Question

A soccer ball is kicked toward the goal. The height of the ball is modeled by the function h(t) = −16t2 + 48t where t equals the time in seconds and h(t) represents the height of the ball at time t seconds. What is the axis of symmetry, and what does it represent? (2 points) t = 1.5; it takes 1.5 seconds to reach the maximum height and 3 seconds to fall back to the ground t = 1.5; it takes 1.5 seconds to reach the maximum height and 1.5 seconds to fall back to the ground t = 3; it takes 3 seconds to reach the maximum height and 3 seconds to fall back to the ground t = 3; it takes 3 seconds to reach the maximum height and 6 seconds to fall back to the ground

Answer 1

1. The function
`h(t)=-16t^(2)+48t` is a parabola of the form
`ax^(2)+bx`. The the formula for the axis of symmetry of a parabola is
`x= (-b)/(2a)`. We can infer from our function that
`a=-16` and
`b=48`, so lets replace those values in our formula:

`x= (-b)/(2a)`

`x= (-48)/(-2(16))`

`x= (-48)/(-32)`

`x= (3)/(2)`

`x=1.5`
We can conclude that to the left of the line of symmetry the ball is reaching its maximum height, and to the right of the line of symmetry the ball is falling.

2. Lets check how much time the ball takes to reach its maximum height and return to the ground. To do that we are going to set the height equal to zero:

`0=-16t^(2)+48t`

`-16t(t-3)=0`

`t=0` or
`t-3=0`

`t=0` or
`t=3`
From our previous point we know that the ball reaches its maximum time at
`t=1.5`, which means that it takes 1.5 seconds to reach the maximum height and 1.5 seconds to fall back to the ground.