The image appears to show a math problem about finding the vertex form of a parabola. The problem states:

"To turn the function $y = x^2 + 6x$ into vertex form, you can rewrite it as $y = (x + 3)^2 - 9$. Then, lower the parabola left and right by 2 units to complete where the new vertex and axis of symmetry are."

Let's go through how to complete the square to rewrite the given equation $y = x^2 + 6x$ into vertex form and find the vertex and axis of symmetry.

Step-by-Step Solution

1. Original Equation: $y = x^2 + 6x$

2. Complete the Square:

   • Add and subtract the square of half the coefficient of $x$ inside the equation:
   • Half of 6 is 3, and $3^2 = 9$.
   • Rewrite the equation: $y = x^2 + 6x + 9 - 9$
   • Factor the perfect square: $y = (x + 3)^2 - 9$

3. Adjusting for Vertex Shift:

   • According to the instructions, lower the parabola left and right by 2 units. This suggests shifting the graph vertically.
   • So, adjust $y = (x + 3)^2 - 9$ by adding or subtracting the required amount to shift the graph.

   If it's a vertical shift:

   • Lowering it means subtracting 2: $y = (x + 3)^2 - 9 - 2 = (x + 3)^2 - 11$.

4. Vertex Form of the Equation:

   • Now, $y = (x + 3)^2 - 11$ is in vertex form.
   • Vertex: $(-3, -11)$
   • Axis of Symmetry: $x = -3$

Let me know if you would like any further details or have any questions!

Follow-up Questions:

1. How do you identify the vertex from a parabola in vertex form?
2. What are the steps to find the axis of symmetry of a parabola?
3. Can you describe the effect of shifting a parabola vertically or horizontally?
4. What would happen to the vertex if you shifted the parabola by a different amount?
5. How can completing the square help in graphing quadratic functions?

Tip:

Completing the square is a powerful technique for converting standard quadratic equations into vertex form, making it easier to identify key features like the vertex and axis of symmetry.