A football is punted into the air. After 1 second, the ball is 48 feet in the air. After 2 seconds, the football reaches its maximum height of 64 feet. Write the equation of the parabola in vertex form.(1 point)
Responses
y=−16(x−2)2+64
y is equal to negative 16 times open paren x minus 2 close paren squared plus 64
y=−0.01(x−64)2+2
y is equal to negative 0 point 0 1 times open paren x minus 64 close paren squared plus 2
y=(x−2)2+64
y is equal to open paren x minus 2 close paren squared plus 64
y=−2(x−1)2+48
y is equal to negative 2 times open paren x minus 1 close paren squared plus 48
y=−2(x−1)2+48
To write the equation of the parabola in vertex form, we need to identify the vertex of the parabola and the value of 'a,' which determines the shape of the parabola.
Given the information provided:
- After 1 second, the ball is 48 feet in the air.
- After 2 seconds, the ball reaches its maximum height of 64 feet.
The vertex of the parabola represents the maximum height, and the x-coordinate of the vertex is the time at which the ball reaches its peak. In this case, the ball reaches its maximum height after 2 seconds, so the x-coordinate of the vertex is 2.
Since the vertex is (2, 64), we can substitute these values into the vertex form of the equation, which is y = a(x - h)^2 + k, where (h, k) represents the vertex coordinates.
Substituting (2, 64) into the equation, we have y = a(x - 2)^2 + 64.
Now we need to determine the value of 'a'. Since the parabola opens downward due to the ball being punted into the air, 'a' will be negative. We know that after 1 second, the ball is 48 feet in the air, so substituting these values, we get 48 = a(1 - 2)^2 + 64.
Simplifying this equation, we have 48 = a - 64, which gives us a = 112.
Substituting the value of 'a' back into the equation, we have y = -112(x - 2)^2 + 64.
Therefore, the equation of the parabola in vertex form is y = -112(x - 2)^2 + 64.