To complete the square for the quadratic equation \( y = -x^2 + 8x + 7 \) and determine the vertex and the axis of symmetry, we follow these steps:
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Rearrange the equation: Start with the quadratic in the standard form: \[ y = -x^2 + 8x + 7 \]
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Factor out the coefficient of \( x^2 \): Since the coefficient of \( x^2 \) is negative, we factor out \(-1\): \[ y = - (x^2 - 8x) + 7 \]
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Complete the square: We need to take the \( x^2 - 8x \) part and complete the square. To do this, take half of the coefficient of \( x \) (which is \(-8\)), square it, and add and subtract that value inside the parentheses. Half of \(-8\) is \(-4\), and squaring it gives \(16\): \[ y = - (x^2 - 8x + 16 - 16) + 7 \] This simplifies to: \[ y = - ((x - 4)^2 - 16) + 7 \] \[ y = - (x - 4)^2 + 16 + 7 \] \[ y = - (x - 4)^2 + 23 \]
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Identify the vertex: The vertex form of a quadratic equation is given by \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From our completed square, we can see that \( h = 4 \) and \( k = 23 \). Therefore, the vertex is at the point \( (4, 23) \).
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Determine the axis of symmetry: The axis of symmetry can be found from the \( x \)-coordinate of the vertex. This means that the axis of symmetry is the vertical line given by the equation \( x = 4 \).
In conclusion, the vertex of the quadratic function \( y = -x^2 + 8x + 7 \) is at the point four, twenty-three, and the axis of symmetry is the vertical line x equals four.