To complete the square for the quadratic function \( y = -x^2 + 8x + 7 \), follow these steps:
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Factor out the coefficient of \( x^2 \) (which is -1) from the \( x^2 \) and \( x \) terms: \[ y = -1(x^2 - 8x) + 7 \]
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Complete the square for the expression in parentheses. To do this, take half of the coefficient of \( x \) (which is -8), square it, and add and subtract this inside the parentheses: \[ \text{Half of } -8 = -4 \quad \text{and } (-4)^2 = 16 \] Now rewrite the equation: \[ y = -1(x^2 - 8x + 16 - 16) + 7 \] This simplifies to: \[ y = -1((x - 4)^2 - 16) + 7 \]
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Distribute the -1: \[ y = -1(x - 4)^2 + 16 + 7 \] Simplifying further gives: \[ y = -1(x - 4)^2 + 23 \]
Now we can identify the vertex and the axis of symmetry:
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The vertex form of the equation is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Here, \( h = 4 \) and \( k = 23 \). Thus, the vertex is: \[ (4, 23) \]
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The axis of symmetry can be found from the vertex, which is the vertical line \( x = h \): \[ x = 4 \]
In summary:
- Vertex: \( (4, 23) \)
- Axis of symmetry: \( x = 4 \)