Quadratic Models (DP IB Applications & Interpretation (AI): SL): Revision Note

Quadratic models

What are the parameters of a quadratic model?

• A quadratic model is of the form $f\left(x\right)=a{x}^2+bx+c$

• The value of $c$ represents the value of the function when $x=0$

  • This is the value of the function when the independent variable is not present

  • This is usually referred to as the initial value

• The value of $a$ has the biggest impact on the rate of change of the function

  • If $a$ has a large absolute value then the rate of change varies rapidly

  • If $a$ has a small absolute value then the rate of change varies slowly

• The maximum (or minimum) of the function occurs when $x=-\frac{b}{2a}$

  • This is given in the formula booklet as the axis of symmetry

What can be modelled as a quadratic model?

• If the graph of the data resembles a $\cup$ or $\cap$ shape

• These can be used if the graph has a single maximum or minimum

• Examples include:

  • H(t) is the vertical height of a football t seconds after being kicked

  • A(x) is the area of rectangle of length x cm that can be made with a 20 cm length of string

What are possible limitations of a quadratic model?

• A quadratic model has either a maximum or a minimum but not both

  • This means one end is unbounded

  • In real-life this might not be the case

    • The function might have both a maximum and a minimum

    • To overcome this you can decide on an appropriate domain so that the outputs are within a range

• Quadratic models are symmetrical

  • This might not be the case in real-life