4K - Applications and modelling with quadratics

Modelling with quadratics

Many real-world problems and scenarios can be modelled with quadratic functions, including the flight path (arc) of a ball, the path of water from a hose or fountain and the construction design of some bridges.

Figure 1 - The parabolic shape of water fountains

How to model with quadratics

When modelling with any function, we need to know the rule that describes the situation - the rule may be provided or you may have to determine the rule from a series of points. Once you have the rule, you can be asked a variety of questions about the model, such as:

• What is the functions value at a certain point?
  • Evaluate the expression f(a).
• When does the function reach a certain value?
  • solve the equation f(x) = b for x.
• When does the function reach the maximum or minimum value?
  • Find the turning point.
• Other more complex questions that use skills and knowledge of quadratics can also be asked, remember:
  • Factorising or using the general quadratic formula will find x-intercepts.
  • The discriminant can be used to determine how many solutions exist.
  • Inequalities can be used asked for, in which case the answer will be an interval.

Note: The function may be defined by a single letter such as f or h.

Optimisation problems with quadratics

Optimisation involves finding the maximum or minimum value of a function which corresponds to an interesting real-world results. For example, the flight of a ball can be modelled by a quadratic function, as such we can find the maximum height of the ball by locating the turning point of the function.

Practical domains

Practical domain exist everywhere in the real world. Common examples include:

• Length, area and volume are all positive quantities
• Time is a positive quantity

When our model includes length, area, volume, time or another quantity with a restriction on its domain, we need to take this into consideration in our answers. Particular care of the practical domain should be taken when graphing quadratics in modelling questions.

4K - VIDEO EXAMPLE 1:

Jeremy throws a ball up into the air. The height (in meters) of the ball, after t seconds, can be modelled by the function:

4K - VIDEO EXAMPLE 2:

Geordie has 1600 meters of fencing available and wants to fence off a rectangular paddock which boarders a straight river; as such no fence is needed along this stretch. What are the dimensions of the largest possible paddock?