1.6 Graphical Transformations: Transformations: Functions that map points to points. There are two types of transformations: 1) Rigid transformations include horizontal translations (shifts), vertical translations (shifts), reflections (flipping the graph over the x-­‐axis or y-­‐axis) or a combination of these. The basic shape and size of the graph remains the same. 2) Nonrigid transformations distort the shape of the graph and horizontal or vertical stretches or shrinks. Vertical and Horizontal Transformations A vertical translation is a shift of the graph up or down. A horizontal translation is a shift of the graph left or right. EX: EX: Describe how the graph of y = x 2 is transformed by the following equations: a) y = x 2 + 5 2
b) y = ( x − 3) EX: Each of the graphs below shows the graph of y = x 3 (dotted graph ) and a vertical or horizontal translation. Write an equation for the translated function. Reflections Across Axes The points (x, y) and (x, -­‐y) are reflections of each other across the x-­‐axis. The points (x, y) and (-­‐x, y) are reflections of each other across the y-­‐axis. Two points (or graphs) that are symmetric with respect to a line are reflections of each other across that line. EX: Find the equation for the reflection of f (x) =
5x − 9
across each axis. x2 + 3
Stretches and Shrinks EX: Given the equation f (x) = x 3 − 16x , find equations for the follwing transformations: a) Vertical stretch by a factor of 3 b) Horizontal shrink by a factor of ½. Combining Transformations Transformations can be performed one after another. In some cases the order in which the transformations occur may make a difference in the resulting equation. EX: