A very common exercise in high school mathematics is to plot transformations of some standard functions. For example, to plot we may start with a standard sine curve and apply the following transformations in turn:
- squeeze it by a factor of 5 in the -direction
- shift it left by
- stretch it by a factor of 2 in the -direction
- shift it down by
This leads to the plot shown.
For sine and cosine graphs an alternative is to plot successive peaks/troughs of the curve and interpolate accordingly. For example, to plot we may proceed as follows.
- Since has a peak at , solve to find as a point where there is a peak at . Hence plot the point .
- Since the angular frequency is 5, the period is and we may plot successive peaks spaced apart from the point .
- Troughs will be equally spaced halfway between the peaks at (at ). Then join the dots with a sinusoidal curve.
- Additionally and intercepts may be found by setting and respectively. We find that the intercepts are at and intercept is .
The first approach is more generalisable to plotting other functions. Instead of thinking of the graph transforming, we also may consider it as a change of coordinates. For example, if we translate the parabola so that its turning point is at , this is equivalent to keeping the parabola fixed and shifting axes so that the new origin is at with respect to the old coordinates. This is illustrated below where the black coordinates are modified to the red ones. The parabola has equation under the black coordinates and or under the red coordinates.
As another example suppose we take a unit circle and stretch it by a factor of in the direction and a factor of in the direction. This is the equivalent of changing scale so that the -axis is squeezed by and the -axis is squeezed by .
Under this stretching of the circle or squeezing of axes, the unit circle equation transforms to that of the ellipse .
More generally, by stretching a Cartesian graph by in the -direction and in the -direction, then shifting it along the vector , we obtain the equation
This uses the fact that and are the inverses of and respectively. Note that if or are less than 1, the stretch becomes a squeezing of the graph, while or correspond to a reflection in the or axes.
We can extend this idea to the rotation of a graph. Suppose for example we wish to rotate the hyperbola by 45 degrees anti-clockwise. This is equivalent to a rotation of the axes by 45 degrees clockwise and the matrix corresponding to this linear transform is
(Here the columns of the change of basis matrix correspond to where the basis vectors (1,0) and (0,1) map to under a 45 degree clockwise rotation.)
In other words we replace with and with in the equation and obtain or .
Here is the same transformation applied to the parabola to obtain or :
If a graph is affinely transformed (by an invertible map) so that maps to and maps to followed by a shift along the vector , then this is equivalent to the coordinates shifting by and then transforming under the inverse mapping :
Here are some special cases of this formula:
- rotation of the graph by anti-clockwise: (the example was done above)
- reflection of the graph in :
- reflection of the graph in where : verify that so
- reflection of the graph in where : this is equivalent to a reflection in the line followed by a shift along the vector so
- reflection of the graph in (special instance of the previous case with ):