# Sketching graphs

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.

## Sketching a quadratic graph using factors

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

### Example

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 .

### Example

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 .

## Sketching a quadratic using 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 in completed square form.

### Example

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.

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 -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 .

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