Effects of Parameters on Functions: Vertical and Horizontal Transformations of Parabolas

1. EFFECTS OF PARAMETERS ON FUNCTIONS

2. Parameter ‘a’ increases from 1 to 2 f(x) = 2x2 f(x) = x2 Parabola stretches vertically

3. f(x) = x2 Parameter ‘a’ decreases from 1 to f(x) = x2 Parabola compresses vertically

4. Parameter ‘a’ changes from 1 to -1 f(x) = -x2 f(x) = x2 Parabola inverts vertically

5. Parameter ‘a’ moves from -1 to -2 f(x) = -2x2 f(x) = -x2 Parabola stretches vertically

6. 2 -2 x 2 1 -1 1 -1 0 -2 x 0 1 2 g(x) f(x) 4 0 0 1 8 2 4 8 If f(x) = x2 and g(x) = ax2, then the ordered pairs for g(x) can be determined by applying the following adjustment to those from f(x). f(x) = x2 g(x) = 2x2

7. Parameter ‘b’ increases from 1 to 2 Function compresses horizontally

8. Parameter ‘b’ decreases from 1 to 0.5 Function stretches horizontally

9. Parameter ‘b’ changes from 1 to -1 Parabola inverts horizontally

10. Parameter ‘b’ increases from 1 to 2 f(x) = |2x| f(x) = |x| Function compresses horizontally

11. Parameter ‘b’ decreases from 1 to ½ f(x) = |½x| f(x) = |x| Function stretches horizontally

12. 4 x 2 -4 -2 0 2 x -2 -1 0 1 2 2 4 0 f(x) 4 2 4 4 0 2 g(x) Impact of parameter ‘b’ -Horizontal Scale change As ‘b’ moves further from zero, the function compresses horizontally As ‘b’ moves closer to zero, the function stretches horizontally If parameter ‘b’ changes its sign, the graph will invert horizontally If f(x) = |x| and g(x) = |bx|, then the ordered pairs for g(x) can be determined by applying the following adjustment to those from f(x). f(x) = |x| g(x) = |2x|

13. Parameter ‘h’ increases from 0 to 4 Function translates 4 units to the right

14. Parameter ‘h’ decreases from 0 to -2 f(x) = |x + 2| f(x) = |x| Function translates 2 units to the left

15. Parameter ‘h’ decreases from 0 to -7 Function translates horizontally 7 units to the left

16. 4 x 2 -4 -2 0 2 x -6 -4 -2 0 2 2 4 0 f(x) 4 2 4 4 0 2 g(x) Impact of parameter ‘h’ -Horizontal Translation As ‘h’ increases from zero, the function translates to the right As ‘h’ decreases from zero, the function translates to the left If f(x) = |x| and g(x) = |x - h|, then the ordered pairs for g(x) can be determined by applying the following adjustment to those from f(x). f(x) = |x| g(x) = |x + 2|

17. Parameter ‘k’ increases from 0 to 2 f(x) = x2 + 2 f(x) = x2 Parabola translates vertically up 2 units

18. Parameter ‘k’ decreases from 0 to -7 Function translates vertically 7 units down

19. Parameter ‘k’ increases from 0 to 3 Function translates 3 units up

20. 4 x 4 -4 -2 0 2 x -4 -2 -0 2 2 4 4 0 f(x) 6 4 4 6 2 2 g(x) Impact of parameter ‘k’ -Vertical Translation As ‘k’ increases from zero, the function translates up As ‘k’ decreases from zero, the function translates down If f(x) = |x| and g(x) = |x| + k, then the ordered pairs for g(x) can be determined by applying the following adjustment to those from f(x). f(x) = |x| g(x) = |x| + 2

21. Parameter ‘a’, ‘h’ and ‘k’ all change f(x) = 2(x – 4)2 - 3 f(x) = x2 Parabola stretches vertically, translates to the right and translates down

22. Parameter ‘a’, ‘b’ and ‘h’ all change Function stretches vertically, inverts horizontally and translates 3 to the right.

23. x 0 -1 1 4 2 9 -6 x 3 f(x) 0 1 2 3 4 6 8 g(x) 10 Impact of parameters ‘a’, ‘b’, ‘h’ and ‘k’