Consider the relationship between y = f(x) and y = f (kx).

Question

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.