Strategy for Modelling Functions (DP IB Maths: AI HL)

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Author

Dan

Last updated

3 July 2023

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Modelling with Functions

What is a mathematical model?

A mathematical model simplifies a real-world situation so it can be described using mathematics
- The model can then be used to make predictions
- Be aware that extrapolating (making predictions outside of the range of the data) is not considered to be accurate
Assumptions about the situation are made in order to simplify the mathematics
Models can be refined (improved) if further information is available or if the model is compared to real-world data

How do I set up the model?

The question could:
- give you the equation of the model
- tell you about the relationship
  - It might say the relationship is linear, quadratic, etc
- ask you to suggest a suitable model
  - Use your knowledge of each model
  - E.g. if it is compound interest then an exponential model is the most appropriate
You may have to determine a reasonable domain
- Consider real-life context
  - E.g. if dealing with hours in a day then $0 \leq t < 24$
  - E.g. if dealing with physical quantities (such as length) then $x > 0$
- Consider the possible ranges
  - If the outcome cannot be negative then you want to choose a domain which corresponds to a range with no negative values
  - Sketching the graph is helpful to determine a suitable domain

Which models do I need to know?

Linear
Piecewise (linear & non-linear)
Quadratic
Cubic
Exponential
Natural logarithmic
Logistic
Direct variation
Inverse variation
Sinusoidal

Examiner Tip

You need to be familiar with the format of the different types of equations and the general shape of the graphs they produce, you need to always be thinking "does my answer seem appropriate for the given situation?"
Sketching graphs is key
- Make sure that you use your GDC to plot the relevant function(s)
- Sometimes you may have to play around with the zoom function or the axes to make sure that you are focused on the relevant domain

Worked example

A cliff has a height $h$ metres above the ground. A stone is projected from the edge of the cliff and it travels through the air until it hits the ground and stops. The vertical height, in metres, of the stone above the ground $t$ seconds after being thrown is given by the function:

$h (t) = 95 + 6 t - 5 t^{2}$ .

State the initial value of

h

2-3-6-ib-ai-sl-modelling-functions-a-we-solution

Determine the domain of

h (t)

2-3-6-ib-ai-sl-modelling-functions-b-we-solution

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Finding Parameters

What do I do if some of the parameters are unknown?

For some models you can use your knowledge to find unknown parameters directly from the information given
- For a linear model $f (x) = m x + c$
  - $m$ is the rate of change, or gradient
  - $c$ is the initial value
- For a quadratic model, $f (x) = a x^{2} + b x + c$
  - $x = \frac{- b}{2 a}$ is the axis of symmetry (this is given in the formula booklet) and is the $x$ -value of the minimum/ maximum point
  - $c$ is the initial value
- For a cubic model, $f (x) = a x^{3} + b x^{2} + c x + d$
  - $d$ is the initial value
- For an exponential model, $f (x) = k a^{x} + c$
  - $k + c$ is the initial value
  - $y = c$ is the horizontal asymptote, so $c$ is a boundary of the model
- For a sinusoidal model $f (x) = a \sin (b x) + d$
  - $a$ is the amplitude
  - $y = d$ is the principal axis
  - $\frac{360}{b}$ is the period
A general method is to form equations by substituting in given values
- You can form multiple equations and solve them simultaneously using your GDC
  - You could be expected to solve a system of up to three simultaneous equations of three unknowns
- This method works for all models
The initial value is the value of the function when $x$ (or the independent variable) is 0
- This is often one of the parameters in the equation of the model

Examiner Tip

It can save you time in exams to know the properties of functions listed above that allow you to find parameters directly from the information given

Worked example

The temperature, $T ℃$ , of a cup of coffee is monitored. Initially the temperature is 80°C and 5 minutes later it is 40°C. It is suggested that the temperature follows the model:

$T (t) = k a^{- t} + 16, t \geq 0$

where $t$ is the time, in minutes, after the coffee has been made.

State the value of