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Turning point form
This is when the function is in the form The graph is the curve which has been translated p units to the right, stretched parallel to the y-axis by scale factor r, then translated q upwards. In this form the maximum or minimum turning point has coordinates (p,q). This is the reason:
If r is positive, then since is never negative, the least possible value of f(x) is given when Hence f(x) has a minimum value of q, which occurs when
If r is negative, then since is never negative, the greatest possible value of f(x) is given when Hence f(x) has a maximum value of q, which occurs when
Remember that is the line of symmetry of the curve.
Completing the square
Writing a quadratic in the form is known as completing the square.
This is demostrated in the following examples.

f

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