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

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

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

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

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

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)

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