Respuesta :
A curve that is representable with a quadratic function f(x) = a·x² + b·x + c, is a parabola
- The function h(x) = -0.03·(x - 14)² + 10.83 models the second jump of the red kangaroo
The reason the above function is correct is as follows:
The known parameters are;
The given function that models the jump of the kangaroo is h(x) = -0.03·(x - 14)² + 6
The horizontal distance traveled (in feet) = x
The height (in feet) = h(x)
The position the kangaroo lands when it jumps from a higher location = 5 feet farther away
Required:
To write a function that models the second jump
Solution:
Let y₀ represent the added height, we have;
The coordinates of the vertex of the parabola given in vertex form, y = a·(x - h)² + k is (h, k)
Therefore;
The coordinates of vertex of h(x) = -0.03·(x - 14)² + 6, is (14, 6)
The highest point of the parabola is 6 feet, and 14 feet is the midpoint of the parabola
Therefore, by symmetry of a parabola, we have;
The maximum range of the parabola is 2 × 14 feet = 28 feet
From the new height, y₀, the new range is 28 feet + 5 feet = 33 feet
From the new height, the second jump, we have;
The height of the parabola, h(x) = y₀ + (-0.03·(x - 14)² + 6)
The height when the kangaroo lands at 33 feet away is h(33) = 0 feet
The initial height of the kangaroo = y₀ feet
Therefore, we get;
h(33) = 0 = y₀ + (-0.03·(33 - 14)² + 6) = y - 4.83
y₀ - 4.83 = 0
y₀ = 4.83
The initial height of the kangaroo, y₀ = 4.83
Therefore, for the second jump, we have;
h(x) = 4.83 + (-0.03·(x - 14)² + 6) = -0.03·(x - 14)² + 10.83
Which gives;
The function that models the second jump is h(x) = -0.03·(x - 14)² + 10.83
Learn more about parabolic functions here;
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