The graph of any function may be transformed either by shifting, stretching/compressing, or reflection.

For shifting and stretching/compressing, there are two types: horizontal and vertical.

A graph may also be reflected either over the x-axis or the y-axis.

1. Vertical transformations:

A shift may also be referred to as a translation. In order to vertically translate the graph of y = f(x) by c units upward, c has to be added to the function. The function now becomes y = f(x) + c.

For a downward translation of c units, the function becomes y = f(x) – c. Note that in this case, c is subtracted from the function y = f(x).

In general, a vertical translation means that every point (x, y) on the graph of y = f(x) is transformed to (x, y + c) on the graph of y = f(x) + c. On the other hand, every point (x, y) on the graph of y = f(x) is transformed to (x, y – c) on the graph of y = f(x) – c.

2. Horizontal transformations:

Horizontal translations of the function y = f(x) are dealt with in a different manner.

When the function is shifted c units to the right, x becomes (x – c) so that the new function is y = f(x – c). When the same function y = f(x) is translated c units to the left, the new function becomes y = f(x + c).

Another way of looking at this is to remember that a horizontal translation means that every point (x, y) on the graph of y = f(x) is transformed to (x + c, y) on the graph of y = f(x – c). On the other hand, every point (x, y) on the graph of y = f(x) is transformed to (x – c, y) on the graph of y = f(x + c).

2. Vertical Stretching and Compression:

The next type of transformation is vertical stretching and compression.

If y = f(x) represents the graph of the original function as mentioned above, then a graph affected by vertical stretching or compression is expressed as y = cf(x). It should be noted that when 0 < c < 1, a vertical shrinking of the graph of y = f(x) is observed. Graphically, a vertical shrinking "pulls" the graph of y = f(x) toward the x-axis.

When c > 1 in the function y = cf(x), the graph is "pushed" away from the x-axis (vertical stretching). The x-intercept remains the same as the original function in both cases.

Another way to think of vertical shrinking and compressing is that every point (x, y) on the graph of y = f(x) is transformed to (x, cy) on the graph of y = cf(x).

3. Horizontal Stretching and Compression:

Next, one has to analyze horizontal stretching and compression.

If y = f(x) represents the graph of the original function as mentioned above, then a graph affected by horizontal stretching or compression is expressed as y = f(cx). It should be noted that when 0 < c < 1, a horizontal stretching of the graph of y = f(x) is observed. Graphically, a horizontal stretching "pulls" the graph of y = f(x) away from the y-axis.

When c > 1 in the function y = f(cx), the graph is "pulled" toward the y-axis (horizontal shrinking). The y-intercept remains the same as the original function in both cases.

In addition, horizontal shrinking and compressing means that every point (x, y) on the graph of y = f(x) is transformed to (x/c, y) on the graph of y = f(cx).

4. Reflection:

The last type of transformation to look at is the reflection. It is fairly straightforward to understand!

When the original graph of y = f(x) is reflected across the x-axis, the function of the reflected graph becomes y = –f(x). On the other hand, when the same function is reflected across the y-axis, the function of the reflected graph is y = f(–x). That’s it!

Remember that the transformations mentioned above may be combined within the same function so that one graph can be shifted, stretched, and reflected!