Plotting a graph takes time. Often mathematicians just want to know the key features.

These are: shape, location and some key points (such as where the graph crosses the axes or turning points).

Aim to recognise the equations and graphs of quadratics, cubics, reciprocals, exponentials and circles.

If a quadratic equation can be factorised, the factors can be used to find where the graph crosses the -axis.

Sketch

The quadratic factorises to give so the solutions of the equation are and .

The graph of crosses the -axis at and .

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

To find where the graph crosses the -axis, work out when = 0. So:

so the graph crosses the -axis at .

Sketch

The quadratic factorises to give so the only solution of the equation = 0 is .

The graph of touches the -axis at .

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

To find where the graph crosses the -axis, work out when :

so the graph crosses the -axis at .

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

Sketch .

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

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 minimum turning point at (3, -5).

When , . So the graph crosses the -axis at (0, 4).

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

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

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).

The turning point could also be found by using symmetry as it will have an value halfway between and , so when .