Answer:
<h2>C. Paige's soccer ball</h2>
Step-by-step explanation:
To solve this problem, we just need to graph Viola's function to know the altitude she reached, and then compare it with Paige's altitude.
So, we know that Viola's function is
, as an extra solution, we can calculate the vertex of this quadratic function, that will give us Viola's max altitude.
![V(v_{x};v_{y})\\V(\frac{-b}{2a};f(v_{x} ))](https://tex.z-dn.net/?f=V%28v_%7Bx%7D%3Bv_%7By%7D%29%5C%5CV%28%5Cfrac%7B-b%7D%7B2a%7D%3Bf%28v_%7Bx%7D%20%29%29)
Where <em>b </em>and <em>a </em>are from the general quadratic function:
.
So, in this case, <em>a = -3; b = 6. </em>Replacing this values, we find the vertex:
![V(\frac{-b}{2a};f(v_{x} ))](https://tex.z-dn.net/?f=V%28%5Cfrac%7B-b%7D%7B2a%7D%3Bf%28v_%7Bx%7D%20%29%29)
![v_{x}= \frac{-b}{2a}=\frac{-6}{2(-3)}=1](https://tex.z-dn.net/?f=v_%7Bx%7D%3D%20%5Cfrac%7B-b%7D%7B2a%7D%3D%5Cfrac%7B-6%7D%7B2%28-3%29%7D%3D1)
Then, we use the function to calculate the vertical coordinate of the vertex:
![v_{y}=f(1)=-3(1)^{2}+6(1)+3=-3+6+3=6](https://tex.z-dn.net/?f=v_%7By%7D%3Df%281%29%3D-3%281%29%5E%7B2%7D%2B6%281%29%2B3%3D-3%2B6%2B3%3D6)
Therefore, the max altitude reached by Viola is 6 meters.
Now, from the graph, we observe that Paige's max altitude is around 7.5 meters.
Therefore, Paige reached the higher height. The answer is C.