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), \( x \) seconds after Amir threw it, is modeled by \( h(x)=-(x+1)(x-7) \) What is the maximum height that the ball will reach? meters

To find the maximum height that the ball will reach, we need to analyze the quadratic function given by \[ h(x) = -(x + 1)(x - 7). \] ### Step 1: Expand the function First, let's expand the function to put it in standard quadratic form \( h(x) = ax^2 + bx + c \). \[ h(x) = -((x + 1)(x - 7)) = -(x^2 - 7x + x - 7) = -x^2 + 6x + 7. \] ### Step 2: Identify the coefficients From the expanded form \( h(x) = -x^2 + 6x + 7 \), we can identify the coefficients: - \( a = -1 \) - \( b = 6 \) - \( c = 7 \) ### Step 3: Find the vertex The maximum height of a quadratic function occurs at the vertex. The x-coordinate of the vertex can be found using the formula: \[ x = -\frac{b}{2a}. \] Substituting the values of \( a \) and \( b \): \[ x = -\frac{6}{2 \cdot -1} = \frac{6}{2} = 3. \] ### Step 4: Calculate the maximum height Now, we substitute \( x = 3 \) back into the height function \( h(x) \) to find the maximum height: \[ h(3) = -3^2 + 6 \cdot 3 + 7. \] Calculating this step by step: 1. \( -3^2 = -9 \) 2. \( 6 \cdot 3 = 18 \) 3. Adding these values: \( -9 + 18 + 7 = 16 \). Thus, the maximum height that the ball will reach is \[ \boxed{16} \text{ meters}. \]