1. Pre-AP Algebra 2 Goal(s):
Analyze the effects on a graph when the parameters of the equation are changed
Vertical shifts
Horizontal shifts
Reflections
Vertical stretches or compressions
Horizontal stretches or compressions

2. Vertical Shifts
Given a graph f(x):
Compare it to f(x) + 3:

3. Vertical Shifts
Given a graph f(x):
Compare it to f(x) – 2:

4. Vertical Shifts
In general, a graph f(x) + k is the graph of f(x) shifted up(+) or down(-) k units.
So f(x) + 5 is shifted up 5 units from f(x) and
f(x) – 8 is shifted down 8 units from f(x).

5. Horizontal Shifts
Given a graph f(x):
Compare it to f(x + 3):

6. Horizontal Shifts
Given a graph f(x):
Compare it to f(x – 1):

7. Horizontal Shifts
In general, a graph f(x - h) is the graph of f(x) shifted left(+) or right(-) h units.
So f(x + 5) is shifted left 5 units from f(x) and
f(x – 8) is shifted right 8 units from f(x).

8. Reflection over the x-axis
Given a graph f(x):
Compare it to -f(x):
In general, a graph
-f(x) is the graph of f(x) reflected over the x-axis.

9. Vertical Stretch or Compression
Given a graph f(x):
Compare it to 2f(x):

10. Vertical Stretch or Compression
Given a graph f(x):
Compare it to 1 3 f(x):

11. Vertical Stretch or Compression
In general, a graph af(x) is the graph of f(x) vertically stretched or compressed. If a<1, there is a compression
If a >1, there is a stretch
So 1 2 f(x) is vertically compressed by a factor of ½ and
4f(x) is vertically stretched by a factor of 4

12. Horizontal Stretch or Compression
Given a graph f(x):
Compare it to f(2x):

13. Horizontal Stretch or Compression
Given a graph f(x):
Compare it to f( 1 3 x):

14. Horizontal Stretch or Compression
In general, a graph f(bx) is the graph of f(x) horizontally stretched or compressed.
If b<1, there is a compression
If b >1, there is a stretch
So f( 1 2 x) is horizontally compressed by a factor of ½ and
f(4x) is horizontally stretched by a factor of 4

15. Function Transformations
Given a function, f(x) the following are general transformations of the graph of the function: -af(b(x-h))+k
h  horizontal shift (left or right)
b  horizontal stretch or compression
a  vertical stretch or compression
(in front of function)  reflection over x-axis
k  vertical shift (up or down)