https://file.aiquickdraw.com/mathbot/file/17266761757751y5069d5.jpg

The task is to describe how to transform the graph of the function \( f(x) = x^2 \) into the given graph \( g(x) \) and find a formula for \( g(x) \). Based on the image, we have four transformations of the quadratic function \( f(x) = x^2 \) that need to be analyzed.

### a) Transformation in the first graph:
- The vertex of the parabola has moved up by 2 units.
- Transformation: This is a **vertical shift** upwards by 2 units.
- Function term: \( g(x) = x^2 + 2 \).

### b) Transformation in the second graph:
- The parabola has been shifted downward by 2 units.
- Transformation: This is a **vertical shift** downwards by 2 units.
- Function term: \( g(x) = x^2 - 2 \).

### c) Transformation in the third graph:
- The parabola has been reflected across the x-axis, meaning it opens downward.
- Transformation: This is a **vertical reflection** and a **shift downward by 2 units**.
- Function term: \( g(x) = -x^2 - 2 \).

### d) Transformation in the fourth graph:
- The vertex has been shifted to the right by 1 unit and upwards by 2 units.
- Transformation: This is a **horizontal shift** to the right by 1 unit and a **vertical shift** upwards by 2 units.
- Function term: \( g(x) = (x - 1)^2 + 2 \).

Would you like more details on any of these transformations, or do you have any specific questions?

### Related Questions:
1. How do you perform a horizontal shift on a quadratic function?
2. What effect does a reflection over the x-axis have on the graph of a function?
3. How would you combine both horizontal and vertical shifts in a function transformation?
4. Can we reflect a parabola over the y-axis instead? What would the equation look like?
5. How does a vertical stretch or compression affect the graph of a quadratic function?

### Tip:
To identify transformations easily, always start by looking at the vertex shift (horizontal or vertical) and whether the parabola has been reflected or stretched.

Beschreiben Sie, wie man den Graphen der Funktion g aus dem Graphen der Funktion f mit f(x) = x^2 erhält, und geben Sie einen Funktionsterm für g an.

Learn how to transform the quadratic function f(x) = x^2 into different forms by applying shifts, reflections, and stretches. This problem covers key graph transformation concepts suitable for high school students.

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