Quadratic Graphs (Cambridge O Level Additional Maths)

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

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