To complete the square for the quadratic equation \( y = -x^2 + 8x + 7 \), we can follow these steps.
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Factor out the coefficient of \( x^2 \) (which is -1) from the first two terms:
\[ y = -(x^2 - 8x) + 7 \]
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Complete the square inside the parentheses. To do this, we take the coefficient of \( x \) (which is -8), divide it by 2, and square it.
\[ \left( \frac{-8}{2} \right)^2 = 16 \]
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Add and subtract this square inside the parentheses:
\[ y = -\left(x^2 - 8x + 16 - 16\right) + 7 \]
Simplifying this gives:
\[ y = -\left((x - 4)^2 - 16\right) + 7 \]
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Distribute the negative sign and simplify:
\[ y = -(x - 4)^2 + 16 + 7 \] \[ y = -(x - 4)^2 + 23 \]
Now we have the equation in vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.
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Identify the vertex and axis of symmetry:
- The vertex \( (h, k) \) is \( (4, 23) \).
- The axis of symmetry is the line \( x = h \), which is \( x = 4 \).
Final Results:
- Vertex: \( (4, 23) \)
- Axis of Symmetry: \( x = 4 \)