To complete the square for the quadratic equation \( y = -x^2 + 8x + 7 \), we will rearrange it and manipulate it to find the vertex form.
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Start with the equation: \[ y = -x^2 + 8x + 7 \]
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Factor out the coefficient of \( x^2 \): We will factor out \(-1\) from the terms involving \(x\): \[ y = - (x^2 - 8x) + 7 \]
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Complete the square: To complete the square inside the parentheses \( x^2 - 8x \), take the coefficient of \( x \) (which is \(-8\)), divide by 2 (which gives \(-4\)), and square it (which gives \(16\)). Add and subtract this square inside the parentheses: \[ y = - \left(x^2 - 8x + 16 - 16\right) + 7 \] Simplifying gives: \[ y = - \left((x - 4)^2 - 16\right) + 7 \]
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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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Identify the vertex and axis of symmetry: Now that we have the equation in vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex, we can see that:
- The vertex is \((4, 23)\).
- The axis of symmetry is the vertical line given by \( x = h = 4 \).
Conclusion
The vertex of the quadratic equation \( y = -x^2 + 8x + 7 \) is \( (4, 23) \) and the axis of symmetry is \( x = 4 \).