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Amir stands on a balcony and throws a ball to his dog, who is at ground level. The ball's height (in meters above the ground), xx seconds after Amir threw it, is modeled by h(x)=−(x+1)(x−7)h(x)=- (x+1)(x-7). What is the maximum height that the ball will reach?

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Q. Amir stands on a balcony and throws a ball to his dog, who is at ground level. The ball's height (in meters above the ground), xx seconds after Amir threw it, is modeled by h(x)=−(x+1)(x−7)h(x)=- (x+1)(x-7). What is the maximum height that the ball will reach?
  1. Identify Quadratic Function: The given function for the ball's height is h(x)=−(x+1)(x−7)h(x) = -(x+1)(x-7). To find the maximum height, we need to find the vertex of the parabola represented by this quadratic function. Since the coefficient of the x2x^2 term is negative, the parabola opens downwards, and the vertex will give us the maximum height.
  2. Find Roots: The quadratic function is in factored form. To find the xx-coordinate of the vertex, we can use the fact that the vertex lies exactly halfway between the roots of the quadratic equation. The roots are the values of xx for which h(x)=0h(x) = 0, which are x=−1x = -1 and x=7x = 7.
  3. Calculate Midpoint: To find the midpoint between x=−1x = -1 and x=7x = 7, we calculate the average of the two roots: (−1+7)/2=6/2=3(-1 + 7) / 2 = 6 / 2 = 3. Therefore, the xx-coordinate of the vertex is x=3x = 3.
  4. Find Maximum Height: Now that we have the x-coordinate of the vertex, we can find the y-coordinate, which represents the maximum height, by substituting x=3x = 3 into the function h(x)h(x). So, h(3)=−(3+1)(3−7)=−4×−4=16h(3) = -(3+1)(3-7) = -4 \times -4 = 16.
  5. Find Maximum Height: Now that we have the xx-coordinate of the vertex, we can find the yy-coordinate, which represents the maximum height, by substituting x=3x = 3 into the function h(x)h(x). So, h(3)=−(3+1)(3−7)=−4×−4=16h(3) = -(3+1)(3-7) = -4 \times -4 = 16.The maximum height that the ball will reach is 1616 meters.

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