**Answer:**

t = 3; It takes the ball 3 seconds to reach the maximum height and 6 seconds to fall back to the ground

**Step-by-step explanation:**

we have

This is a vertical parabola open downward

The vertex represent a maximum

where

t represent the time in seconds

h(t) represent the height of the ball

**What is the axis of symmetry, and what does it represent?**

we know that

The equation of the axis of symmetry of a vertical parabola is equal to the x-coordinate of the vertex

Convert the quadratic equation into vertex form

**step 1**

Factor -16

**step 2**

Complete the square

**step 3**

Rewrite as perfect squares

The vertex is the point (3,144)

That means ----> It takes the ball 3 seconds to reach the maximum height of 144 units

The x-coordinate of the vertex is 3

therefore

The equation of the axis of symmetry is

**step 4**

Find how long take to the ball hit the ground

we know that

When the ball hit the ground the height is equal to zero

so

For h(t)=0

Solve for t

take square root both sides

so

therefore

t = 3; It takes the ball 3 seconds to reach the maximum height and 6 seconds to fall back to the ground