's motion, with the "a" value indicating it reaches a maximum height before it starts its descent! Now, let's take a peek at how high-flying balls became part of our everyday fun. Ever wonder why we even throw balls into the sky? Historically, the act of throwing items dates back to ancient civilizations using it for sports, hunting, or even religious rituals. Fast forward to sports like basketball, baseball, and volleyball, where throwing and catching became not just skills but a way of life, invoking competition and camaraderie. Jumping to today, you know those moments when you’re at a picnic and excitedly toss a ball to a friend? That simple action demonstrates physics in action! When you throw a ball, you're applying the principles of projectile motion, which can help in sports, physics experiments, or just having a good time outdoors. Understanding the trajectory can give your throws more accuracy and style, proving physics can make you the star of the field!