The coordinates of the turning point and the equation of the line of symmetry can be found by writing the quadratic expression in completed square form.

Find the equation of the line of symmetry and the coordinates of the turning point of the graph of .

Writing in completed square form gives .

Squaring positive or negative numbers always gives a positive value. The lowest value given by a squared term is 0, which means that the minimum value of the term is given when . This also gives the equation of the line of symmetry for the quadratic graph.

The value of when is -5. This value is always the same as the constant term in the completed square form of the equation.

So the graph of has a line of symmetry with equation and a turning point at (3, -5)

- Question
Sketch the graph of , labelling the points of intersection and the turning point.

The coefficient of is positive, so the graph will be a positive U-shaped curve with a minimum turning point.

Factorising gives and so the graph will cross the x-axis at and .

The constant term in the equation is -3, so the graph will cross the y-axis at (0, -3).

Writing in completed square form gives , so the coordinates of the turning point are (1, -4).