1.6 - Order for Applying Transformations

1 Using multiple Transformations to Graph Quadratic Functions September 16, 2012 MCF3M S. Inrig Page 1 of 1 Order for Applying Transformations You will recall that the basic ( parent ) quadratic function is 2)(xxf=, which describes a parabola that opens upward and has its vertex at the origin (0,0). We have seen three kinds of Transformations in section : kxxf+=2)(, which represents a translation ( shift ) of the entire graph upward (if k is positive) or downward (if k is negative), by k units 2)()(hxxf =, which represents a translation ( shift ) of the entire graph to the right (if h is positive) or left (if h is negative, which changes the sign following x to a + !) 2)(axxf=, which causes: o a reflection about the x axis if a is negative o a vertical stretch by a factor of a (if the absolute value of a is greater than 1), or a vertical compression by a factor of a (if the absolute value of a is less than 1) What if more than one of the above Transformations is to be applied?

2 Does the Order in which they are applied matter? Yes, it does. Note the following: 1. Horizontal and vertical Transformations are independent of each other. The parameter h affects only the horizontal position of the graph; the parameters a and k affect only the vertical aspects of the graph (direction of opening, stretch/compression, and position). This means that you can apply the horizontal transformation (if any) and then the vertical ones, or vice-versa. (The textbook examples do the horizontal Transformations first.) 2. We have considered only one kind of horizontal transformation in this course, so there is no question of Order for horizontal Transformations . However, there are three possible Transformations in the vertical direction: o reflection about the x axis (determined by the sign of a) o vertical stretch or compression (determined by the magnitude, or absolute value, of a) o vertical translation (up or down), determined by k So what Order should be used for these?

3 Note that a function which involves only vertical Transformations can be written in the form kaxxf+=2)(. If we were evaluating the expression on the right of the equals sign (using BEDMAS), we would square the x value first, then multiply by a, and finally add the value of k. Notice that the a value is used before the k value when we do the evaluation. The same thing applies with vertical Transformations : we consider the effect of a first, then k. This means that we o do reflection (if necessary) and stretching/compression first (it doesn t matter whether we do the reflection before stretching/compression, or afterward) o do the translation last For an example of how to do multiple vertical Transformations , see the textbook, pages 51-53. Note that you may need to rearrange a given equation to get it in the form khxaxf+ =2)()(, before Applying Transformations (see example 4 on page 55).