x^2 shift left 2 units, reflected across the y axis, shift down 5 units, vertical scaling by a factor of 4, reflection across the x-axis

We are given several transformations to apply to the function \( f(x) = x^2 \). Let's apply them step by step in the following order:

1. **Shift left 2 units**: Shifting a function left by 2 units means replacing \( x \) with \( x + 2 \). So, the function becomes:
   \[
   f(x) = (x + 2)^2
   \]

2. **Reflect across the y-axis**: Reflecting across the y-axis means replacing \( x \) with \( -x \). Now, the function becomes:
   \[
   f(x) = (-x + 2)^2 = (2 - x)^2
   \]

3. **Shift down 5 units**: Shifting down means subtracting 5 from the function. Thus, the function becomes:
   \[
   f(x) = (2 - x)^2 - 5
   \]

4. **Vertical scaling by a factor of 4**: Scaling vertically means multiplying the entire function by 4. This results in:
   \[
   f(x) = 4((2 - x)^2 - 5) = 4(2 - x)^2 - 20
   \]

5. **Reflect across the x-axis**: Reflecting across the x-axis means multiplying the entire function by \( -1 \). The final function is:
   \[
   f(x) = -4(2 - x)^2 + 20
   \]

Thus, the transformed function is:

\[
f(x) = -4(2 - x)^2 + 20
\]

### Would you like further explanation or have any questions?

Here are 5 related questions to enhance understanding:

1. How does reflecting a function across the x-axis affect its graph?
2. What is the general form of a parabola after a horizontal shift and reflection?
3. How would you shift a function right by 3 units instead of left?
4. What does vertical stretching or compressing do to a quadratic function?
5. How can we recognize the transformations of a function from its equation alone?

**Tip:** Always apply horizontal transformations (shifts and reflections) before vertical transformations (scaling, shifts, or reflections) to avoid confusion.

x^2 shift left 2 units, reflected across the y-axis, shift down 5 units, vertical scaling by a factor of 4, reflection across the x-axis

Learn how to apply multiple transformations to the quadratic function x^2, including shifts, reflections across both axes, and vertical scaling. Follow a detailed step-by-step solution for a high school-level algebra problem. Perfect for students in Grades 9-12.

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