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The given function \( f(x) = x^2 - 2x \) undergoes a series of transformations to form \( g(x) \). Let's break down each transformation step:

1. **Translation 2 units left**: This modifies the \( x \)-term in the function by replacing \( x \) with \( (x + 2) \). The new function after this step becomes:
   \[
   f(x+2) = (x+2)^2 - 2(x+2)
   \]
   Expanding this:
   \[
   (x+2)^2 - 2(x+2) = x^2 + 4x + 4 - 2x - 4 = x^2 + 2x
   \]

2. **Translation 3 units up**: To translate the graph up by 3 units, we add 3 to the function:
   \[
   f(x+2) + 3 = (x^2 + 2x) + 3 = x^2 + 2x + 3
   \]

3. **Reflection in the \( y \)-axis**: Reflecting the graph in the \( y \)-axis involves replacing \( x \) with \( -x \):
   \[
   g(x) = (-x)^2 + 2(-x) + 3 = x^2 - 2x + 3
   \]

Thus, the rule for \( g(x) \) is:
\[
g(x) = x^2 - 2x + 3
\]

Would you like further details or explanations on any step?

Here are 5 follow-up questions you may consider:

1. How does reflecting a graph in the \( y \)-axis affect its symmetry?
2. What is the effect of translating a graph horizontally by a certain number of units?
3. Can you describe how vertical translations affect the range of a function?
4. How would the transformations change if we reflected the graph in the \( x \)-axis instead?
5. How do you expand binomials like \( (x + 2)^2 \) algebraically?

**Tip**: When performing multiple transformations, always follow the order of operations: apply horizontal transformations, then reflections, and finally vertical translations.

The graph of g is a translation 2 units left and 3 units up, followed by a reflection in the y-axis of the graph of f(x) = x^2 - 2x. Write a rule for g.

Learn how to apply translations and reflections to the quadratic equation f(x) = x^2 - 2x. This problem covers reflections over the y-axis and translations, suitable for high school students.

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