To complete the square for the quadratic function \( y = -x^2 + 8x + 7 \), we follow these steps:
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Factor out the coefficient of \( x^2 \): Since the coefficient of \( x^2 \) is \(-1\), we factor it out of the terms involving \( x \): \[ y = -1(x^2 - 8x) + 7 \]
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Complete the square: To complete the square for the expression \( x^2 - 8x \), we find the term that completes the square. We take half of the coefficient of \( x \) (which is \(-8\)), square it, and add/subtract that value inside the parentheses: \[ \left( \frac{-8}{2} \right)^2 = (-4)^2 = 16 \] Add and subtract \( 16 \) inside the parentheses: \[ y = -1(x^2 - 8x + 16 - 16) + 7 \] This simplifies to: \[ y = -1((x - 4)^2 - 16) + 7 \] Distributing the \(-1\) gives: \[ y = -1(x - 4)^2 + 16 + 7 \] \[ y = -1(x - 4)^2 + 23 \]
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Identify the vertex: The vertex form of a quadratic function is given by \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From our completed square: \[ y = -1(x - 4)^2 + 23 \] So, the vertex is: \[ (4, 23) \]
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Determine the axis of symmetry: The axis of symmetry is the vertical line that passes through the vertex. For the vertex \((4, 23)\), the axis of symmetry is: \[ x = 4 \]
Final Answer:
- Vertex: \((4, 23)\)
- Axis of symmetry: \(x = 4\)
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