# Finding the turning point and the line of symmetry - Higher

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.

### Example

Find the equation of the line of symmetry and the coordinates of the turning point of the graph of The coefficient of is positive, so the graph will be a positive U-shaped curve.

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 turning point of the graph is given when  is also the equation of the line of symmetry

When , so the turning point has coordinates (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.

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

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

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