Suppose the ball was 3.24 feet above ground when it was hit, and that it reached a maximum height of approximately 102.51 feet when it had traveled a ground distance of approximately 214.26 feet. The ball lands after traveling a ground distance of approximately 432 feet.
Find an equation of the form y = C(x-z1)(x-z2) where z1 and z2 are the zeros (or roots) of the quadratic polynomial (or x-intercepts of the graph) and C is a scaling constant that needs to be determined.
Thanks!
I guess we are assuming that the path will be a parabola
make the following sketch
on the x-axis, label 3 points A, B , and C
where A is where the ball is hit,
B is the maximum point, and
C is the place where the Ball hits the ground
sketch the parabola, and continue it to the left of A to hit the origin at O
so C is (217,74)
AB = 214.26
but B must be the midpoint of OC
so BC = 432-214.26 = 217.74
then OA = 217.74-214.26 = 3.48 OC = 432+3.48 = 435.48
so parabola =
y = C(x-0)(x-435.48)
but (217.74 , 102.51) lies on it (the vertex)
102.74 = C(217.74)(-217.74)
C = -.00216
y = -.00216x(x-435.48)
check:
if x = 3.48 , the height should be 3.24 ft
y = -.00216(3.48)(-432)
= 3.247 YEAHHHH!
To find the equation of the quadratic polynomial in the form y = C(x - z1)(x - z2), we need to determine the values of z1, z2, and C.
Given that the ball reaches a maximum height of approximately 102.51 feet when it has traveled a ground distance of approximately 214.26 feet, we can deduce the coordinates of the vertex (the maximum point) of the parabolic path of the ball. The x-coordinate of the vertex is the average of the x-intercepts (zeros) of the quadratic equation.
The ball lands after traveling a ground distance of approximately 432 feet. Assuming the ball lands at the x-coordinate of the vertex, we can calculate the distance from the initial point to the x-coordinate of the vertex by subtracting the distance traveled from the initial height (3.24 feet) from the distance to the landing point (432 feet). This gives us 428.76 feet as the distance from the initial point to the vertex.
Now, let's calculate the x-coordinate of the vertex (the average of the x-intercepts) by using the relation that the distance to the vertex is half of the total distance traveled:
x_vertex = (initial_x + landing_x) / 2
= (0 + 432) / 2
= 216 feet
Therefore, the x-coordinate of the vertex is 216 feet.
Next, to find the zeros (x-intercepts), we subtract the distance traveled from the initial point to the vertex from the x-coordinate of the vertex. This gives us the two distances from the x-coordinate of the vertex to the x-intercepts:
distance_to_z1 = x_vertex - initial_x
= 216 - 0
= 216 feet
distance_to_z2 = landing_x - x_vertex
= 432 - 216
= 216 feet
Since the ball was initially hit from an elevated position, the vertex will have a positive y-coordinate. Therefore, the zeros (x-intercepts) will be symmetric about the x-coordinate of the vertex.
Now we have all the required values to write the equation of the quadratic polynomial:
y = C(x - z1)(x - z2)
Substituting the values we found:
y = C(x - 216)(x - 216)
Simplifying the equation, we get the final equation:
y = C(x - 216)^2, where C is the scaling constant that needs to be determined.