To complete the square for the quadratic function \( y = -x^2 + 8x + 7 \), we'll follow these steps:
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Factor out the coefficient of \( x^2 \) from the first two terms. Since the coefficient is negative, we'll factor out \(-1\):
\[ y = -1(x^2 - 8x) + 7 \]
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Complete the square for the expression inside the parentheses (\( x^2 - 8x \)). To complete the square, 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 squared is \(16\).
\[ y = -1(x^2 - 8x + 16 - 16) + 7 \]
This allows us to rewrite the equation as:
\[ y = -1((x - 4)^2 - 16) + 7 \]
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Distribute the \(-1\):
\[ y = -1(x - 4)^2 + 16 + 7 \]
Simplifying gives us:
\[ y = -1(x - 4)^2 + 23 \]
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Identify the vertex and axis of symmetry: The vertex form of a parabola is given by \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
In this case, the vertex is \((4, 23)\).
The axis of symmetry is given by the line \( x = h \). Therefore, the axis of symmetry is \( x = 4 \).
Final results:
- Vertex: \( (4, 23) \)
- Axis of symmetry: \( x = 4 \)