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Transformation of the graph of a quadratic function

Question:

How to move the graph of y = 2x2 to get the graph of y = 2x2 + 12x + 17?

Solution:

Write the function y = 2x2 into the vertex form.

y = 2x2 = 2(x - 0)2 + 0

The vertex of the graph of y = 2x2 is (0, 0), axis of symmetry is x = 0 which is y-axis. Because the coefficient of the x square term is positive, so the graph is open upward.

Use the completing square formula to find the vertex of the quadratic function y = 2x2 + 12x + 17

y = 2x2 + 12x + 17 = 2(x2 + 6x) + 17 = 2(x2 + 6x + 33 - 33) + 17 = 2(x2 + 6x + 32) - 18 + 17 = 2(x + 3)2 - 1

Therefore, vertex of the graph of the function y = 2x2 + 12x + 17 is (-3, -1), axis of symmetry is x = -3, and the graph is upward.

So, we need to move the graph of y = 2x2 left 3 units then down 1 unit to get the graph of y = 2x2 + 12x + 17.