Transforming functions

Transforming graphs of functions Graphs can be transformed by translating, reflecting, stretching or rotating them. The equation of the transformed graph is related to the equation of the original graph. y = f(x) When investigating transformations it is useful to distinguish between functions and graphs. For example, to investigate transformations of the function f(x) = x2, the equation of the graph of y = x2 can be written as y = f(x). y = 3 – f(x + 2)

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. What do you notice? This is the graph of y = f(x) – 3 and this is the graph of y = f(x) – 7. What do you notice? Teacher notes Ask students to complete a table of values comparing functions. They could write the values of x in one row (e.g. from –5 to 5) and write the corresponding values of x2, x2 + 4, and x2 – 3 in the rows beneath. This will help them to understand why the function is translated in this way. 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 also be investigated for other graphs and functions using the activities at the end of this section. Mathematical practices 7) Look for and make use of structure. Students should notice patterns in how the shape of a graph changes as different transformations are applied. The graph of y = f(x) + a is the graph of y = f(x) translated vertically by a units. Write a table of values comparing these functions.

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). What do you notice? This is the graph of y = f(x + 2), and this is the graph of y = f(x + 3). What do you notice? Teacher notes Ask students to complete a table of values comparing these functions. They could write the values of x in one row (e.g. from –5 to 5) and write the corresponding values of x2 – 3, (x – 1)2 – 3, and (x + 2)2 – 3 in the rows beneath. This will help them to understand why the function is translated in this way. 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. Mathematical practices 7) Look for and make use of structure. Students should notice patterns in how the shape of a graph changes as different transformations are applied. The graph of y = f(x + a ) is the graph of y = f(x) translated horizontally by –a units. Write a table of values comparing these functions.

Reflections across the x-axis Here is the graph of y = x2 –2x – 2, where y = f(x). y This is the graph of y = –f(x). x What do you notice? The graph of y = –f(x) is the graph of y = f(x) reflected across the x-axis. Here is the table of values: Teacher notes Ask students to complete similar tables of values for other functions. Establish that the graph of y = –f(x ), is a reflection of y = f(x) across the x-axis. This can be investigated for other graphs and functions using the activities at the end of this section. Mathematical practices 7) Look for and make use of structure. Students should notice patterns in how the shape of a graph changes as different transformations are applied. x 1 2 3 4 5 f(x) –3 –2 1 6 13 –f(x) 3 2 –1 –6 –13

Reflections across the y-axis Here is the graph of y = x2 –2x – 2, where y = f(x). y This is the graph of y = f(–x). x What do you notice? The graph of y = f(–x) is the graph of y = f(x) reflected across the y-axis. Here is the table of values: Teacher notes Ask students to complete similar tables of values for other functions. Establish that the graph of y = f(–x ), is a reflection of y = f(x) across 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 across the y-axis. For example, the graph of y = x2. These functions are called even functions. Mathematical practices 7) Look for and make use of structure. Students should notice patterns in how the shape of a graph changes as different transformations are applied. x –2 –1 1 2 f(x) 6 1 –2 –3 –2 –f(x) –2 –3 –2 1 6

Vertical stretch and compression Here is the graph of y = x2 –2x + 3, where y = f(x). This is the graph of y = 2f(x). y What do you notice? 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. Teacher notes When a < 1, the graph is compressed. 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. Ask students to complete a table of values comparing these functions to help them understand the changes. They could write the values of x in one row (e.g. from –5 to 5) and write the corresponding values of x2 – 3, 2(x2 – 3), and ½(x2 – 3) in the rows beneath. This will help them to understand why the function is translated in this way. They could also look at multiplication by a negative value, to see how the vertical stretching and compression discussed on this slide relates to reflection across the y axis. Mathematical practices 7) Look for and make use of structure. Students should notice patterns in how the shape of a graph changes as different transformations are applied. x The graph of y = af(x) is the graph of y = f(x) stretched parallel to the y-axis by scale factor a. What happens when a < 1?

Horizontal stretch and compression Here is the graph of y = x2 + 3x – 4, where y = f(x). y This is the graph of y = f(2x). x What do you notice? 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. Teacher notes When a < 1, the graph is stretched. Ask students to complete a table of values comparing these functions to help them understand the changes. They could write the values of x in one row (e.g. from –5 to 5) and write the corresponding values of x2 – 3, 2(x2 – 3), and ½(x2 – 3) in the rows beneath. This will help them to understand why the function is translated in this way. They could also look at multiplication by a negative value, to see how the vertical stretching and compression discussed on this slide relates to reflection across the y axis. Students 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 students 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. Mathematical practices 7) Look for and make use of structure. Students should notice patterns in how the shape of a graph changes as different transformations are applied. The graph of y = f(ax) is the graph of y = f(x) compressed parallel to the x-axis by scale factor . 1 a What happens when a < 1?

Combining transformations We can now look at what happens when we combine any of these transformations. For example, since all quadratic curves have the same basic shape, any quadratic curve can be obtained by performing a series of transformations on the curve y = x2. Write down the series of transformations that must be applied to the graph of y = x2 to give the graph y = 2x2 + 4x – 1. Complete the square to distinguish the transformations: 2x2 + 4x – 1 = 2(x2 + 2x) – 1 = 2((x + 1)2 – 1) – 1 = 2(x + 1)2 – 3

Combining transformations solution These are the transformations that must be applied to y = x2 to give the graph y = 2x2 + 4x – 1: y = x2 1. Translate –1 units horizontally. y = (x + 1)2 2. Stretch by a scale factor of 2 vertically. y = 2(x + 1)2 Teacher notes Stress the importance of performing the transformations in the correct order. For example, if we translated (x + 1)2 by –3 units in the y-direction we would have (x + 1)2 – 3. If we then stretched this by a scale factor 2 in the y-direction we would have 2((x + 1)2 – 3) = 2(x + 1)2 – 6. Students could look at different orders of applying the transformations to see which orders have the same resulting function. Mathematical practices 6) Attend to precision. Students should take care to combine transformations in a correct order. 3. Translate –3 units vertically. y = 2(x + 1)2 – 3 These transformations must be performed in the correct order.