The graphs of all quadratic functions, \(y=ax^2 + bx+c\), produce U shaped graphs, which are called parabolas, which open either upwards or downwards. These parabolas have an axis of symmetry, which is a vertical line that divides the graph into two symmetrical parts. This axis intersects a very important point in the graph called the vertex of the graph. This point is important because it is either the lowest point (if the graph opens upwards) or highest point (if the graph opens downwards) in the graph.

When we graph a quadratic function it should contain all of the following features:

• Axis of symmetry (use a dashed line).
• Vertex (labeled).
• Minimum or maximum point.
• Intersection points with \(x\) axis.

The value of \(a\) in the functions  \(y=ax^2 + bx+c\), affects the graph in several ways.

• If \(a > 0\), the graph opens upwards and has a minimum at the vertex.
• If \(a < 0\), the graph opens downwards and has a maximum at the vertex.
• If \(\left| a \right| > 1\), the graph is said to be narrow compared to \(f(x) = {x^2}\)
• If \(0 < \left| a \right| < 1\), the graph is said to be wide compared to \(f(x) = {x^2}\).