The Graph-Translation Theorem

1. Section 6-3
The Graph-Translation Theorem

2. In-Class Activity
p. 356. Use a graphing calculator

3. In-Class Activity
p. 356. Use a graphing calculator
1.a.

4. In-Class Activity
p. 356. Use a graphing calculator
1.a. 1.b. Moved 8 units right

5. In-Class Activity
p. 356. Use a graphing calculator
1.a. 1.b. Moved 8 units right
2.a.

6. In-Class Activity
p. 356. Use a graphing calculator
1.a. 1.b. Moved 8 units right
2.a. 1.b. Moved 4 units left

7. In-Class Activity
p. 356. Use a graphing calculator
1.a. 1.b. Moved 8 units right
2.a. 1.b. Moved 4 units left
3.a. The translation that maps y1 = x2 onto y2 = (x - h)2 is a
horizontal translation of h units.

8. In-Class Activity
p. 356. Use a graphing calculator

9. In-Class Activity
p. 356. Use a graphing calculator
4.a.

10. In-Class Activity
p. 356. Use a graphing calculator
4.a. 4.b. Moved 3 units up

11. In-Class Activity
p. 356. Use a graphing calculator
4.a. 4.b. Moved 3 units up
5.a.

12. In-Class Activity
p. 356. Use a graphing calculator
4.a. 4.b. Moved 3 units up
5.a. 5.b. Moved 6 units down

13. In-Class Activity
p. 356. Use a graphing calculator
4.a. 4.b. Moved 3 units up
5.a. 5.b. Moved 6 units down
6.a. The translation that maps y1 = x2 onto y2 = x2 + k is a
vertical translation of k units.

14. Compare
y = x2 y = (x - 2)2 + 4 y = (x + 6)2 + 3
Vertex:

15. Compare
y = x2 y = (x - 2)2 + 4 y = (x + 6)2 + 3
Vertex: (0, 0)

16. Compare
y = x2 y = (x - 2)2 + 4 y = (x + 6)2 + 3
Vertex: (0, 0) (2, 4)

17. Compare
y = x2 y = (x - 2)2 + 4 y = (x + 6)2 + 3
Vertex: (0, 0) (2, 4) (-6, 3)

18. Graph-Translation Theorem

19. Graph-Translation Theorem
In an equation using x and y, the following are the same:

20. Graph-Translation Theorem
In an equation using x and y, the following are the same:
1. Replace x with x - h and y with y - k

21. Graph-Translation Theorem
In an equation using x and y, the following are the same:
1. Replace x with x - h and y with y - k
2. Apply Th,k to the original base equation

22. Graph-Translation Theorem
In an equation using x and y, the following are the same:
1. Replace x with x - h and y with y - k
2. Apply Th,k to the original base equation
***Th,k is the translation of h horizontal units and k
vertical units

23. Example 1
a. Find the image of y = 5x2 under T− 2
3 ,6

24. Example 1
a. Find the image of y = 5x2 under T− 2
3 ,6
y = 5(x

25. Example 1
a. Find the image of y = 5x2 under T− 2
3 ,6
y = 5(x + 2/3)2

26. Example 1
a. Find the image of y = 5x2 under T− 2
3 ,6
y = 5(x + 2/3)2 + 6

27. Example 1
a. Find the image of y = 5x2 under T− 2
3 ,6
y = 5(x + 2/3)2 + 6
b. Graph both equations in your graphing calculator.
What do you notice?

28. Example 1
a. Find the image of y = 5x2 under T− 2
3 ,6
y = 5(x + 2/3)2 + 6
b. Graph both equations in your graphing calculator.
What do you notice?

29. Example 1
a. Find the image of y = 5x2 under T− 2
3 ,6
y = 5(x + 2/3)2 + 6
b. Graph both equations in your graphing calculator.
What do you notice?
The graph moved 2/3 units left
and 6 units up

30. Example 1
a. Find the image of y = 5x2 under T− 2
3 ,6
y = 5(x + 2/3)2 + 6
b. Graph both equations in your graphing calculator.
What do you notice?
The graph moved 2/3 units left
and 6 units up
Vertex:

31. Example 1
a. Find the image of y = 5x2 under T− 2
3 ,6
y = 5(x + 2/3)2 + 6
b. Graph both equations in your graphing calculator.
What do you notice?
The graph moved 2/3 units left
and 6 units up
Vertex: (-2/3, 6)

32. Vertex Form of a Parabola

33. Vertex Form of a Parabola
y = a(x - h)2 + k or y - k = a(x - h)2 gives a horizontal shift of
h units and a vertical shift of k ( Th,k )

34. Vertex Form of a Parabola
y = a(x - h)2 + k or y - k = a(x - h)2 gives a horizontal shift of
h units and a vertical shift of k ( Th,k )
The vertex will be (h, k)

35. Example 2
Graph y - 2 = -2(x + 3)2 by hand.

36. Example 2
Graph y - 2 = -2(x + 3)2 by hand.
x y
-6 -16
-5 -6
-4 0
-3 2
-2 0
-1 -6
0 -16

37. Example 2
Graph y - 2 = -2(x + 3)2 by hand.
x y
-6 -16
-5 -6
-4 0
-3 2
-2 0
-1 -6
0 -16

38. Example 3
Graph y + 3 = 1/2x2 by hand.

39. Example 3
Graph y + 3 = 1/2x2 by hand.
x y
-3 3/2
-2 -1
-1 -5/2
0 -3
1 -5/2
2 -1
3 3/2

40. Example 3
Graph y + 3 = 1/2x2 by hand.
x y
-3 3/2
-2 -1
-1 -5/2
0 -3
1 -5/2
2 -1
3 3/2

41. Axis of Symmetry:

42. Axis of Symmetry: A line through the vertex that shows
symmetry of a parabola; Equation is x = h.

43. Axis of Symmetry: A line through the vertex that shows
symmetry of a parabola; Equation is x = h.
Maximum/Minimum:

44. Axis of Symmetry: A line through the vertex that shows
symmetry of a parabola; Equation is x = h.
Maximum/Minimum: Occurs at the vertex; Minimun
when it opens up (a > 0) and maximum when it opens
down (a < 0)

45. Graph a Parabola by Hand

46. Graph a Parabola by Hand
1. Find the vertex

47. Graph a Parabola by Hand
1. Find the vertex
2. Find the symmetrical values on either side of the vertex

48. Graph a Parabola by Hand
1. Find the vertex
2. Find the symmetrical values on either side of the vertex
3. Fill in the graph

49. Homework

50. Homework
p. 361 #1 - 25
“If the wind will not serve, take to the oars.”
- Latin Proverb