Suppose that the graph of f(x)=x4−x2 is compressed horizontally by a factor of 2 and is then shifted 9 units to the right.

We are given the function \( f(x) = x^4 - x^2 \). The problem asks us to perform two transformations on this function:

1. **Horizontal Compression by a Factor of 2**: A horizontal compression of a function \( f(x) \) by a factor of 2 is represented by replacing \( x \) with \( \frac{x}{2} \). So, the compressed function becomes:
   \[
   f\left(\frac{x}{2}\right) = \left(\frac{x}{2}\right)^4 - \left(\frac{x}{2}\right)^2 = \frac{x^4}{16} - \frac{x^2}{4}
   \]

2. **Shift 9 Units to the Right**: To shift the function \( f(x) \) 9 units to the right, replace \( x \) with \( x - 9 \) in the function. Applying this to the compressed function:
   \[
   f\left(\frac{x - 9}{2}\right) = \left(\frac{x - 9}{2}\right)^4 - \left(\frac{x - 9}{2}\right)^2
   \]
   Expanding the terms:
   \[
   \left(\frac{x - 9}{2}\right)^4 = \frac{(x - 9)^4}{16}, \quad \left(\frac{x - 9}{2}\right)^2 = \frac{(x - 9)^2}{4}
   \]
   Therefore, the transformed function is:
   \[
   f(x) = \frac{(x - 9)^4}{16} - \frac{(x - 9)^2}{4}
   \]

This is the final equation of the transformed function.

Would you like more details on any part of the process?

### Related Questions:
1. How do vertical stretches or compressions affect a function?
2. What is the difference between horizontal and vertical transformations?
3. How do translations affect the graph of a function in general?
4. Can you explain how to expand \( (x-9)^4 \) in detail?
5. What happens to the function if it's shifted vertically instead of horizontally?

### Tip:
When dealing with horizontal transformations, always remember that changes inside the function's argument (like \( x \)) affect the graph in the opposite direction of what you might initially think (compression by 2 means multiplying by \( \frac{1}{2} \), and a right shift by 9 means subtracting 9).

Suppose that the graph of f(x)=x^4−x^2 is compressed horizontally by a factor of 2 and is then shifted 9 units to the right.

Learn how to transform the polynomial function f(x)=x^4−x^2 with a horizontal compression by a factor of 2 and a shift of 9 units to the right. This solution includes key concepts in function transformations and polynomial expansions, suitable for students in Grades 10-12.

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