Quadratic & Cubic Models (DP IB Maths: AI HL)

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Dan

Last updated

20 June 2023

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Quadratic Models

What are the parameters of a quadratic model?

A quadratic model is of the form $f (x) = a x^{2} + b x + c$
The 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 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 $x = - \frac{b}{2 a}$
- 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
- 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 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 graphs are symmetrical
- This might not be the case in real-life

Examiner Tip

Read and re-read the question carefully, try to get involved in the context of the question!
- Imagine what happens to a stone as you throw it from a cliff, what would the path look like?
- What would it be like to manage a toy factory, would you expect profit to rise or fall as you increase the price of the toy?
Sketch a graph of the function being used as the model, use your GDC to help you
If you are completely stuck try “doing something” with the quadratic function – sketch it, factorise it, solve it

Worked example

A company sells unicorn toys. The profit, $£ P$ , made by selling one unicorn toy can be modelled by the function

$P (x) = \frac{1}{10} (- x^{2} + 20 x - 50)$

where $x$ is the selling price of the toy.

Find the selling price which maximises profit. State the maximum profit.

2-3-2-ib-ai-sl-quadratic-models-we-solution

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Cubic Models

What are the parameters of a cubic model?

A cubic model is of the form $f (x) = a x^{3} + b x^{2} + c x + d$
The d 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 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

What can be modelled as a cubic model?

If the graph of the data has exactly one maximum and one minimum within an interval
If the graph is monotonic with no maximum or minimum
- D(t) is the vertical distance below starting point of a bungee jumper t seconds after jumping
- V(x) is the volume of a cuboid of length x cm that can be made with a 200 cm² of cardboard

What are possible limitations of a cubic model?

Cubic graphs have no global maximum or minimum
- This means the function is unbounded
- In real-life this might not be the case
- The function might have a maximum or minimum
- To overcome this you can decide on an appropriate domain so that the outputs are within a range

Examiner Tip

Read and re-read the question carefully, try to get involved in the context of the question!
Always sketch the graph using your GDC to help
Pay particular attention to the domain of the question
- If the domain is given, make sure that you focus only on that section when you sketch the graph
- If the domain is not given, think about whether or not it needs to be restricted based on the context of the question, e.g. can time be negative?

Worked example

The vertical height of a child above the ground, $h$ metres, as they go down a water slide can be modelled by the function

$h (t) = \frac{4}{7} (35 - 12 t + 6 t^{2} - t^{3})$ ,

where $t$ is the time in seconds after the child enters the slide.

State the vertical height of the slide.

2-3-2-ib-ai-sl-cubic-models-a-we-solution

Given that the child reaches the ground at the bottom of the slide, find the domain of the function.

2-3-2-ib-ai-sl-cubic-models-b-we-solution

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