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We are tasked with finding the transformation for the function \( g(x) \) based on the following transformations applied to the function \( f(x) = x^2 - 4x + 1 \):

1. A vertical stretch by a factor of 3.
2. A reflection in the y-axis.
3. A translation 2 units left.

### Step-by-step transformations:

1. **Vertical Stretch by a factor of 3**:
   - This means that every output of the function is multiplied by 3. So, \( f(x) \) becomes:
   \[
   g_1(x) = 3f(x)
   \]
   Therefore, we have:
   \[
   g_1(x) = 3(x^2 - 4x + 1) = 3x^2 - 12x + 3
   \]

2. **Reflection in the y-axis**:
   - To reflect a function over the y-axis, we replace \( x \) with \( -x \). Applying this to \( g_1(x) \), we get:
   \[
   g_2(x) = 3(-x)^2 - 12(-x) + 3 = 3x^2 + 12x + 3
   \]

3. **Translation 2 units to the left**:
   - A horizontal translation by 2 units to the left means replacing \( x \) with \( x + 2 \). Applying this to \( g_2(x) \), we get:
   \[
   g(x) = 3(x + 2)^2 + 12(x + 2) + 3
   \]
   Expanding this:
   \[
   g(x) = 3(x^2 + 4x + 4) + 12(x + 2) + 3
   \]
   \[
   g(x) = 3x^2 + 12x + 12 + 12x + 24 + 3
   \]
   Simplifying:
   \[
   g(x) = 3x^2 + 24x + 39
   \]

Thus, the rule for \( g(x) \) is:

\[
g(x) = 3x^2 + 24x + 39
\]

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Do you want further details on any of the steps? Here are some related questions for further exploration:

1. How do you find the reflection of a function over the x-axis?
2. What changes occur when a function is translated vertically instead of horizontally?
3. How do you determine the effect of a vertical stretch on a quadratic function’s vertex?
4. What happens to the graph of a function when you apply multiple transformations in different orders?
5. Can you reverse these transformations to return to the original function?

**Tip**: Always expand and simplify carefully when applying transformations like translations and stretches.

Let the graph of g be a vertical stretch by a factor of 3 and a reflection in the y-axis, followed by a translation 2 units left of the graph of f(x) = x^2 - 4x + 1. Write a rule for g.

This problem demonstrates how to apply a vertical stretch, reflection in the y-axis, and translation to the quadratic function f(x) = x^2 - 4x + 1. Learn the step-by-step solution involving key algebraic transformations. Suitable for high school students.

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