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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

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cerverusdante
1. The function [tex]h(t)=-16t^{2}+48t[/tex] is a parabola of the form [tex]ax^{2}+bx[/tex]. The the formula for the axis of symmetry of a parabola is [tex]x= \frac{-b}{2a} [/tex]. We can infer from our function that [tex]a=-16[/tex] and [tex]b=48[/tex], so lets replace those values in our formula:
[tex]x= \frac{-b}{2a} [/tex]
[tex]x= \frac{-48}{-2(16)} [/tex]
[tex]x= \frac{-48}{-32} [/tex]
[tex]x= \frac{3}{2} [/tex]
[tex]x=1.5[/tex]
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:
[tex]0=-16t^{2}+48t[/tex]
[tex]-16t(t-3)=0[/tex]
[tex]t=0[/tex] or [tex]t-3=0[/tex]
[tex]t=0[/tex] or [tex]t=3[/tex]
From our previous point we know that the ball reaches its maximum time at [tex]t=1.5[/tex], which means that it takes 1.5 seconds to reach the maximum height and 1.5 seconds to fall back to the ground.
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The axis of the symmetry of a parabola is a vertical line that divides the parabola in two equal parts. The axis of the symmetry is 3/2 which represents that the maximum height (32 units) reached by the ball is after 3/2 seconds.

Given information-

The height of the ball is modeled with the function,

[tex]h(t) = -16t^2 + 48t [/tex]

Here t is the time in seconds.

Axis of symmetry

The axis of the symmetry of a parabola is a vertical line that divides the parabola in two equal parts.

As the roots of the parabola divides it into the two equal part. Therefore equate the equation to the 0 for finding the roots of the equation.

[tex]\begin{aligned}\\ -16t^2 + 48t &=0\\ -16t(t-3)&=0\\ \end[/tex]

Thus one roots of the equation is,

[tex]\begin{aligned}\\ -16t&=0\\ t&=0\\ \end[/tex]

Another root,

[tex]\begin{aligned}\\ t-3&=0\\ t&=0\\ \end[/tex]

The roots of the equation are (0,3). The axis of the parabola is the half of the sum of the roots of the equation of that parabola. Thus,

[tex]t=\dfrac{0+3}{2} \\ t=\dfrac{3}{2} \\[/tex]

Put this value of t in the given equation to find the height.

[tex]h(t) = -16\times(\dfrac{3}{2}) ^2 + 48\times\dfrac{3}{2} \\ h(t) =-36+72\\ h(t)=36[/tex]

Hence the axis of the symmetry is 3/2 which represents that the maximum height (36 units) reached by the ball is after 3/2 seconds.

Learn more about the axis of symmetry here;

brainly.com/question/21589886

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