SOLUTION: I know the answer is y = -16x^2 + 576, but how do I figure out how to write the equation? Write the equation of the quadratic function whose graph is a parabola containing the g

Algebra ->  Functions -> SOLUTION: I know the answer is y = -16x^2 + 576, but how do I figure out how to write the equation? Write the equation of the quadratic function whose graph is a parabola containing the g      Log On


   

Question 1038747: I know the answer is y = -16x^2 + 576, but how do I figure out how to write the equation?
Write the equation of the quadratic function whose graph is a parabola containing the given points.
An egg is dropped from the top of a 576-foot-high building. The ball is 432 feet above the ground after 3 seconds, and it reaches level ground in 6 seconds. The height above the ground is a quadratic function of the time after the ball is thrown. Write the equation of this function.
Found 2 solutions by solver91311, josmiceli:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


A quadratic function has the form

So if you let represent the time since the ball was dropped and represent the height of the ball at that time, then we have three ordered pairs that must be in the solution set of the desired function, namely , , and .

Substituting in the general form of the function for the first point:



which reduces to



Then substituting the second and third points as well as the value we now know for the constant term we get:





Simplifying these two expressions:





Solve the 2X2 system of linear equations to find the coefficients and to put into:



John

My calculator said it, I believe it, that settles it

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Start with the general form of a parabola
which has a maximum and not a minimum
+h+=+-a%2At%5E2+%2B+b%2At+%2B+c+ ( +a+ must be positive )
Each point on the parabola is ( t,h )
----------------------------
Since the ball is dropped and not thrown upward,
The vertex ( peak ) is at +t%5Bv%5D=0+
The formula for the vertex is:
+t%5Bv%5D+=+-b%2F%282a%29+
+0+=+-b%2F%282a%29+, so +b=0+
-----------------------------
The parabola must have the form:
+h+=+-a%2At%5E2+%2B+c+
They give you 2 points:
( 3, 432 )
( 6, 0 )
Use these points to find +a+ and +c+
----------------
+432+=+-a%2A3%5E2+%2B+c+
(1) +-9a+%2B+c+=+432+
and
+0+=+-a%2A6%5E2+%2B+c+
(2) +-36a+%2B+c+=+0+
-----------------------
Subtract (2) from (1)
(1) +-9a+%2B+c+=+432+
(2) +36a+-+c+=+0+
-----------------------
+27a+=+432+
+a+=+16+
and
(2) +-36a+%2B+c+=+0+
(2) +-36%2A16+%2B+c+=+0+
(2) +c+=+576+
---------------------
So, the equation is:
+h+=+-16t%5E2+%2B+576+
Here's the plot:
+graph%28+400%2C+400%2C+-1%2C+8%2C+-70%2C+700%2C+-16x%5E2+%2B+576+%29+