Quadratic Graphs (Cambridge (CIE) O Level Additional Maths): Revision Note

Quadratic Graphs

A quadratic is a function of the form  where  is not zero They are a very common type of function in mathematics, so it is important to know their key features

What does a quadratic graph look like?

• The shape made by a quadratic graph is known as a parabola

• The parabola shape of a quadratic graph can either look like a “∪-shape” or a “∩-shape”

  • A quadratic with a positive coefficient of will be a ∪-shape

  • A quadratic with a negative coefficient of will be a ∩-shape

• A quadratic will always cross the -axis

• A quadratic may cross the -axis twice, once, or not at all

  • The points where the graph crosses the -axis are called the roots

• If the quadratic is a ∪-shape, it has a minimum point (the bottom of the ∪)

• If the quadratic is a ∩-shape, it has a maximum point (the top of the ∩)

• Minimum and maximum points are both examples of turning points

How do I sketch a quadratic graph?

• We could create a table of values for the function and then plot it accurately

  • However we often only require a sketch to be drawn, showing just the key features

• The key features needed to be able to sketch a quadratic graph are

  • the overall shape

    • ∪-shape graphs occur when (positive quadratic)

    • ∩-shape graphs occur when  (negative quadratic)

  • the -intercept(s), these are also known as the roots (there may be none!)

    • roots are found by setting the quadratic function (or ) equal to zero

    • i.e. solve 

    • if there are no (real) solutions (i.e. no roots), the graph does not intersect the -axis

      • the discriminant can be used to determine whether a quadratic function has 0, 1 or 2 roots

  • the -intercept

    • this is found by setting  in the quadratic function

    • so for  the coordinates of the -intercept will be 

  • the minimum or maximum point (turning point)

    • sometimes a rough idea of where this should lie is enough

    • sometimes the specific coordinates of the turning point will be needed

    • when required the coordinates of the turning point can be found by either completing the square or differentiation

      • in cases where the quadratic has just one root, the graph will touch (rather than cross) the -axis and so this will be the turning point

Sketching Graphs by Completing the Square

How does completing the square help me sketch graphs?

• Completing the square can quickly tell us the coordinates of the turning point on a quadratic graph

• This is based on the fact that a squared term (e.g.  ) cannot be negative

• STEP 1
  Complete the square - rewrite  in the form 

• STEP 2
  Deduce the -coordinate of the turning point

• for all values of 

  • Therefore it's minimum value is 0, and this occurs when 

The -coordinate is 

• STEP 3
  Deduce the -coordinate of the turning point

• 

  • Therefore

The -coordinate is 

• STEP 4
  The turning point has coordinates This can be considered when sketching the graph of the quadratic function

• Note that the turning point could be a maximum or minimum point - this will depend on the value of 

  • is the coefficient of the  term

  • If  is positive, the graph is - shaped and will have a minimum point

  • If  is negative, the graph is - shaped and will have a maximum point

    

How do I use the graph of a quadratic function to find its range?

• The range of a quadratic function will be shown on its graph by the values  takes

  • i.e.  the turning point from a quadratic graph will determine its range

• For the quadratic function  whose graph has a minimum point

  • the range of the  will be 

• For the quadratic function  whose graph has a maximum point 

  • the range of  will be 

• If there any restrictions on the domain of then they could affect the range of