1. 0-2: Smart Graphing Objectives: Identify symmetrical graphs
Identify odd/even functions
Sketch the graphs of functions
using translations, reflections &
dilations
© 2002 Roy L. Gover Modified by Mike Efram 2004

2. Definition
Point Symmetry: Two points, P & P’ are symmetric with respect to a point M if M is the midpoint of M P P’

3. ...For a graph to have point symmetry with respect to a point M, M must be the midpoint of every set of points P & P’ on the graph. Examples...

4. Example
Point Symmetry
Consider:

5. Example
Point Symmetry: M M

6. Definition
A graph that is symmetrical with the point (0,0) is symmetric with respect to the origin.

7. Definition
A function f(x) is symmetric with respect to the origin if and only if
f(-x)=-f(x)

8. Example f(x)=x3 is symmetric with the origin because
f(-x)=-f(x). ie f(-2)=-8 & f(2)=8,therefore f(-2)=-f(2)

9. Is f(x)=x2 symmetric with respect to the origin?
Try This
Is f(x)=x2 symmetric with respect to the origin? No

10. Important Idea
Graphs that have line symmetry can be folded along the line of symmetry so that the two halves match exactly.

11. Examples of Line Symmetry

12. Symmetry with respect to x=0 ( y-axis ) exists if and only if:
Definition
Symmetry with respect to x=0 ( y-axis ) exists if and only if:
f(x)=f(-x)
Example: f(x)=x2-3

13. Important Idea
Symmetry is useful in graphing functions. If you graph part of the function and understand the symmetry, the rest of the graph can be sketched.

14. Definition
Even Functions are functions symmetric with the y axis. They have exponents that are all even.

15. Definition
Odd functions are functions symmetric with the origin. They have exponents that are all odd.

16. Try This Are the following functions even, odd or neither: Even Odd

17. Summary
Odd functions:
f(-x) = -f(x)
Symmetry with origin (0, 0)

18. Summary
Even functions:
f(x) = f(-x)
Symmetry with y-axis

19. Definition Reflections: the mirror image of a graph. Example f(x)=x2

20. Try This
Without using a graphing calculator, graph f(x)=-x3 using its parent graph as a starting point.

21. Solution

22. Definition
Translation: the sliding of a graph vertically or horizontally without changing its size or shape.

23. Examples Vertical Translations Horizontal Translations f(x)=x2+3

24. Try This Write the equation of this graph based on its parent graph.
Hint: a vertical & horizontal translation is required.

25. Try This Write the equation of this graph based on its parent graph.
Hint: a reflection & horizontal translation is required.

26. Try This
Without using your calculator, sketch the graph of:

27. Definition
Dilation: changing a graph’s size. Making it either smaller or larger. Examples:

28. Example The graph of f(x) is pictured at the right. Sketch a graph of:
b) f(x+3)-2
c) -f(x-3)-2
d) 2f(x+2)+3
a) f(x+3)