Quadratic Functions (DP IB Analysis & Approaches (AA)): Revision Note

Quadratic Functions & Graphs

What are the key features of quadratic graphs?

• A quadratic graph can be written in the form where 

• The value of affects the shape of the curve

  • If is positive the shape is concave up

  • If is negative the shape is concave down

• The -intercept is at the point

• The zeros or roots are the solutions to 

  • These can be found by

    • Factorising

    • Quadratic formula

    • Using your GDC

  • These are also called the -intercepts

  • A quadratic graph can have 0, 1 or 2 -intercepts

    • This is determined by the value of the discriminant

• There is an axis of symmetry at 

  • This is given in your formula booklet

  • If there are two -intercepts then the axis of symmetry goes through their midpoint

    • E.g. If there are roots at and then the axis of symmetry is at

• The vertex lies on the axis of symmetry

  • It can be found by completing the square

  • The -coordinate of the vertex is 

  • The -coordinate can be found using your GDC or by calculating when 

  • If is positive then the vertex is the minimum point

  • If is negative then the vertex is the maximum point

What are the equations of a quadratic function?

• 

  • This is the general form

  • It clearly shows the -intercept

  • The axis of symmetry is

    • This is given in the formula booklet

• 

  • This is the factorised form

  • It clearly shows the roots and

  • The axis of symmetry is 

• 

  • This is the vertex form

  • It clearly shows the vertex

  • The axis of symmetry is therefore 

  • It clearly shows how the function can be transformed from the graph 

    • Vertical stretch by scale factor ­

    • Translation by vector 

How do I find an equation of a quadratic?

• If you have the roots and ,

  • Write in factorised form

  • You will need a third point on the curve to substitute in to find the value of

• If you have the vertex ,

  • Write in vertex form 

  • You will need a second point on the curve to find the value of

• If you have three random points , and ,

  • Write in the general form 

  • Substitute the three points into the equation, one at a time

  • Form and solve a system of three linear equations to find the values of , and