Transforming graphs of functions Graphs can be transformed by translating, reflecting, stretching or rotating them. The equation of the transformed graph will be related to the equation of the original graph. When investigating transformations it is most useful to express functions using function notation. For example, suppose we wish to investigate transformations of the function f(x) = x2. The equation of the graph of y = x2, can be written as y = f(x).

Vertical translations Here is the graph of y = x2, where y = f(x). y This is the graph of y = f(x) + 1 x and this is the graph of y = f(x) + 4. This is the graph of y = f(x) – 3 and this is the graph of y = f(x) – 7. Establish that for y = f(x) + a, if a is positive the curve y = f(x) is translated a units upwards. If a is negative, the curve y = f(x) is translated a units downwards. This can be investigated for other graphs and functions using the activities at the end of this section. The graph of y = f(x) + a is the graph of y = f(x) translated by the vector . a

Horizontal translations Here is the graph of y = x2 – 3, where y = f(x). y This is the graph of y = f(x – 1), x and this is the graph of y = f(x – 4). This is the graph of y = f(x + 2), and this is the graph of y = f(x + 3). Establish that for f(x + a), if a is negative the curve is translated a units to the right (in the positive horizontal direction). If a is positive, the curve is translated a units to the left (in the negative horizontal direction). This can be investigated for other graphs and functions using the activities at the end of this section. The graph of y = f(x + a ) is the graph of y = f(x) translated by the vector . –a

Reflections in the x-axis Here is the graph of y = x2 –2x – 2, where y = f(x). y x This is the graph of y = –f(x). The graph of y = –f(x) is the graph of y = f(x) reflected in the x-axis. Establish that the graph of y = –f(x ), is a reflection of y = f(x) in the x-axis. This can be investigated for other graphs and functions using the activities at the end of this section.

Reflections in the y-axis Here is the graph of y = x3 + 4x2 – 3 where y = f(x). y x This is the graph of y = f(–x). The graph of y = f(–x) is the graph of y = f(x) reflected in the y-axis. Establish that the graph of y = f(–x ), is a reflection of y = f(x) in the y-axis. This can be investigated for other graphs and functions using the activities at the end of this section. The graphs of some functions remain unchanged when reflected in the y-axis. For example, the graph of y = x2. These functions are called even functions.

Stretches in the y-direction Here is the graph of y = x2, where y = f(x). This is the graph of y = 2f(x). y This graph is produced by doubling the y-coordinate of every point on the original graph y = f(x). This has the effect of stretching the graph in the vertical direction. Demonstrate that the distance from the x-axis to the curve y = f(2x) is always double the distance from the x-axis to the curve y = f(x). For example, the point (2,4) becomes (2,8) and the point (–1, 1) becomes (–1, 2). The x-coordinate stays the same in each case and the y-coordinate doubles. This can be investigated for other graphs and functions using the activities at the end of this section. x The graph of y = af(x) is the graph of y = f(x) stretched parallel to the y-axis by scale factor a.

Stretches in the x-direction Here is the graph of y = x2 + 3x – 4, where y = f(x). y This is the graph of y = f(2x). x This graph is produced by halving the x-coordinate of every point on the original graph y = f(x). This has the effect of compressing the graph in the horizontal direction. Pupils should notice that the intersection on the y-axis has not changed and that the graph has been compressed (or squashed) horizontally. Demonstrate that the distance from the y-axis to the curve y = f(2x) is always half the distance from the y-axis to the curve y = f(x). Ask pupils to predict what the graph of y = f(½x) would look like. This is probably the most difficult transformation to visualize. Using the activities on the next few slides will help. The graph of y = f(ax) is the graph of y = f(x) stretched parallel to the x-axis by scale factor . a 1