Question

matthew and johan are conducting an experiment. matthew throws a ball upward with an initial velocity of 19.6 meters per second from a height of 2 meters. the boys determine that the function ( h(x)=-9.8 x^{2}+19.6 x + 2 ) can be used to model the scenario, where ( h(x) ) represents the height in meters of the ball ( x ) seconds after it is thrown. identify the vertex and determine what it represents in the context of the ball. a the vertex is ( (1,11.8) ). after 11.8 seconds, the ball reaches its maximum height of 1 meter. b the vertex is ( (1,2) ). the ball lands on the ground after 1 second. it lands 2 meters away from where it was launched. c the vertex is ( (1,7.8) ). after 1 second, the ball reaches its maximum height of 7.8 meters. d the vertex is ( (1,11.8) ). the ball lands on the ground after 1 second. it lands 11.8 meters away from where it was launched. e the vertex is ( (1,11.8) ). after 1 second, the ball reaches its maximum height of 11.8 meters.

Explanation:

Step1: Encontrar la coordenada \(x\) del vértice

Para una función cuadrática \(y = ax^{2}+bx + c\), la fórmula para la coordenada \(x\) del vértice es \(x=-\frac{b}{2a}\).
En la función \(h(x)=-9.8x^{2}+19.6x + 2\), \(a=-9.8\) y \(b = 19.6\).
Sustituyendo en la fórmula: \(x=-\frac{19.6}{2\times(-9.8)}=\frac{-19.6}{-19.6}=1\).

Step2: Encontrar la coordenada \(y\) del vértice

Sustituir \(x = 1\) en la función \(h(x)=-9.8x^{2}+19.6x + 2\).
\(h(1)=-9.8\times(1)^{2}+19.6\times1 + 2=-9.8+19.6 + 2=11.8\).

Step3: Interpretar el vértice en el contexto

Como \(a=-9.8<0\), la parábola se abre hacia abajo y el vértice \((1,11.8)\) representa el máximo de la función. Aquí, \(x\) es el tiempo en segundos y \(h(x)\) es la altura en metros. Entonces, después de \(1\) segundo, la pelota alcanza su altura máxima de \(11.8\) metros.

Answer:

E. The vertex is \((1, 11.8)\). After \(1\) second, the ball reaches its maximum height of \(11.8\) meters.