# 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 .

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 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).