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## Quadratic Function Example 3

The question is how to move the graph of y = (1/2)x^{2} to get the graph of y = (1/2)(x - 3)^{2} - 2?

The function of y = (1/2)x^{2} can be written to y = (1/2)(x - 0)^{2} - 0, so its vertex is (0, 0), axis of symmetry is x = 0 which is y-axis, and the graph is open upward because of the coefficient of x square term, a > 0.

The vertex of y = (1/2)(x - 3)^{2} - 2 is (3, -2), the axis of symmetry is x = 3, and the graph is open upward because of the coefficient of the x square term, a > 0. Because both functions have the same coefficient of the x square term a, so we can move the graph of y = (1/2)x^{2} to get the graph of y = (1/2)(x - 3)^{2} - 2.

Note: If the quadratic function is written in the form y = a(x - h)^{2} + k, then the vertex is (h, k), the axis of symmetry is x = h.

The graph of the quadratic function y = ax^{2} + bx + x, where the coefficient a can not be zero, is the set of points (x, y) that satisfy the quadratic function
y = ax^{2} + bx + c.

To draw the graph of y = (1/2)x^{2}, because the coefficient a is larger than zero, so the graph is open upward. So x = 0 is the minimum value of the quadratic function. We let x = 0 be the middle value of the set of x.

x | ... | -2 | -1 | 0 | 1 | 2 | ... | |
---|---|---|---|---|---|---|---|---|

y | ... | 2 | 1/2 | 0 | 1/2 | 2 | ... |

So we move the graph of y = (1/2)x^{2} right 3 units to get the graph of y = (x - 3)^{2}, then move the graph down 2 units to get the graph of y = (1/2)(x - 3)^{2} - 2.