Graphs of quadratic functions

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

The graph of the quadratic function y = ax^2 + bx + c has a minimum turning point when a \textgreater 0 and a maximum turning point when a a \textless 0 . The turning point lies on the line of symmetry.

Graph of y = ax2 + bx + c

The graph of the quadratic function y = ax^2 + bx + c has a minimum turning point when a > 0 and a maximum turning point when a < 0. The turning point lies on the line of symmetry.

Finding points of intersection

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

If the graph of the quadratic function y = ax^2 + bx + c crosses the x-axis, the values of x at the crossing points are the roots or solutions of the equation ax^2 + bx + c = 0 . If the equation ax^2 + bx + c = 0 has just one solution (a repeated root) then the graph just touches the x-axis without crossing it. If the equation ax^2 + bx + c = 0 has no solutions then the graph does not cross or touch the x-axis.

Finding roots graphically

When the graph of y = ax^2 + bx + c 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 y = x^2 -x – 4 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-3-2-1012345
y82-2-4-4-22816

Plot these points and join them with a smooth curve.

The roots of the equation y = x^2 -x – 4 are the x-coordinates where the graph crosses the x-axis, which can be read from the graph: x = -1.6 and x=2.6 (1 dp)

The roots of the equation y = x^2 -x – 4 are the x-coordinates where the graph crosses the x-axis, which can be read from the graph: x = -1.6 and x=2.6 (1 dp).