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Math 140 1.7: Graphing Techniques Transformations Trigonometry Here is a summary of the transformations of y = f(x). See Summary of Graphing Techniques on page 74. 1. Vertical translation (or shift): y f ( x) k For k > 0, The graph of y f ( x) k is the parent graph, y f (x) , shifted up k units. The graph of y f ( x) k is the parent graph, y f (x) , shifted down k units; 2. Horizontal translation (or shift): y f ( x h) For h > 0, The graph of y f ( x h) is the parent graph, y f (x) , shifted left h units. The graph of y f ( x h) is the parent graph, y f (x) , shifted right h units; 3. Reflections: a.) Reflection about the x-axis: The graph of y f (x) is the reflection of the parent graph, y f (x) , about the x–axis. b.) Reflection about the y-axis: The graph of y f ( x) is the reflection of the parent graph, y f (x) , about the y–axis. 4. Vertical Stretch or Compression: y af ( x) The graph of y af ( x) is the graph of the parent graph, y f (x) , … Stretched vertically by a multiplication factor of a for a 1 , or Compressed vertically by a multiplication factor of a for 0 a 1 . For a < 0, the graph is also reflected across the x–axis. 5. Horizontal Stretch or Compression: y f (ax) The graph of y f (ax) is the graph of the parent graph, y f (x) , … Compressed horizontally by a multiplication factor of Stretched horizontally by a multiplication factor of 1 a 1 a for a 1 , or for 0 a 1 . For a < 0, the graph is also reflected across the y–axis. Order of Operations for Transformations 1. Stretch/Compress 2. Reflect 3. Translate Hint: When reflecting in the y–axis AND translating horizontally, factor out a –1. See p. 75, Example 6, also #65, #66 from homework. Vertical and Horizontal Translations Vertical Stretching and Compressing Horizontal Stretching and Compressing y f (x) - bold y f ( x) 3 - thin y f ( x 4) - dotted y f (x) - bold y 2 f ( x) - thin y 12 f ( x) - dotted y f (x) - bold y f (2 x) - thin y f ( 12 x) - dotted For problems 39 – 67: Graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the parent function and show all stages. Be sure to show at least three key points on the final graph. Find the domain and range of each function. Verify your results with a graphing utility. Remember: (1) Stretch/Compress; (2) Reflect; (3) Translate (40) f x x 2 4 (44) h x x 1 (48) g x (50) h x 3 2 x 1 x 2 (60) g x 3 x 1 3 (46) f x x 2 3 3 (56) h x (like 63) f x x 2 3 2 1 2 x (64) f x 4 x 1 Hole at (1, 0) (66) g x 4 2 x Finding the Cube Root Key MATH, then 4: Hole at (2, 0) 3 ( Finding the Absolute Value Key MATH, then arrow key to the right to get the NUM menu. Then 1: abs( OR 2nd CATALOG, then the first item on the list will be abs( , once it is highlighted hit ENTER. Question: What about f(x) = (2x)3 ? Do I think of the x as getting cut in half, or rewrite the function as f(x) = 8x3, and think of the y–values as getting multiplied by 8. Answer: Regardless of how you conceptualize f(x) the net effect on the graph is the same.