Let's analyze the math problems in the image.

1. Determine the vertex form of each quadratic function.

The three quadratic equations in standard form are:

1. $y = x^2 + 2x - 1$
2. $y = x^2 - 2x + 2$
3. $y = x^2 - x - 2$

We need to match these with the vertex forms provided:

• A. $y = (x - 1)^2 + 1$
• B. $y = (x - 1)^2 - 3$
• C. $y = (x + 1)^2 - 2$

Conversion from standard form to vertex form:

The vertex form of a quadratic is given by $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex.

1. $y = x^2 + 2x - 1$:

   • Completing the square: $y = (x + 1)^2 - 2$
   • This matches C.

2. $y = x^2 - 2x + 2$:

   • Completing the square: $y = (x - 1)^2 + 1$
   • This matches A.

3. $y = x^2 - x - 2$:

   • Completing the square: $y = (x - 1/2)^2 - \text{(adjust terms)}$ Since this is not in the set of choices, it is possible there is a typo or another type of matching test expected.

2. Determine the equation of the axis of symmetry:

For each quadratic in vertex form $y = a(x-h)^2 + k$, the axis of symmetry is $x = h$.

• For $y = (x - 3)^2 + 5$, the axis of symmetry is $x = 3$, which corresponds to B.
• For $y = (x + 2)^2 - 7$, the axis of symmetry is $x = -2$, which corresponds to E.

Would you like more details on any specific problem or concept?

Here are 5 questions that expand on the current material:

1. How do you complete the square to convert a quadratic equation to vertex form?
2. What is the significance of the axis of symmetry in a parabola?
3. How do you derive the vertex from a quadratic equation in standard form?
4. What is the geometric interpretation of the vertex form of a quadratic equation?
5. How does the value of $a$ in $y = a(x-h)^2 + k$ affect the shape of the parabola?

Tip: When solving for the vertex form, always remember that the expression inside the square must match $(x - h)$, and any remaining constant terms go outside.