1. Vertical shifts (up) A familiar example: Vertical shift up 3:
y-values each increase by 3
Vertical shift up 3:
graph is shifted up 3 units

2. More vertical shifts (down)
Original curve:
y-values each decrease by 5
Vertical shift down 5:
graph is shifted down 5 units

3. Horizontal shifts (right)
Original curve:
Horizontal shift right 3:
y-values are shifted to the right 3 units
graph is shifted right 3 units

4. More horizontal shifts (left)
Original curve:
y-values are shifted to the left 4 units
Horizontal shift Left 4:
graph is shifted left 4 units

5. Summary of vertical and horizontal shifts
Given a function g whose graph is known, and a positive number k, the graph of the function f is:
graph of g, shifted up k units
graph of g, shifted down k units
CAUTION: the signs here may be counter-intuituve!
graph of g, shifted right k units
graph of g, shifted left k units

6. Reflections about the x-axis
Original curve:
y-values each replaced by their opposite
undefined if x < 0
Reflected about x-axis:

7. Reflections about the y-axis
Original curve:
mirror image of y-values
undefined if x < 0
Reflected about y-axis:
undefined if x > 0
domain:
domain:

8. Summary of reflections
Given a function g whose graph is known, the graph of the function f is:
graph of g, reflected about the x-axis
domain of f is domain of g
graph of g, reflected about the y-axis
domain of f is "opposite" of domain of g
i.e. if domain of g is [a,b] then domain of f is [-b,-a]

9. Vertical stretching
A cubic polynomial:
each y-value doubles

10. Vertical Shrinking
The same cubic:
each y-value shrinks by 1/3

11. Horizontal Stretching
The same cubic:
Y-values are stretched out from the center

12. Horizontal shrinking
The same cubic

13. Summary of stretching and shrinking
Given a function g whose graph is known, and a positive number c, the graph of the function f is:
graph of g, stretched vertically
graph of g, shrunk vertically
graph of g, shrunk horizontally
graph of g, stretched horizontally