Complete the square to determine the vertex and axis of symmetry of y=−x^2+8x+7 (4 points)

Answer 1

To complete the square for the quadratic equation \( y = -x^2 + 8x + 7 \), we will follow these steps:

1. Factor out the coefficient of \( x^2 \) (which is \(-1\) in this case) from the first two terms:

   \[ y = - (x^2 - 8x) + 7 \]

2. Complete the square inside the parentheses. To do this, we take the coefficient of \( x \) (which is \(-8\)), divide it by 2 to get \(-4\), and square it to get \(16\). We then add and subtract this value inside the parentheses:

   \[ y = - \left( x^2 - 8x + 16 - 16 \right) + 7 \]

   Simplifying this, we have:

   \[ y = - \left( (x - 4)^2 - 16 \right) + 7 \]

3. Distribute the negative sign:

   \[ y = - (x - 4)^2 + 16 + 7 \]

   Combine like terms:

   \[ y = - (x - 4)^2 + 23 \]

Now, the function is in the vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. From our equation \( y = - (x - 4)^2 + 23 \), we can identify:

• The vertex \( (h, k) = (4, 23) \).
• The axis of symmetry is given by the line \( x = h \), which, in this case, is \( x = 4 \).

Summary

• Vertex: \( (4, 23) \)
• Axis of symmetry: \( x = 4 \)

Answer 2

shorter
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Answer 3

Sure! Here’s the shorter version:

To complete the square for \( y = -x^2 + 8x + 7 \):

1. Factor out \(-1\): \[ y = - (x^2 - 8x) + 7 \]

2. Complete the square: \[ y = - \left( (x - 4)^2 - 16 \right) + 7 \implies y = - (x - 4)^2 + 23 \]

Vertex: \( (4, 23) \)

Axis of symmetry: \( x = 4 \)