Describe how changes to a function affect its graph.

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