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.
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 $h = -16(t - 2)^2 + 64$.
To find the equation of the parabola in vertex form, we need the vertex coordinates.
Since the maximum height is reached after 2 seconds, the vertex coordinates are (2, 64).
The vertex form equation for a parabola is given by:
y = a(x - h)^2 + k
Where (h, k) represents the vertex coordinates.
Therefore, the equation can be written as:
y = a(x - 2)^2 + 64
Now, we need to find the value of "a" to complete the equation.
We have another point on the parabola, which is (1, 48).
Substituting the values of (x, y) into the equation, we get:
48 = a(1 - 2)^2 + 64
48 = a(-1)^2 + 64
48 = a + 64
-16 = a
So the value of "a" is -16.
The final equation of the parabola in vertex form is:
y = -16(x - 2)^2 + 64
To write the equation of a parabola in vertex form, we need to know the vertex coordinates (h, k) and the coefficient of the squared term (a).
In this case, the vertex coordinates (h, k) are (2, 64), as given.
To find the coefficient (a), we need to determine the relationship between time (t) and the height of the football (h).
We know that it took the football 1 second to reach a height of 48 feet, and after 2 seconds, it reached its maximum height of 64 feet.
Let's calculate the average rate of change between these two points to find the value of "a":
Average Rate of Change = Change in Height / Change in Time
Average Rate of Change = (64 - 48) / (2 - 1)
Average Rate of Change = 16 / 1
Average Rate of Change = 16
Since the coefficient (a) determines the rate of change of the parabola, we can set it equal to the average rate of change we just calculated:
a = 16
Now we can write the equation of the parabola in vertex form:
h(t) = a(t - h)^2 + k
Plugging in the vertex coordinates (h, k) = (2, 64) and the coefficient (a) = 16, we get:
h(t) = 16(t - 2)^2 + 64
Therefore, the equation of the parabola in vertex form is h(t) = 16(t - 2)^2 + 64.