https://file.aiquickdraw.com/mathbot/file/1726080859303ambygz7s.jpg

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.

The graph in the figure is a horizontal and/or vertical shift of the graph of y = x^2. Find a formula for the function graphed in the figure.

Analyze the graph showing a horizontal and vertical shift of the quadratic function y = x^2. Learn how to derive the formula g(x) = (x + 2)^2 + 1 by identifying the vertex shift. This lesson is ideal for students in Grades 9-11 studying algebra and graph transformations.

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