The document discusses various graphing techniques including stretching and shrinking graphs vertically or horizontally, reflecting graphs across axes, translating graphs vertically or horizontally, and identifying even, odd, and neither types of functions. It provides examples of how to determine if a graph is symmetric with respect to axes or the origin. Combinations of transformations are also discussed.

1. 2.7 Graphing Techniques
Chapter 2 Graphs and Functions

2. Concepts and Objectives
 Graphing Techinques
 Stretching and shrinking a graph
 Reflecting a graph
 Even and odd functions
 Translations

3. Stretching and Shrinking
 Compare the graphs of gx and hx with their parent
graphs:
  2g x x
• Narrower
• Same x-intercept
• Vertical stretching

4. Stretching and Shrinking (cont.)
 
1
2
h x x
• Wider
• Same x-intercept
• Vertical shrinking

5. Stretching and Shrinking (cont.)
 These graphs show the distinction between
and :
 y af x
 y f ax
 y f x
 2y f x
 y f x
 2y f x
• Same x-intercept
• Different y-intercept
• Different x-intercept
• Same y-intercept
vertical stretching horizontal shrinking

6. Stretching and Shrinking (cont.)
Generally speaking, if a > 0, then
• If a > 1, then the graph of y = afx is a vertical stretching of
the graph of y = fx.
• If 0 < a < 1, then the graph of y = afx is a vertical shrinking
of the graph of y = fx.
Vertical Stretching or Shrinking
• If 0 < a < 1, then the graph of y = fax is a horizontal
stretching of the graph of y = fx.
• If a > 1, then the graph of y = fax is a horizontal shrinking of
the graph of y = fx.
Horizontal Stretching or Shrinking

7. Reflecting a Graph
 Forming a mirror image of a graph across a line is called
reflecting the graph across the line.
 Compare the reflected graphs and their parent graphs:
y x
y x 
y x
y x 
 over x-axis
over y-axis 

8. Reflecting a Graph (cont.)
• The graph of y = –fx is the same as the graph of y = fx reflected
across the x-axis.
• The graph of y = f–x is the same as the graph of y = fx reflected
across the y-axis.
Reflecting Across An Axis
• The graph of an equation is symmetric with respect to the y-axis
if the replacement of x with –x results in an equivalent equation.
• The graph of an equation is symmetric with respect to the x-axis
if the replacement of y with –y results in an equivalent equation.
Symmetry With Respect to an Axis

9. Symmetry
 Example: Test for symmetry with respect to the x-axis
and the y-axis.
a) b)2
4y x  2 4x y 

10. Symmetry
 Example: Test for symmetry with respect to the x-axis
and the y-axis.
a) b)2
4y x  2 4x y 
 2
4y x  
2
4y x 
2
4y x  
Symmetric about the
x-axis, not symmetric
about the y-axis.

11. Symmetry
 Example: Test for symmetry with respect to the x-axis
and the y-axis.
a) b)2
4y x  2 4x y 
 2
4y x  
2
4y x 
2
4y x  
 2 4x y  
2 4x y  
 2 4x y  
2 4x y 
Symmetric about the
x-axis, not symmetric
about the y-axis.
Not symmetric about
either axis.

12. Symmetry (cont.)
 Another kind of symmetry occurs with a graph can be
rotated 180° about the origin, with the result coinciding
exactly with the original graph.

13. Symmetry (cont.)
 Example: Is the following graph symmetric with respect
to the origin?
• The graph of an equation is symmetric with respect to the origin
if the replacement of both x with –x and y with –y results in an
equivalent equation.
Symmetry With Respect to the Origin
3
y x
 3
y x  
3
y x  
3
y x

14. Even and Odd Functions
Example: Decide whether each function is even, odd, or
neither.
a)
• A function f is called an even function if f–x = fx for all x in the
domain of f. (Its graph is symmetric with respect to the y-axis.)
• A function f is called an odd function if f–x = –fx for all x in the
domain of f. (Its graph is symmetric with respect to the origin.)
Even and Odd Functions
  4 2
8 3f x x x 
     4 2
8 3f x x x    
4 2
8 3x x     : evenf x f x 

15. Even and Odd Functions (cont.)
b) c)  3
6 9f x x x 
     3
6 9f x x x    
3
6 9x x  
 3
6 9x x  
   : oddf x f x  
  2
3 5f x x x 

16. Even and Odd Functions (cont.)
b) c)
 What do you notice about the exponents and the
function’s being called even, odd, or neither?
  3
6 9f x x x 
     3
6 9f x x x    
3
6 9x x  
 3
6 9x x  
   : oddf x f x  
  2
3 5f x x x 
     2
3 5f x x x    
2
3 5x x 
   f x f x 
   f x f x  
neither

17. Translations
 Notice the difference between the graphs of
and and their parent graphs.
 y f x c 
 y f x c 
 f x x  f x x
  3g x x    3h x x 

18. Translations (cont.)
• If a function g is defined by gx = fx + k, where k is
a real number, then the graph of g will be the same
as the graph of f, but translated |k| units up if k is
positive or down if k is negative.
Vertical Translations
• If a function g is defined by gx = fx – h, where h is a
real number, then the graph of g will be the same as
the graph of f, but translated |h| units to the right if
h is positive or left if h is negative.
Horizontal Translations

19. Summary
Function Graph Description
y = fx Parent function
y = afx Vertical stretching (a > 1) or shrinking (0 < a < 1)
y = fax Horizontal stretching (0 < a < 1) or shrinking (a > 1)
y = –fx Reflection about the x-axis
y = f–x Reflection about the y-axis
y = fx + k Vertical translation up (k > 0) or down (k < 0)
y = fx – h Horizontal translation right (h > 0) or left (h < 0)
Note that for the horizontal translation, fx – 4 would translate the
function 4 units to the right because h is positive.

20. Combinations
 Example: Given a function whose graph is y = fx,
describe how the graph of y = –fx + 2 – 5 is different
from the parent graph.

21. Combinations
 Example: Given a function whose graph is y = fx,
describe how the graph of y = –fx + 2 – 5 is different
from the parent graph.
The graph is translated 2 units to the left, 5 units down,
and the entire graph is reflected across the x-axis.

22. Classwork
 2.7 Assignment (College Algebra)
 Page 270: 2-14 (even); page 256: 24-44 (×4);
page 244: 50-56 (even)
 2.7 Classwork
 Quiz 2.6