To find the range of the quadratic function \( y = -x^2 - 14x - 48 \), we can follow these steps:
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Identify the direction of the parabola: The coefficient of \( x^2 \) is negative (\(-1\)), which means the parabola opens downwards.
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Find the vertex: The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = -1 \) and \( b = -14 \). \[ x = -\frac{-14}{2 \cdot -1} = \frac{14}{-2} = -7 \]
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Calculate the y-coordinate of the vertex: Substitute \( x = -7 \) back into the original function to find \( y \). \[ y = -(-7)^2 - 14(-7) - 48 \] \[ y = -49 + 98 - 48 \] \[ y = 1 \]
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Determine the range: Since the parabola opens downwards and the maximum value occurs at the vertex (\( y = 1 \)), the range of the function is all values less than or equal to 1.
Therefore, the range of the function \( y = -x^2 - 14x - 48 \) is: \[ \text{Range: } (-\infty, 1] \]