How Do You Find the Parameters of a Model Using Simultaneous Equations or Given Data Points?

Substitute each given data point into the model to create one equation in the unknown parameters. Form enough independent equations to match the number of unknown parameters, then solve the resulting simultaneous equations, usually using a GDC.

This is part of SL 2.6: Modelling skills. A parameter is a constant that determines a model's shape or position, such as $a$, $b$, and $c$ in $y=ax^2+bx+c$.

For example, suppose a quadratic model $y=ax^2+bx+c$ passes through $(0,1)$, $($, and .

Substitute each point:

$$c=1$$ $$a+b+c=6$$ $$4a+2b+c=15$$

Solve this system using the GDC simultaneous-equation solver:

$$a=2,\qquad b=3,\qquad c=1$$

Therefore, the model is

$$y=2x^2+3x+1.$$

The same method applies when a model has a different form. For an exponential model $y=ka^x$, the point with $x=0$ gives $y=k$, so an initial value can identify immediately. A second point can then determine . Before solving, check that the equations are independent: repeated information does not provide an extra constraint. After solving, write the parameter values back into the original equation, retain full calculator precision during calculations, and round only requested final quantities.

Finally, check the model by substituting the original $x$-values and confirming that it produces the corresponding $y$-values. In context, also consider whether the model has a reasonable domain and whether using it outside the data range would involve unreliable extrapolation.

A common misconception is that the coordinates themselves are the parameters. They are not: each coordinate pair supplies values of $x$ and $y$, while the parameters remain unknown until the equations are solved.

Exam technique: On either GDC-active Paper 1 or Paper 2, show the substituted equations before giving the calculator solution. Examiners need evidence of the correct set-up, and a modelling response may also require interpretation or evaluation of the model's suitability.