To complete the square for the quadratic equation \( y = -x^2 + 8x + 7 \), we will format it in the vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.
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Start with the original equation: \[ y = -x^2 + 8x + 7 \]
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Factor out the coefficient of \( x^2 \) from the first two terms: \[ y = -(x^2 - 8x) + 7 \]
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Complete the square:
- Take the coefficient of \( x \) (which is \(-8\)), halve it (\(-4\)), and then square it (\(16\)).
- Add and subtract this square inside the parentheses.
\[ y = -\left(x^2 - 8x + 16 - 16\right) + 7 \] \[ y = -\left((x - 4)^2 - 16\right) + 7 \]
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Distribute the negative sign and simplify: \[ y = - (x - 4)^2 + 16 + 7 \] \[ y = - (x - 4)^2 + 23 \]
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Write in vertex form: \[ y = -1(x - 4)^2 + 23 \]
Now, we can identify the vertex and the axis of symmetry.
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Vertex (h, k): The vertex is \( (h, k) = (4, 23) \).
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Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex, given by the line \( x = h \): \[ x = 4 \]
Summary:
- The vertex of the parabola is \( (4, 23) \).
- The axis of symmetry is \( x = 4 \).
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