Types of Graphs (Edexcel IGCSE Maths A): Revision Note

Types of Graphs

What types of graphs do I need to know?

• You need to be able to recognise, sketch, and interpret the following types of graph:

  • Linear ()

    • or

  • Quadratic ()

    • 

  • Cubic ()

    • 

  • Reciprocal ()

    • 

    • 

• You must also be able to recognise the three basic trigonometric graphs, covered in a separate section

Where are the asymptotes on reciprocal graphs?

• An asymptote is a line on a graph that a curve becomes closer to but never touches

  • These may be horizontal or vertical

• The reciprocal graph, (where is a constant)

  • does not have a y-intercept

  • and does not have any roots

• This graph has two asymptotes

  • A horizontal asymptote at the x-axis: 

    • This is the limiting value when the value of x gets very large (or very negative)

  • A vertical asymptote at the y-axis: 

    • This is the value that causes the denominator to be zero

• The reciprocal graph, (where and are both constants)

  • is the same shape as

  • but is shifted upwards by units

    • would be shifted down by 3 units

  • This means the horizontal asymptote also shifts up by units

    • The vertical asymptote remains on the y-axis

• The graph of is similar to but has two key differences

  • is steeper than

  • is always positive, even when is negative

What does the graph of a cubic look like?

• A cubic is a function of the form

  • and are constants

  • It is sometimes referred to as a polynomial of degree (order) 3

• In general the graph of a cubic will take one of the four forms

  • All are smooth curves

• The exact form of a particular cubic will depend on:

  • The number (and value) of roots (-axis intercepts)

  • The -axis intercept

  • The sign of the coefficient of the term ()

    • If the graph is a positive cubic ('starts' in the bottom left, 'ends' in the top right)

    • If the graph is a negative cubic ('starts' in the top left, 'ends' in the bottom right)

  • The turning points

• Cubics can have two turning points

  • a maximum point and a minimum point

• However, note that the graphs of and :

  • Do not have a maximum or minimum (turning points)

  • Only cross the -axis once, at