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.

Answer 1

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 =

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 $h = -16(t - 2)^2 + 64$.