To complete the square for the quadratic equation \( y = -x^2 + 8x + 7 \), we will rearrange the equation and factor it appropriately.
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Start with the equation: \[ y = -x^2 + 8x + 7 \]
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Factor out the coefficient of \( x^2 \): Since the coefficient of \( x^2 \) is negative, we will factor out \(-1\): \[ y = - (x^2 - 8x) + 7 \]
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Complete the square: To complete the square for the expression \( x^2 - 8x \):
- Take half of the coefficient of \( x \) (which is -8), giving us \(-4\), and then square it: \[ (-4)^2 = 16 \]
- Add and subtract this square inside the parentheses: \[ y = - (x^2 - 8x + 16 - 16) + 7 \]
- Simplify it: \[ y = - ( (x - 4)^2 - 16 ) + 7 \] \[ y = - (x - 4)^2 + 16 + 7 \] \[ y = - (x - 4)^2 + 23 \]
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Identify the vertex and axis of symmetry: The equation is now in the vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, \( a = -1 \), \( h = 4 \), and \( k = 23 \).
Thus, the vertex is: \[ (4, 23) \]
The axis of symmetry is the vertical line that passes through the vertex: \[ x = 4 \]
In conclusion, the vertex of the parabola is \((4, 23)\), and the axis of symmetry is \(x = 4\).