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## Sketching Quadratic Graphs

It is often not necessary to plot quadratics by drawing up a table of x values and corresponding y values. Often a few points will suffice to obtain the main features of the graph and sketch it. The main features of a quadratic graph are:

The vertex. This is the maximum or minimum of the graph. A quadratic has eithe a maximum or a minimum, and there is only one of them. If the equation of the graph isand then the vertex is a maximum and the coordinates of the vertex are

The graph above left has a vertex atThis is a maximum becauseThe graph above right has a vertex at

The equation of the line of symmetry. For a quadratic graphthe equation of the line of symmetry isFor(above left), this gives

The– intercept. This is easily found by substitutingin the equation of the graphForwe obtain

The- intercept(s), if they exist. Not every quadratic has- intercepts. Forintercepts to exist, the equationmust have real solutions. Since the solutions are given bywe must have that:

for a unique solution. The graph touches the– axis but does not cross it.

for two distinct solutions. In this case the two solutions are given byand

The- intercepts for the graphareand

If the quadratic factorises then we may find the intercepts by setting each factor equal to 0. For examplefactorises asand