The equation you're working with is:

$y = (x + 4)^2 + 2$

This is a quadratic equation in vertex form. The general form of a quadratic equation is $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

From the equation $y = (x + 4)^2 + 2$, we can deduce that:

• The vertex of the parabola is at $(-4, 2)$.
• Since the coefficient of $(x + 4)^2$ is positive (it’s 1), the parabola opens upwards.

Now, comparing the two graphs:

• The first graph has a vertex at $(-4, 2)$ and opens upwards, which matches the equation given.
• The second graph has a vertex at $(-2, 2)$, which does not match the given equation.

Therefore, the first graph represents the quadratic function $y = (x + 4)^2 + 2$.

Would you like further explanation or have additional questions on this?

Here are some related questions you might find helpful:

1. How can I find the vertex of any quadratic equation?
2. What happens when the coefficient of $(x + h)^2$ in a quadratic equation is negative?
3. How do you determine the direction of the parabola (up or down)?
4. How can you determine the equation of a quadratic function from a graph?
5. How does the form of a quadratic function relate to its graph?

Tip: You can always find the vertex of a quadratic function written in standard form $ax^2 + bx + c$ by using the formula $x = \frac{-b}{2a}$ to find the x-coordinate, and then substitute it into the function to find the y-coordinate.