1. 5.2 Exploring Translations of Quadratic Relations
Recall, last time:
y = - 𝟏 𝟓 x2
y = x2
y = -9x2
Now:

2. Translations of Quadratic Relations
In order to figure out how y = x2 moved around the grid, we need a reference point
The simplest point to keep track of is the vertex
This will come in handy later
Here, the blue graph of y = x2 is translated up 1 unit

3. Translation of the form y = x2 + k
k is a vertical translation
The red curve is y = x2 + 1
The curve on the right has been vertically translated down 3 units
y = x2 - 3

4. Translation of the form y = (x-h)2
h is a horizontal translation
**Note! You move y=x2 in the opposite direction of the sign
i.e. if (x + h)2, translate to the left
If (x – h)2, translate to the right
For the y = x2 curve on the right, it has been horizontally translated right 2 units
y = x2 became y = (x – 2)2

5. Translation of the form y = (x-h)2
This parabola has been horizontally translated left 4 units relative to y = x2
It has equation
y = (x + 4)2

6. Combinations: y = (x-h)2 + k
Combinations occurs when a vertical and horizontal translation occur simultaneously
Value of h
Value of k
Equation
Relationship to y = x2
Vertex
Left/Right
Up/Down
y = x2
N/A
(0, 0) 1 2
y = (x - 1)2 + 2
Right 1
Up 2
(-1, 2) -2 -4
y = (x + 2)2 - 4
Left 2
Down 4
(-2, -4)

7. In Summary…
The graph is y = (x – h)2 + k is just the graph of y = x2 that is translated vertically and horizontally
Translations are also known as shifts
Vertical shifts are up or down
Horizontal shifts are left or right