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When we talk about projectile motion, we're discussing objects that are thrown or launched into the air and are affected by the force of gravity. Imagine a baseball being thrown straight up; it follows a predictable path known as a parabolic trajectory. The physics behind this can be understood with a quadratic function, which describes the height of the object over time. In our exercise, the baseball's height is given by the equation \( h = -16t^2 + 32t \). Here, \( t \) is the time in seconds, and \( h \) is the height in feet. The equation considers two main phenomena:

• The upward motion initially due to the throw, represented by \(+32t\), which increases the height over time.
• The downward pull of gravity is captured by \(-16t^2\), causing the object to ascend more slowly and eventually descend.

Since the force of gravity is constant, the path of the baseball is symmetric: it rises to a peak and then falls back down at the same rate. Understanding the fundamental concept of projectile motion is essential when analyzing questions like what height the ball reaches at different times.

The vertex of a parabola is a crucial concept in understanding quadratic equations. It represents the highest or lowest point on a graph. In the context of our baseball example, the vertex tells us the maximum height reached during its flight. Because the parabola opens downwards (due to the negative coefficient of \( t^2 \) in \( h = -16t^2 + 32t \)), this vertex corresponds to the maximum height. To find the vertex's \( t \)-value, which indicates when the maximum height occurs, we use the formula \( t = \frac{-b}{2a} \) from the equation \( ax^2 + bx + c \). Here,

• \( a = -16 \)
• \( b = 32 \)
• \( c = 0 \)

Plugging in these values, we determine \( t = \frac{-32}{2(-16)} = 1 \text{ sec} \). This means that at 1 second, the baseball reaches its zenith. To calculate how high it goes, substitute \( t = 1 \) back into the original equation: \( h = -16(1)^2 + 32(1) = 16 \text{ feet} \). The vertex not only marks the peak height but also reveals critical details about the trajectory's shape.

The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). In our task, we want to find specific times when the baseball reaches a height of 7 feet. To do this, we set our equation \( h = -16t^2 + 32t \) equal to 7 and solve for \( t \) through the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, the parameters are:

• \( a = -16 \)
• \( b = 32 \)
• \( c = -7 \)

Substitute them to get: \[ t = \frac{-32 \pm \sqrt{32^2 - 4(-16)(-7)}}{-32} \] Solve this to find two possible values for \( t \): \( \frac{1}{4} \text{ sec} \) and \( \frac{7}{4} \text{ sec} \). This means the baseball is at 7 feet when it rises at \( \frac{1}{4} \text{ sec} \) and descends at \( \frac{7}{4} \text{ sec} \). The quadratic formula helps us navigate through problems involving quadratic equations by providing a direct way to find their solutions.