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 y = x^2 - 6x + 4.

Writing y = x^2 - 6x + 4 in completed square form gives y = (x - 3)^2 - 5.

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 (x - 3)^2 - 5 is given when x = 3. This also gives the equation of the line of symmetry for the quadratic graph.

The value of y when x = 3 is -5. This value is always the same as the constant term in the completed square form of the equation.

So the graph of y = x^2 - 6x + 4 has a line of symmetry with equation x = 3 and a turning point at (3, -5)

Question

Sketch the graph of y = x^2 - 2x - 3, labelling the points of intersection and the turning point.

The coefficient of x^2 is positive, so the graph will be a positive U-shaped curve with a minimum turning point.

Factorising y = x^2 - 2x - 3 gives y = (x + 1)(x - 3) and so the graph will cross the x-axis at x = -1 and x = 3.

The constant term in the equation y = x^2 - 2x - 3 is -3, so the graph will cross the y-axis at (0, -3).

Writing y = x^2 - 2x - 3 in completed square form gives y = (x - 1)^2 - 4, so the coordinates of the turning point are (1, -4).

A graph of y = x^2 – 2x – 3, with the points of intersection and the turning point labelled.