1. What is Function Dilation?

Function dilation is a transformation that stretches or compresses a function horizontally. The general form is \( f'(x) = f(kx) \), where k is the scale factor. When |k| > 1, the function compresses horizontally; when 0 < |k| < 1, it stretches.

2. How Does the Calculator Work?

The calculator applies the dilation formula:

Where:

• \( f(x) \) — Original function
• \( k \) — Scale factor (non-zero real number)
• \( f'(x) \) — Transformed function

Explanation: The transformation replaces every x in the original function with kx, effectively scaling the input by the factor k.

3. Importance of Function Dilation

Details: Understanding function dilation is crucial in signal processing, image manipulation, and when analyzing how changes in input scale affect output behavior in mathematical models.

4. Using the Calculator

Tips: Enter the original function using standard mathematical notation (e.g., "x^2" for quadratic, "sin(x)" for sine). The scale factor must be non-zero.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dilation and other transformations?
A: Dilation specifically refers to scaling, while other transformations include translation (shifting) and reflection (flipping).

Q2: How does dilation affect the graph of a function?
A: Horizontal dilation by factor k compresses the graph horizontally if |k|>1, or stretches it if 0<|k|<1. Negative k also reflects the graph across the y-axis.

Q3: Can I use this for any type of function?
A: Yes, the dilation transformation applies to all functions, though the visual effect may differ based on the function's properties.

Q4: What about vertical dilation?
A: Vertical dilation would be represented as f'(x) = a·f(x), which is a different transformation not covered by this calculator.

Q5: How is this used in real-world applications?
A: Function dilation is fundamental in signal processing (time-scaling), image resizing, and adjusting mathematical models to different scales.