All quadratic functions have the same type of curved graphs with a line of symmetry.

The graph of the quadratic function has a minimum turning point when and a maximum turning point when a . The turning point lies on the line of symmetry.

Graph of y = ax2 + bx + c

Finding points of intersection

Roots of a quadratic equation ax2 + bx + c = 0

If the graph of the quadratic function crosses the x-axis, the values of at the crossing points are the roots or solutions of the equation . If the equation has just one solution (a repeated root) then the graph just touches the x-axis without crossing it. If the equation has no solutions then the graph does not cross or touch the x-axis.

Finding roots graphically

When the graph of is drawn, the solutions to the equation are the values of the x-coordinates of the points where the graph crosses the x-axis.

Example

Draw the graph of and use it to find the roots of the equation to 1 decimal place.

Draw and complete a table of values to find coordinates of points on the graph.

 x y -3 -2 -1 0 1 2 3 4 5 8 2 -2 -4 -4 -2 2 8 16

Plot these points and join them with a smooth curve.

The roots of the equation are the x-coordinates where the graph crosses the x-axis, which can be read from the graph: and (1 dp).