Question
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.
Answers
Answered by
GPT 3.5
Since the ball is in the air, we're dealing with a downward-opening parabola. Let's call the time in seconds $t$ and the height in feet $h$.
We can find the equation of the parabola in vertex form by using the vertex and another point on the parabola.
We know that the vertex is at $(2, 64)$, so we have the values $h = 64$ and $t = 2$.
Now let's plug these values into the vertex form equation:
$$h = a(t - k)^2 + h$$
Given that the vertex is $(2, 64)$, $k = 2$, and $h = 64$, the equation becomes:
$$h = a(t - 2)^2 + 64$$
Since the ball is 48 feet in the air after 1 second, we can use this point $(t, h)$ as $(1, 48)$.
Plugging these values into the equation gives us:
$$48 = a(1 - 2)^2 + 64$$
Simplifying, we find:
$$48 = a + 64$$
Subtracting 64 from both sides:
$$-16 = a$$
Now we can substitute this value of $a$ back into the equation:
$$h = -16(t - 2)^2 + 64$$
So the equation of the parabola in vertex form is $h = -16(t - 2)^2 + 64$.
We can find the equation of the parabola in vertex form by using the vertex and another point on the parabola.
We know that the vertex is at $(2, 64)$, so we have the values $h = 64$ and $t = 2$.
Now let's plug these values into the vertex form equation:
$$h = a(t - k)^2 + h$$
Given that the vertex is $(2, 64)$, $k = 2$, and $h = 64$, the equation becomes:
$$h = a(t - 2)^2 + 64$$
Since the ball is 48 feet in the air after 1 second, we can use this point $(t, h)$ as $(1, 48)$.
Plugging these values into the equation gives us:
$$48 = a(1 - 2)^2 + 64$$
Simplifying, we find:
$$48 = a + 64$$
Subtracting 64 from both sides:
$$-16 = a$$
Now we can substitute this value of $a$ back into the equation:
$$h = -16(t - 2)^2 + 64$$
So the equation of the parabola in vertex form is $h = -16(t - 2)^2 + 64$.