Chapter 1

1. Chapter 1 1-4 Shifting, reflecting and stretching graphs

4. objectives • The student will be able to: • Recognize graphs of parent functions • Use vertical and horizontal shifts and reflections to graph functions • Use nonrigid transformations

5. Parent functions • What are parent functions? • Answer: A parent function is the simplest function with the defining characteristic of the family. Functions in the same family are transformations of the parent functions.

6. Examples of parent functions

7. Vertical and horizontal shift • Let c be a positive real number. Vertical and horizontal shifts in the graph of are represented as follows: • Vertical shifts c units upward: • Vertical shift c units downward: • Horizontal shift c units to the right: • Horizontal shift c units to the left:

8. Example 1 • Many functions have graphs that are simple transformations of the graphs of parent functions. • Example 1: • Sketch the graph of and .

9. Example 1 continue • If we graph . • What happen to the original graph? • It translate the original graph one unit up. This is what we called a vertical shift or vertical translation.

10. Example 2 • Example 2: • Sketch the graph of and .

11. Example 2 continue • If we graph . • What happen to the original graph? • It translate the original graph one unit to the right. This is what we called a Horizontalshiftor vertical translation.

12. Shifts in the graphs of a function • Lets compare the graph of each function to the parent function and describe the transformation. • A) • Shifting 8 units downs • B) • Shifting 5 units to the right • C) • Shifting 4 units up

13. Student Practice • Do problems 7 and 8 from book page 47

14. Finding equations from graphs Example #3 Each graph is a transformation of the graph f. . Find the equation of the graphs

15. Student Guided practice • Lets do problems 30 and 32 from page 48

16. Reflecting Graphs • Another type of transformation is called reflection. • There are two types of reflection across the coordinate axis: • Reflection in the x-axis h(x)=-f(x) • Reflection in the y-axis h(x)=f(-x)

17. Example 4 Lets look at our normal quadratic function

18. Example 4 continue • What happen when we add a negative to the • So now we have This is what we called reflection across the x-axis

19. Example 5 • Now lets change x in the • So now we have • Since is the same line that is what we called reflection across the y-axis

20. Student Guided practice Work on reflection worksheet

21. Example 6 Lets compare the following equations to the equation

22. NONRIGID TRANSFORMATION What are rigid transformations?Answer: horizontal shifts, vertical shifts, and reflections are called rigid transformations. These transformations change only the position of the graph in the coordinate plane. What are nonrigid transformations? Answer: Are those that cause a distortion- a change in the shape of the original graph. Like vertical and horizontal stretches and compressions.

23. NonRigid transformations types • For y = f(x) and the real number c, • • A vertical stretch is represented by g(x) = cf(x) , • where c > 1 . • • A vertical shrink is represented by g(x) = cf(x) , • where 0 < c < 1 . • • A horizontal shrink is represented by h(x) = f(cx) , • where c > 1 . • • A horizontal stretch is represented by h(x) = f(cx) , where 0 < c < 1 .

24. Homework • Do problems 10-12 ,35-46 odd numbers • From pages 47 and 48.

25. Closure • Today we learned about transformations of functions and rigid and nonrigid transformations.