We had this question on our A Level assignment: Rewrite the quadratic expression 2x2 - 12x + 23 in the form A(x + B)2 + C. Hence state the coordinates of the turning point of the graph y = 2x2 - 12x + 23. I had no problem with the first part, which is completing the square. I did the second part using dy/dx = 0, but our teacher said that was the wrong way, we should realise the word "hence" means use the answer to the first part. I don't know what she means. Can you explain? |
You say you had no problem with the first part, so we assume you got as far as
y = 2(x - 3)2 + 5
What you were expected to realise is that the "completed square" form of the quadratic shows how the graph of y = 2x2 - 12x + 23 can be built up from the basic graph of y = x2 in three stages. These are:
These three stages correspond to three graphical transformations, which you should recognise as being of the following forms:
So in your question, the graph of basic parabola y = x2 should be firstly translated 3 units parallel to the x-axis, then stretched vertically scale factor 2, and finally translated 5 units parallel to the y-axis. This is shown in the following animation:
The turning point has thus moved from the origin to the point (3,5).