The peak height reached at 144 feet, and the graph passing through points (0, 120), (1, 144), (2, 120), gives;
1. The vertex form of a quadratic equation is f(x) = a•(x - h)² + k
Where;
a = The leading coefficient
(h, k) = The coordinates of the vertex.
Let t represent the time of travel of the ball and let h represent the height.
From the given table, we have;
f(t) = a•(t - h)² + k
f(0) = 120 = a•(0 - h)² + k = a•h² + k
120 = a•h² + k
f(1) = 144 = a•(1 - h)² + k = a•(h² - 2•h + 1) + k
f(1) = a•(h² - 2•h + 1) + k
f(1) - f(0) = 144 - 120 = 24 = -((2•h - 1)•a
24 = -((2•h - 1)•a
f(2) = 120 = a•(2 - h)² + k
f(2) - f(1) = -24 = -((2•h - 3)•a
(f(1) - f(0)) - (f(2) - f(1)) = 48 = -2•a
Therefore;
a = -24
24 = -((2•h - 1)•a
24 = -((2•h - 1)×(-24)
-1 = -((2•h - 1)
2•h - 1 = 1
h = 2/2 = 1
h = 1
120 = a•(2 - h)² + k
120 = (-24)•(2 - 1)² + k
k = 120 + 24 = 144
Which gives;
The vertex form is y = -24•(x-1)² + 144
From the vertex form, we have;
y = -24•(x-1)² + 144 = -24•(x²-2•t+1) + 144
y = -24•(x²-2•tlx+1)+ 144 = -24•x²+48•x - 24 + 144
y = -24•x²+48•x + 120
y = -24•x² + 48•x - 24 + 144 = -24•x²+ 48•x + 120
y = -24•x² + 48•x + 120
The height of the ball after 1.5 seconds is
y = -24×1.5² + 48×1.5 + 120 = 138
- The height of the ball at t = 1.5 seconds is 138 feet
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