When we modify the parent function f(x) = x^4 to f(2x), we are essentially applying a transformation that affects the shape and behavior of the graph. Specifically, this transformation is a horizontal compression or “squeezing” of the function’s graph.
To understand this effect, let’s break it down:
- Original Function: The graph of f(x) = x^4 is a parabola that opens upwards, centered around the y-axis. It rises slowly for negative and positive values of x, and its growth accelerates as x moves further from zero.
- Transformed Function: By changing to f(2x) = (2x)^4 = 16x^4, we alter the input of the function. This change means we are effectively halving the input value needed to achieve the same output.
This transformation results in the following changes:
- Horizontal Compression: The graph of f(2x) gets compressed horizontally by a factor of 2. This means that the function will reach its maximum and minimum values more quickly as x increases or decreases. For example, while the original function f(x) might reach the height of 8 at x=2, the transformed function f(2x) will reach the same height at x=1.
- Output Scaling: The function also grows more rapidly than the original for larger values of x due to the factor of 16 in f(2x).
In conclusion, the transition from f(x) = x^4 to f(2x) leads to a function that is horizontally compressed by a factor of 2, resulting in faster growth and changed coordinate points for the function’s outputs. This has implications for the graph’s appearance, making it steeper and narrower, which could be visualized through a graphing tool for an even clearer understanding.