To complete the square for the quadratic equation \(y = -x^2 + 8x + 7\), we follow these steps:
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Factor out the coefficient of \(x^2\): Since the coefficient of \(x^2\) is \(-1\), we'll factor that out from the first two terms: \[ y = -1(x^2 - 8x) + 7 \]
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Complete the square: To complete the square inside the parentheses, we take half of the coefficient of \(x\) (which is \(-8\)), square it, and add/subtract that value inside the parentheses.
Half of \(-8\) is \(-4\), and squaring \(-4\) gives \(16\). We then add and subtract \(16\) inside the parentheses: \[ y = -1(x^2 - 8x + 16 - 16) + 7 \] Rearranging this gives: \[ y = -1((x - 4)^2 - 16) + 7 \]
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Distribute and simplify: \[ y = -1(x - 4)^2 + 16 + 7 \] \[ y = -1(x - 4)^2 + 23 \]
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Identify the vertex: The vertex form of a parabola is given by \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. From our equation: \[ h = 4, \quad k = 23 \] Therefore, the vertex is \((4, 23)\).
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Find the axis of symmetry: The axis of symmetry is the vertical line that passes through the vertex. The equation is given by: \[ x = h \] Thus, the axis of symmetry is: \[ x = 4 \]
Summary
- Vertex: \((4, 23)\)
- Axis of Symmetry: \(x = 4\)
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