Complete the square to determine the vertex and axis of symmetry of y=x2+8x+7

Question

𝑦
=
𝑥
2
+
8
𝑥
+
7
. Show all the steps.

Answer 1

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

1. Rewrite the equation: \[ y = x^2 + 8x + 7 \]

2. Move the constant term to the other side: \[ y - 7 = x^2 + 8x \]

3. Complete the square on the right side:

   • Take the coefficient of \( x \) (which is 8), divide it by 2, and square it: \[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \]
   • Add and subtract this square inside the equation: \[ y - 7 = x^2 + 8x + 16 - 16 \]
   • This simplifies to: \[ y - 7 = (x + 4)^2 - 16 \]

4. Simplify the equation: \[ y - 7 + 16 = (x + 4)^2 \] \[ y + 9 = (x + 4)^2 \] \[ y = (x + 4)^2 - 9 \]

5. Identify the vertex and axis of symmetry:

   • The equation is now in the vertex form \( y = a(x - h)^2 + k \) where \((h, k)\) is the vertex.
   • From \( y = (x + 4)^2 - 9 \), we can see that:
     • \( h = -4 \)
     • \( k = -9 \)
   • Therefore, the vertex is \((-4, -9)\).
   • The axis of symmetry can be found from the vertex \( x = h \), which gives us: \[ x = -4 \]

Summary:

• The vertex of the parabola represented by the function \( y = x^2 + 8x + 7 \) is \((-4, -9)\).
• The axis of symmetry is the vertical line \( x = -4 \).