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 \(y = ax^2 + bx + c\) is a smooth curve with one turning point. The turning point lies on the line of symmetry.

Graph of y = ax² + bx + c

A graph showing the turning point when a > 0 and 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	-1	0	1	2	3	4	5
y	8	2	-2	-4	-4	-2	2	8	16

x
-3
-2
-1
0
1
2
3
4
5

y
8
2
-2
-4
-4
-2
2
8
16

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).

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Solving quadratic equations - AQAGraphs of quadratic functions