A. Function Transformers

1. A. Function Transformers Pre-Calculus 30

2. PC30.7 • Extend understanding of transformations to include functions (given in equation or graph form) in general, including horizontal and vertical translations, and horizontal and vertical stretches. • PC30.8 • Demonstrate understanding of functions, relations, inverses and their related equations resulting from reflections through the: x-axis, y-axis, line y=x

3. Key Terms • Transformations • Mapping • Translations • Image Point • Reflection • Invariant Point • Stretch • Inverse of a Function • Horizontal Line Test

5. 1. Translations • PC30.7 • Extend understanding of transformations to include functions (given in equation or graph form) in general, including horizontal and vertical translations, and horizontal and vertical stretches.

6. 1. Translations • First off, a transformation is when a functions equation is altered resulting in any combination of location, shape and/or orientation changes of the graph • Every point on the original graph corresponds to a point on the transformed graph • The relationship between the points is called mapping

7. Mapping Notation is a way to show the relation between the original function and the transformed function. original (x,y) translation (x,y+3) • Mapping Notation: (x,y)  (x,y+3)

8. Translation is a type of transformation • A translation can move a graph left, right, up and down. • In a translation the location of the graph changes but not the shape or orientation.

9. Lets look at a quick example to see how a translation works and what it looks like in an equation • Graph: y=x2 , y-2=x2 , y=(x-5)2

11. Now before we graph the following 3 functions let’s predict what we think will happen? • Graph: y=x2 , y+1=x2 , y=(x+3)2

13. So with vertical and horizontal translations we shift the graph of a function vertically and/or horizontally by applying one or both of the changes to the equation • Vertical Shift: y-k=f(x) • Horizontal Shift: y=f(x-h) • Both: y-k=f(x-h)

14. Example 1 Sketch a graph of

15. Example 2

16. Key Ideas p. 12

17. Practice • Ex. 1.1 (p.12) #1-14 #1-13 odds, 17-19

18. 2. Reflect and Stretch • PC30.7 • Extend understanding of transformations to include functions (given in equation or graph form) in general, including horizontal and vertical translations, and horizontal and vertical stretches. • PC30.8 • Demonstrate understanding of functions, relations, inverses and their related equations resulting from reflections through the: x-axis, y-axis, line y=x

19. 2. Reflect and Stretch • A Reflections of a functions graph is the mirror image in a line called the Line of Reflection • Reflections do not change the shape of the graph but does change the orientation of the graph

20. When output of a function is multiplied by -1 the result is y=-f(x) • Vertical Reflection (reflect in x-axis) • (x,y)(x,-y) • Line of reflection=x-axis

21. When input of a function is multiplied by -1 the result is y=f(-x) • Horizontal Reflection (reflect in y-axis) • (x,y)(-x,y) • Line of reflection=y-axis

22. Example 1

23. A Stretchchanges the shape of a graph but not its location • A vertical stretch can make the function shorter or taller bc the stretch multiplies or divides the y-values by a constant while the x is unchanged

24. Shorter (vert compression) • (x,y)(x,y) • Use IaI because the negative is used in reflection • Taller (vert expansion) • (x,y)(x,ay) • Use IaI because the negative is used in reflection

25. A Horizontal Stretch can make the function narrower or wider because the stretch multiplies or divides the x-values by a constant while the y-values are unchanged

26. Narrower (horiz compression) • (x,y)(x, y) • Use IbIbecause the negative is used in reflection • Wider (horiz expansion) • (x,y)(bx, y) • Use IbIbecause the negative is used in reflection

27. If the a or b values are negative there would also be a reflection.

28. Example 2

29. Example 3 a) b)

30. Example 4

31. Key Ideas p. 27

33. Practice • Ex. 1.2 (p.28) #1-12 #1-6, 7-9 odds in each, 10-12, 15, 16

34. 3. Combining Transformations • PC30.7 • Extend understanding of transformations to include functions (given in equation or graph form) in general, including horizontal and vertical translations, and horizontal and vertical stretches. • PC30.8 • Demonstrate understanding of functions, relations, inverses and their related equations resulting from reflections through the: x-axis, y-axis, line y=x

35. 3. Combining Transformations

36. Multiple transformations can be applied to a function using the General Transformation Model y-k=af(b(x-h)) or y=af(b(x-h)) +k • The same order of operations are used as when you are working with numbers (BEDMAS) • So multiplying and dividing (stretches, reflections) are done first then add and subtract (translations)

37. Steps to graph combinations: 1. Horizontal stretch and reflect in the y-axis (if b<0) 2. Vertical stretch and reflect in the x-axis (if a<0) 3. Horizontal and/or vertical Translations (h and k)

38. Lets look at the transformations in mapping notation for y=af(b(x-h)) +k

40. Example 1

41. Example 2

42. Example 3

43. Key Ideas p.38

44. Practice • Ex. 1.3 (p.38) #1-12 odds in each with multiple parts #3-16 odds in each with multiple parts

45. 4. Inverse Functions • PC30.8 • Demonstrate understanding of functions, relations, inverses and their related equations resulting from reflections through the: x-axis, y-axis, line y=x

46. 4. Inverse Functions

47. The Inverse of a Function y=f(x) is denoted y=f -1(x) if the inverse is a function. • The -1 is not an exponent because f represents a function, not a variable. (just like in sin -1(x))

48. The inverse of a function reverses the processes represented by that function. • For example, the process of squaring a number is reversed by taking the square root. Taking the reciprocal of a number is reversed by taking the reciprocal again.

49. For example, for f(x)=2x+1 we are multiplying by 2 and adding 1. • What would the inverse be?

50. To determine the inverse of a function, interchange the x and y coordinates Function  Inverse (x,y)  (y,x) y=f(x)  x=f(y) reflect in the line y=x