The image you uploaded contains a math problem in German. I'll extract the text from the image, solve it step by step, provide an answer statement, and explain the solution. Let me analyze the content first.

Extracted Problem:

The image contains the following mathematical problem:

1. Der Abstand zur Spiegelachse beträgt 4 cm.

• a) Spiegeln Sie die Funktion $f(x) = x^2$ an der Geraden $x = 2$ und geben Sie die neue Funktion an.
• b) Wie groß ist der Abstand dieser neuen Parabel von ihrer Symmetrieachse?

Solution:

Part (a): Spiegeln Sie die Funktion $f(x) = x^2$ an der Geraden $x = 2$.

To reflect the function $f(x) = x^2$ over the line $x = 2$, we need to consider how the reflection works.

1. Reflection Process:

   • The original function is $f(x) = x^2$.
   • The line of reflection is $x = 2$.
   • To find the reflected function, we use the formula for reflection $x' = 2x_0 - x$, where $x_0$ is the line of reflection.

2. Deriving the Reflected Function:

   • $f(x) = (2 - x)^2$.
   • Expanding this gives: $f'(x) = (2 - x)^2 = (x - 2)^2 = x^2 - 4x + 4$

   Thus, the reflected function is $f'(x) = x^2 - 4x + 4$.

Part (b): Abstand der neuen Parabel von ihrer Symmetrieachse.

1. Symmetry Axis of the Original Parabola:

   • The original parabola $f(x) = x^2$ has its axis of symmetry at $x = 0$.

2. Symmetry Axis of the Reflected Parabola:

   • The reflected parabola $f'(x) = x^2 - 4x + 4$ has its axis of symmetry at $x = 2$, because the equation of the axis of symmetry of a parabola in the form $ax^2 + bx + c$ is $x = -\frac{b}{2a}$.

3. Distance Between Symmetry Axes:

   • The distance between the symmetry axis of the original parabola $x = 0$ and the reflected parabola $x = 2$ is simply $2 - 0 = 2$ units.

Final Answer:

• The reflected function is $f'(x) = x^2 - 4x + 4$.
• The distance between the symmetry axes of the original and reflected parabolas is 2 units.

Erklärung:

• Durch die Spiegelung der Funktion $f(x) = x^2$ an der Geraden $x = 2$ wird der Graph der Funktion entlang der x-Achse verschoben, wobei die neue Funktion $f'(x) = (x - 2)^2$ entsteht. Dies führt zu einer Verschiebung der Symmetrieachse um 2 Einheiten nach rechts. Daher beträgt der Abstand der neuen Parabel von der Symmetrieachse der ursprünglichen Parabel 2 Einheiten.

---

Would you like more details, or do you have any questions? Here are five related questions for further practice:

1. What is the general method for reflecting any function across a vertical line $x = a$?
2. How would the reflection of $f(x) = x^2$ change if reflected over the line $x = 0$?
3. How can you determine the axis of symmetry for any quadratic function?
4. What is the geometric interpretation of the distance between two parallel lines?
5. How does the reflection of a function change its vertex?

Tip: When reflecting a function over a vertical line, the distance of each point from the line remains the same but is on the opposite side of the line.