The graph shown is of the function $y = x^2$, which is a standard parabola with its vertex at the origin $(0, 0)$.

The task is to translate this graph to match the equation $y = (x + 4)^2 + 2$. Here's how the transformation happens:

• Horizontal Translation: The $(x + 4)$ term indicates a horizontal shift. Specifically, the graph is shifted 4 units to the left. In general, $(x + h)^2$ shifts the graph $h$ units horizontally, and since $h = -4$, the graph moves left by 4 units.

• Vertical Translation: The $+2$ outside the square indicates a vertical shift 2 units up. This means the graph moves upwards by 2 units.

So, the vertex of the new parabola will be at $(-4, 2)$ instead of $(0, 0)$.

Steps to graph the translation:

1. Shift the entire graph of $y = x^2$ 4 units to the left.
2. Then, shift the graph 2 units upwards.

The result is a new parabola with vertex $(-4, 2)$, which opens upwards like the original graph of $y = x^2$, but is now translated.

Would you like more details on any of the steps, or do you have any questions?

Here are 5 related questions to think about:

1. What is the effect of negative values outside the squared term, like $y = -x^2$?
2. How would the graph change if the equation was $y = (x + 4)^2 - 2$?
3. What does the general form $y = a(x - h)^2 + k$ represent geometrically?
4. How would the shape of the parabola change if the equation had a coefficient, like $y = 2(x + 4)^2 + 2$?
5. How does a vertical stretch or compression affect the vertex form of a parabola?

Tip: Always pay attention to signs inside and outside the squared term to understand horizontal and vertical shifts correctly!