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 a affects the shape of the curve

  • If a is positive the shape is concave up ∪

  • If a is negative the shape is concave down ∩

• The y-intercept is at the point (0, c)

• The zeros or roots are the solutions to 

  • These can be found by

    • Factorising

    • Quadratic formula

    • Using your GDC

  • These are also called the x-intercepts

  • There can be 0, 1 or 2 x-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 x-intercepts then the axis of symmetry goes through the midpoint of them

• The vertex lies on the axis of symmetry

  • It can be found by completing the square

  • The x-coordinate is 

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

  • If a is positive then the vertex is the minimum point

  • If a 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 y-intercept (0, c)

  • You can find the axis of symmetry by 

    • This is given in the formula booklet

• 

  • This is the factorised form

  • It clearly shows the roots (p, 0) & (q, 0)

  • You can find the axis of symmetry by 

• 

  • This is the vertex form

  • It clearly shows the vertex (h, k)

  • The axis of symmetry is therefore 

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

    • Vertical stretch by scale factor ­a

    • Translation by vector 

How do I find an equation of a quadratic?

• If you have the roots x = p and x = q...

  • Write in factorised form 

  • You will need a third point to find the value of a

• If you have the vertex (h, k) then...

  • Write in vertex form 

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

• If you have three random points (x₁, y₁), (x₂, y₂) & (x₃, y₃) then...

  • Write in the general form 

  • Substitute the three points into the equation

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