To complete the square and find the vertex and axis of symmetry for the quadratic equation \( y = -x^2 + 8x + 7 \), we can follow these steps:
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Rearrange the equation so that we can focus on the \( x \) terms: \[ y = - (x^2 - 8x) + 7 \]
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Complete the square for the expression inside the parentheses. To do this, take half of the coefficient of \( x \) (which is -8), square it, and then add and subtract that square inside the brackets: \[ \text{Half of -8 is } -4. \ \text{Square of -4 is } 16. \] So, we can rewrite the equation as: \[ y = - (x^2 - 8x + 16 - 16) + 7 \] This simplifies to: \[ y = - ((x - 4)^2 - 16) + 7 \]
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Distribute the negative sign and simplify: \[ y = - (x - 4)^2 + 16 + 7 \] \[ y = - (x - 4)^2 + 23 \]
Now we have the vertex form of the quadratic equation, which is: \[ y = - (x - 4)^2 + 23 \]
From this, we can identify the vertex and the axis of symmetry:
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Vertex: The vertex is at the point \( (h, k) \), where \( h = 4 \) and \( k = 23 \). Thus, the vertex is \( (4, 23) \).
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Axis of symmetry: The axis of symmetry is the vertical line that passes through the vertex, given by \( x = h \). Therefore, the axis of symmetry is: \[ x = 4 \]
In summary:
- The vertex of the parabola is \( (4, 23) \).
- The axis of symmetry is \( x = 4 \).
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