Understanding Graph Shifts and Reflections in Mathematics

1. Vertical shifts (up) A familiar example: 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