To complete the square and find the vertex and axis of symmetry for the quadratic function \( y = -x^2 + 8x + 7 \), we can follow these steps:
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Factor out the leading coefficient from the first two terms (if it's not 1): \[ y = -1(x^2 - 8x) + 7 \]
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Complete the square for the expression inside the parentheses:
- Take the coefficient of \( x \) (which is \(-8\)), halve it to get \(-4\), and then square it, resulting in \(16\).
- Add and subtract \(16\) inside the parentheses: \[ y = -1(x^2 - 8x + 16 - 16) + 7 \] Simplifying this, we have: \[ y = -1((x - 4)^2 - 16) + 7 \] \[ y = -1(x - 4)^2 + 16 + 7 \] \[ y = -1(x - 4)^2 + 23 \]
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Identify the vertex: The completed square form is \( y = -1(x - 4)^2 + 23 \), which shows:
- The vertex is at \( (4, 23) \).
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Find the axis of symmetry: The axis of symmetry is the vertical line that passes through the vertex. The equation for the axis of symmetry is: \[ x = 4 \]
In summary, for the quadratic equation \( y = -x^2 + 8x + 7 \):
- The vertex is \( (4, 23) \).
- The axis of symmetry is \( x = 4 \).