How to Shift a Parabola Any Number of Units

This page will show how to shift a parabola any number of units in any direction. Throughout this page, we'll use the parent function y = x2.

Vertical Movements
Shifting any number of units along the y-axis (vertical movement) is easy: just add the amount you want to shift to the function. For example, if you wanted to shift the parabola 5 units down, you would write y = x2 - 5. Shifting up is done through addition; shifting down is done through subtraction.

Horizontal Movement
Shifting along the x-axis (horizontal movement) is a little more complicated; you can't just simply add bx to the equation, because that shifts the parabola down as well (which is what we don't want). For example, the vertex of y = x2 + 6x is (-3, -9), not (-3, -0). If you want to only shift horizontally a distance d units, use the following formula: y = x2 - 2dx + d2. If d is positive, the parabola shifts right; if d is negative, the graph shifts left.

Diagonal Movement
Shifting parabolas diagonally involves both horizontal and vertical movement. For example, if you wanted to shift a parabola left 3 units and up 2 units, here's what you would do:
 * 1) Shift horizontally 3 units. d = -3. y = x2 - 2dx + d2 = x2 - 2(-3)x + (-3)2 = x2 + 6x + 9.
 * 2) Shift vertically 2 units. y = x2 + 6x + 9 + 2 = x2 + 6x + 11.

Demonstration
Here is a demonstration I made of shifting parabolas: https://www.desmos.com/calculator/lzfsolwhft