What does the transformation f(x)↦f(1/4x) do to the graph of f(x)?

The transformation \( f(x) \mapsto f\left(\frac{1}{4}x\right) \) affects the graph of \( f(x) \) by **horizontally stretching** it.

Here's how this works:
- The transformation takes \( f(x) \) and replaces \( x \) with \( \frac{1}{4}x \), which means that for the same output value \( f(x) \), you now need a smaller input \( x \) (since you're dividing by 4).
- As a result, the graph appears to stretch horizontally by a factor of 4.
  
This is because the horizontal compression/expansion happens relative to the reciprocal of the coefficient inside the function argument:
- If the argument inside the function is \( f(kx) \), the graph is horizontally **compressed** by a factor of \( \frac{1}{|k|} \) if \( |k| > 1 \).
- If \( |k| < 1 \), the graph is **stretched**. In your case, \( \frac{1}{4} \) implies a horizontal stretch by a factor of 4.

### Summary:
The transformation \( f(x) \mapsto f\left(\frac{1}{4}x\right) \) stretches the graph of \( f(x) \) **horizontally by a factor of 4**.

Would you like more details or have any questions?

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Here are 5 related questions:
1. What happens if the transformation is \( f(x) \mapsto f(4x) \)?
2. How does a vertical stretch differ from a horizontal stretch in transformations?
3. Can you combine horizontal and vertical transformations in one function? How?
4. What would the transformation \( f(x) \mapsto f(x - 3) \) do to the graph of \( f(x) \)?
5. How does the graph of \( f(x) \mapsto -f(x) \) change the original graph of \( f(x) \)?

**Tip:** Horizontal transformations affect the graph by altering the input, while vertical transformations affect the output directly.

Discover how the transformation f(x) ↦ f(1/4x) affects the graph of a function. Learn about horizontal stretching and how it modifies the graph by a factor of 4. Suitable for students in Grades 9-11, this guide covers key function transformation concepts with examples.

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