SUMMARY OF FUNCTION TRANSFORMATIONS

Transcription

1 SUMMARY OF FUNCTION TRANSFORMATIONS The graph of = Af(B(t +h))+k is a transformation of the graph of = f(t). The transformations are done in the following order: B: The function stretches or compresses horizontall b a factor of 1 B. If B is negative, the function also reflects across the vertical ais. h: The function shifts horizontall b h units. If h > 0, the function shifts left. If h < 0, the function shifts right. A: The function stretches or compresses verticall b a factor of A. If A is negative, the function also reflects across the horizontal ais. k: The function shifts verticall b k units. If k > 0, the function shifts up. If k < 0, the function shifts down. **It can be helpful to remember BhAk as a wa of remembering the ordering. Other orderings are possible; this ordering, however, will work in all cases. Graph the transformations below b doing the following on graphing paper: Graph the basic function used in this transformation. (Eample: f() = ). Use our Librar of Functions Handout if necessar. State the series of transformations and the order in which the occur. Graph the transformation. Check our graph using the interactive GeoGebra applet located on the course webpage. You will also want to check our graph using a few of the input/ output pairs. Hand-sketched eamples are located under Contents. Full solutions to the eercises below are given in the net few pages. Section I: Horizontal and Vertical Shifts (a) g 1 () = ( 5) +1 (b) g () = ++ (c) g 3 () = (+1) 3 (d) g () = 1 + (e) g 5 () = + Section II: Horizontal and Vertical Stretches and Reflections (a) g 1 () = (b) g () = (c) g 3 () = 3 (d) g () = (e) g 5 () = 5 (f) g () = Section III: General Function s (a) g 1 () = 3( ) +5 (b) g () = (+1) 3 3 (c) g 3 () = 3 (d) g () = + (e) g 5 () = +5 (f) g () = (+1)+3 1

2 1. Horizontal and Vertical Shifts 1. Transforming f() = into g 1 () = ( 5) +1: The graph of = g 1 () is in Figure 1. It is obtained b the following transformations: (a) Shift 5 units right (b) Shift 1 unit up Figure Transforming f() = into g () = ++: The graph of = g () is in Figure. It is obtained b the following transformations: (a) Shift units left (b) Shift units up Figure Instructor: A.E.Car Page of 10

3 3. Transforming f() = 3 into g 3 () = (+1) 3 : The graph of = g 3 () is in Figure 3. It is obtained b the following transformations: (a) Shift 1 units left (b) Shift units down Figure Transforming f() = 1 into g () = 1 +: The graph of = g () is in Figure. It is obtained b the following transformations: (a) Shift units right (b) Shift units up Figure Instructor: A.E.Car Page 3 of 10

4 5. Transforming f() = into g 5 () = +: The graph of = g 5 () is in Figure 5. It is obtained b the following transformations: (a) Shift units right (b) Shift units up Figure Instructor: A.E.Car Page of 10

5 . Horizontal and Vertical Stretches, Compressions and Reflections. Transforming f() = into g 1 () = : The graph of = g 1 () is in Figure. It is obtained b the following transformations: (a) Reflect across the horizontal ais Figure Transforming f() = into g () = : The graph of = g () is in Figure 7. It is obtained b the following transformations: (a) Reflect across the vertical ais Figure Instructor: A.E.Car Page 5 of 10

6 . Transforming f() = 3 into g 3 () = 3 : The graph of = g 3 () is in Figure. It is obtained b the following transformations: (a) Stretch verticall b a factor of Figure Transforming f() = into g () = : The graph of = g () is in Figure 9. It is obtained b the following transformations: (a) Stretch verticall b a factor of (b) Reflect across the horizontal ais Figure Instructor: A.E.Car Page of 10

7 10. Transforming f() = into g 5 () = 5 : The graph of = g 5 () is in Figure 10. It is obtained b the following transformations: (a) Compress horizontall b a factor of 1 5. Figure Transforming f() = into g () = : The graph of = g () is in Figure 11. It is obtained b the following transformations: (a) Reflect across the vertical ais (b) Reflect across the horizontal ais (c) Stretch verticall b a factor of Figure Instructor: A.E.Car Page 7 of 10

8 3. General Function s 1. Transforming f() = into g 1 () = 3( ) +5: The graph of = g 1 () is in Figure 1. It is obtained b the following transformations: (a) Shift units right (b) Stretch verticall b a factor of 3 (c) Shift 5 units up Figure Transforming f() = 3 into g () = (+1) 3 3: The graph of = g () is in Figure 13. It is obtained b the following transformations: (a) Shift 1 unit left (b) Reflect across the horizontal ais (c) Shift 3 units down Figure Instructor: A.E.Car Page of 10

9 1. Transforming f() = into g 3 () = 3: The graph of = g 3 () is in Figure 1. It is obtained b the following transformations: (a) Compress horizontall b a factor of 1 (b) Shift 3 units down Figure Transforming f() = into g () = +: The graph of = g () is in Figure 15. It is obtained b the following transformations: (a) Reflect across the vertical ais (b) Shift units up Figure Instructor: A.E.Car Page 9 of 10

10 1. Transforming f() = 1 into g 5() = +5: The graph of = g 5 () is in Figure 1. It is obtained b the following transformations: (a) Stretch verticall b a factor of (b) Shift 5 units up Figure Transforming f() = into g () = (+1)+3: The graph of = g () is in Figure 17. It is obtained b the following transformations: (a) Compress horizontall b a factor of 1 (b) Shift 1 unit left (c) Stretch verticall b a factor of (d) Shift 3 units up and reflect across the vertical ais Figure Instructor: A.E.Car Page 10 of 10