Question

Consider the relationship between y = f(x) and y = f (kx).
What is this type of function transformation called? What does it look like on a graph?

Answer 1

The relationship between \( y = f(x) \) and \( y = f(kx) \) represents a horizontal compression or stretching of the graph of the function \( f(x) \), depending on the value of the constant \( k \).

1. **If \( |k| > 1 \)**: The graph of \( f(kx) \) undergoes a horizontal compression. This means the features of the graph (like peaks, valleys, and intercepts) get closer together. For example, if \( k = 2 \), the graph compresses horizontally, making it narrower.

2. **If \( 0 < |k| < 1 \)**: The graph of \( f(kx) \) will experience a horizontal stretch. This means the features of the graph will be spread farther apart. For instance, if \( k = 0.5 \), the graph stretches horizontally, making it wider.

3. **If \( k < 0 \)**: The graph will be reflected across the y-axis in addition to the compression/stretching effect described above.

When graphing, keep in mind that the basic shape of the function remains the same, but its width and orientation change depending on the value of \( k \). This type of transformation is specifically called a **horizontal scaling** transformation.