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To solve the problem, let's analyze the given information:

1. The parabola intersects the x-axis at 2 points, meaning the equation of the parabola must have two distinct real roots.
2. The axis of symmetry is at \( x = 2 \), which implies that the midpoint of the roots (the x-coordinate of the vertex) must be 2.

### Equation of a Parabola
The general factored form of a quadratic function is:

\[
y = (x - p)(x - q)
\]

where \( p \) and \( q \) are the x-intercepts (roots). The axis of symmetry is the midpoint between these two roots:

\[
\text{Axis of Symmetry} = \frac{p + q}{2}
\]

We are told the axis of symmetry is at \( x = 2 \), so:

\[
\frac{p + q}{2} = 2 \quad \Rightarrow \quad p + q = 4
\]

Now, let's examine each option:

**Option A**: \( y = (x - 2)(x - 2) \)
- The roots are both \( x = 2 \), meaning the parabola only touches the x-axis at \( x = 2 \) (a double root), so it does not intersect the x-axis at two points. This is not the correct option.

**Option B**: \( y = (x - 1)(x + 5) \)
- The roots are \( x = 1 \) and \( x = -5 \). The sum of the roots is \( 1 + (-5) = -4 \), so the axis of symmetry would be at \( \frac{-4}{2} = -2 \), not 2. This is not correct.

**Option C**: \( y = (x + 1)(x - 5) \)
- The roots are \( x = -1 \) and \( x = 5 \). The sum of the roots is \( -1 + 5 = 4 \), so the axis of symmetry is at \( \frac{4}{2} = 2 \), which matches the given condition. This could be the correct equation.

**Option D**: \( y = (x + 2)(x + 2) \)
- The roots are both \( x = -2 \), so this only touches the x-axis at \( x = -2 \), not intersecting at two points. This is not the correct option.

### Conclusion:
The correct answer is **Option C: \( y = (x + 1)(x - 5) \)**, as it has an axis of symmetry at \( x = 2 \) and intersects the x-axis at two points.

Would you like further details or have any questions? Here are five related questions to expand your understanding:

1. What is the vertex form of a parabola, and how can it be derived from the factored form?
2. How do you find the vertex of a parabola given its equation in standard form?
3. What is the relationship between the axis of symmetry and the roots of a quadratic function?
4. How can you determine the direction of a parabola's opening based on its equation?
5. Can a parabola have more than two x-intercepts?

**Tip:** To find the axis of symmetry, always calculate the midpoint between the two roots.

A parabola intersects the x-axis at 2 points and the axis of symmetry is at x = 2. Which of the following could be the equation of the parabola? (Options provided in image)

Learn how to solve a problem where a parabola intersects the x-axis at two points and has an axis of symmetry at x = 2. This involves applying the properties of parabolas and quadratic equations. Suitable for high school students studying Algebra.

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