Given the graph y = f(x), explain and contrast the effect of the constant c on the graphs y = f(cx) and y = cf(x).

Given the graph y = f(x)

The graph y = f(cx), where c is a constant is refered to as horizontal stretch/compression

A horizontal stretching is the stretching of the graph away from the y-axis. A horizontal compression is the squeezing of the graph towards the y-axis.  A compression is a stretch by a factor less than 1.

If | c | < 1 (a fraction between 0 and 1), then the graph is stretched horizontally by a factor of c units.
If | c | > 1, then the graph is compressed horizontally by a factor of c units.

For values of c that are negative, then the horizontal compression or horizontal stretching of the graph is followed by a reflection across the y-axis.

The graph y = cf(x), where c is a constant is referred to as a vertical stretching/compression.

A vertical streching is the stretching of the graph away from the x-axis. A vertical compression is the squeezing of the graph towards the x-axis. A compression is a stretch by a factor less than 1.

If | c | < 1 (a fraction between 0 and 1), then the graph is compressed vertically by a factor of c units.
If | c | > 1, then the graph is stretched vertically by a factor of c units.

For values of c that are negative, then the vertical compression or vertical stretching of the graph is followed by a reflection across the x-axis.