When two or more transformations are combined to form a new transformation, the result is called a sequence of transformations, or a composition of transformations. Remember, that in a composition, one transformation produces an image upon which the other transformation is then performed. Sequences of transformations applied to functions work in a similar manner. When working with composition of transformations, it was seen that the order in which the transformations were applied often changed the outcome. This same potential problem is present when working with a sequence of transformations on functions.
For example, given the function of y = x2, a vertical stretch of 3 followed by a vertical shift of 2 will not produce the same graph as a vertical shift of 2 followed by a vertical stretch of 3.There are, however, certain situations where the order is not important and the same graph will exist regardless of the order in which the transformations are applied.
Be careful not to assume that this will be the case in all problems.To determine when order of sequences of transformations affects function graphs:
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• If two or more of the transformations have a vertical effect on the graph, the order of those transformations will most likely affect the graph.
• If two or more of the transformations have a horizontal effect on the graph, the order of those transformations will most likely affect the graph.
• Vertically oriented transformations and horizontally oriented transformations to not affect one another.Consider the problem f (x) = 2(x + 3) - 1.
The parent function is f (x) = x, a straight line.
It can be seen that the parentheses of the function have been replaced by x + 3, as in f (x + 3) = x + 3. This is a horizontal shift of three units to the left from the parent function.
The multiplication of 2 indicates a vertical stretch of 2, which will cause to line to rise twice as fast as the parent function. The parent has a slope of 1, whereas this new function will have a slope of 2.
The subtraction of 1 indicated a vertical shift of one unit down. There was a pattern to the order in which this problem was analyzed (horizontal shift - - vertical stretch - - vertical shift). This pattern is similar to order of operations. The parentheses were done first, then any multiplication/division, followed by any addition/subtraction. So the Answer is A
