2.  Determine whether the graph of each equation is symmetric wrt the x-axis, the y-axis, the line y = x the line y = -x or none.  1.  2. Warm up

3. Lesson 3-2 Families of Graphs Objective: Identify transformations of simple graphs and sketch graphs of related functions.

4.  A family of graphs is a group of graphs that displays 1 or more similar characteristics.  Parent graph – the anchor graph from which the other graphs in the family are derived. Family of graphs

5. Identity Functions  f(x) =x y always = whatever x is

6. Constant Function  f(x) = c In this graph the domain is all real numbers but the range is c. c

7. Polynomial Functions f(x) = x 2 The graph is a parabola.

8. Square Root Function  f(x)=

9. Absolute Value Function  f(x) =|x|

10. Greatest Integer Function (Step) y=[[x]]

11.  y=x -1 or 1/x Rational Function

12.  A reflection is a “flip” of the parent graph.  If y = f(x) is the parent graph:  y = -f(x) is a reflection over the x-axis  y =f(-x) is a reflection over the y-axis Reflections

13. Parent Graph y =x 3 y=-f(x) y=f(-x)

14.  y=f(x)+c moves the parent graph up c units  y=f(x) - c moves the parent graph down c units Translations

15. Translations f(x) +c 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 f(x) = x 2 f(x) +2= x 2 + 2 f(x) - 5 = x 2 - 5 Vertical Translations

16. Translations  y=f(x+c) moves the parent graph to the left c units  y=f(x – c) moves the parent graph to the right c units

17. Translations y =f(x+c) 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Horizontal Translations f(x) f(x - 5) f(x + 2) 5 2 In other words, ‘+’ inside the brackets means move to the LEFT

18. Translations  y=c f(x); c>1 expands the parent graph vertically (narrows)  y=c f(x); 0<c<1 compresses the parent graph vertically (widens)

19. Translations y=cf(x) 2 4 6 8 x y = f(x) -2-4 -6 10 20 30 -10 -20 -30 0 The graph of cf(x) gives a stretch of f(x) by scale factor c in the y direction. f(x) 2f(x) 3f(x) 0 Points located on the x axis remain fixed. Stretches in the y direction y co-ordinates doubled y co-ordinates tripled

20. 2 4 6 8 x y = f(x) -2-4 -6 10 20 30 -10 -20 -30 0 The graph of cf(x) gives a stretch of f(x) by scale factor c in the y direction. f(x) ½f(x) 1/3f(x) y co-ordinates halved y co-ordinates scaled by 1/3 Translations y = cf(x);0<c<1

21.  y=f(cx); c>1 compresses the parent graph horizontally (narrows)  y=f(cx); 0<c<1 expands the parent graph horizontally (widens) Translations

22. Translations y=f(cx) ½ the x co-ordinate 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 f(x) f(2x) f(3x) The graph of f(cx) gives a stretch of f(x) by scale factor 1/c in the x direction. 1/3 the x co-ordinate Stretches in x

23. 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 f(x) f(1/2x) f(1/3x) The graph of f(cx) gives a stretch of f(x) by scale factor 1/c in the x direction. All x co-ordinates x 3 All x co-ordinates x 2 Stretches in x Translations y=f(cx)

24. Sources  http://mrstevensonmaths.wordpress.com/2011/02/07/t ransformation-of-graphs-2/; August 9,2013 http://mrstevensonmaths.wordpress.com/2011/02/07/t ransformation-of-graphs-2/