https://file.aiquickdraw.com/mathbot/file/1726691035614p0rw0rl2.jpg

The problem is written in Dutch and can be translated as:

"Given is the function \( f(x) = -x^2 + 4x \).
By reflecting it over the line \( y = 3 \), the graph of a function \( g \) is formed.
Find the equation of \( g \)."

### Solution:

1. The reflection is over the line \( y = 3 \), which means the function needs to be vertically reflected across this line. This can be done by subtracting the function values from \( 6 \), since the reflection formula over \( y = 3 \) is given by:

\[
g(x) = 2 \cdot 3 - f(x) = 6 - f(x)
\]

2. Substitute \( f(x) = -x^2 + 4x \) into this formula:

\[
g(x) = 6 - (-x^2 + 4x)
\]
\[
g(x) = 6 + x^2 - 4x
\]

Thus, the equation of the reflected function \( g(x) \) is:

\[
g(x) = x^2 - 4x + 6
\]

Would you like more details on how the reflection works, or do you have any further questions? 

Here are 5 related questions to expand on this topic:
1. What happens when a function is reflected over the x-axis or the y-axis?
2. How does the graph of \( f(x) = -x^2 + 4x \) look before and after reflection?
3. What would the equation be if the function was reflected over a different horizontal line, such as \( y = 2 \)?
4. How can we determine the vertex of the parabolas before and after reflection?
5. Can this method of reflection be applied to non-parabolic functions as well?

**Tip**: When reflecting functions over a horizontal line \( y = c \), the transformation is \( g(x) = 2c - f(x) \), ensuring that the vertical distances from the line are mirrored perfectly.

Given the function f(x) = -x^2 + 4x. Through a reflection over the line y = 3, the graph of a function g is obtained. Determine the equation of g.

Learn how to reflect the quadratic function f(x) = -x^2 + 4x over the horizontal line y = 3, and find the equation of the resulting function g(x). This problem covers algebraic and geometric transformations, and is suitable for students in Grades 9-11.

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