The graph shown in the image is a parabola that looks like a shifted version of the standard parabola $y = x^2$. Let's analyze the transformation.

1. Vertex Shift: The vertex of the standard parabola $y = x^2$ is at $(0, 0)$. However, in this graph, the vertex has been shifted to the point $(-2, 1)$. This implies both a horizontal and vertical shift.

   • A horizontal shift to the left by 2 units would involve the transformation $y = (x + 2)^2$.
   • A vertical shift upwards by 1 unit would involve the transformation $y = (x + 2)^2 + 1$.

Therefore, the formula for the graph is: $g(x) = (x + 2)^2 + 1$

Let me know if you need any further clarifications!

Related Questions:

1. How does changing the value of the constant outside the squared term affect the graph?
2. What happens when you change the sign inside the parentheses with $x$?
3. Can you describe the effect of multiplying $x^2$ by a coefficient?
4. How would the graph change if the vertex were at $(-3, 1)$ instead?
5. What is the difference between vertical and horizontal shifts in graphs?

Tip:

When identifying transformations of graphs, always look for changes in the vertex, as they indicate horizontal and vertical shifts.