Using the equation representing the height of the firework (h = -16t2 + v0t + h0), algebraically determine the extreme value of f(t) by completing the square and finding the vertex. Interpret what the value represents in this situation.
for y = ax^2 + bx + c, the x of the vertex is -b/(2a)
so for h = -16t2 + v0t + h0 , the t of the vertex is -v0/-32 = v0/32
h = -16(v0^2/1024) + (v0)^2 + h0
= -v0/32 + v0^2 + h0
vertex is (v0/32 , -v0/32 + v0^2 + h0)
check my algebra, I did not write it out first.
Recall that for ax^2+bx+c the vertex lies at (-b/2a, (4ac-b^2)/4a)
Now just plug in your coefficients
To find the extreme value of the function f(t) = -16t^2 + v0t + h0, we need to rewrite the equation in vertex form by completing the square. The vertex form of a quadratic equation is given by f(t) = a(t - h)^2 + k, where (h, k) represents the coordinates of the vertex.
Given equation: h = -16t^2 + v0t + h0
Step 1: Divide the entire equation by -16 to simplify the equation:
h/(-16) = t^2 - (v0/16)t - (h0/16)
Step 2: Move the constant term to the right side of the equation:
h/(-16) + (h0/16) = t^2 - (v0/16)t
Step 3: Now, we need to complete the square. Take the coefficient of t, which is -(v0/16), and divide it by 2, then square it:
(-(v0/16) / 2)^2 = (v0/32)^2 = v0^2 / (32)^2
Step 4: Add the square term calculated in Step 3 to both sides of the equation:
h/(-16) + (h0/16) + v0^2 / (32)^2 = t^2 - (v0/16)t + v0^2 / (32)^2
Step 5: Rewrite the right side of the equation as a perfect square trinomial:
h/(-16) + (h0/16) + v0^2 / (32)^2 = (t - v0/(32))^2
Step 6: Simplify the left side of the equation:
(h - h0 + v0^2 / (32)^2) / (-16) = (t - v0/(32))^2
Now we have the equation in vertex form, f(t) = a(t - h)^2 + k, where:
a = -16
h = v0/(32)
k = (h - h0 + v0^2 / (32)^2) / (-16)
From the equation, we can see that the vertex of the parabolic path of the firework is located at (h, k). The extreme value of f(t) corresponds to the highest point on the parabolic trajectory. In this situation, the value of k represents the maximum height reached by the firework.