The following has been covered so far in this chapter:
- how to identify linear graphs from tables of values
- the slope and axis intercepts of a linear graph
- finding the equation of a linear function, and
- sketching linear functions from equations and other given information
These techniques can now be used to solve a range of real life applications. It's all the same mathematics, but this time it will be applied to a given context. Applying linear functions to real life applications is known as linear modelling.
The domain of a linear model
When modelling any real-life scenario with a linear function, a range of values must be considered that makes sense for that situation. In mathematics, this is called the domain.
Worked example
Example 1
A helicopter flies for $4$4 hours at a constant speed of $315$315 km/h before reaching its destination. If $D$D represents the distance in kilometres, and $t$t represents the time elapsed in hours, what is the domain?
Think: This model will stop being valid when the helicopter stops flying, so the domain will only include $t$t-values for when the helicopter is flying, where $t$t is the time in hours.
Do: The domain for this linear model would be written as $0\le t\le4$0≤t≤4. To interpret this domain, we read it as, the values of $t$t start at $0$0 hours and end at $4$4 hours.
Practice questions
Question 1
The amount of medication $M$M (in milligrams) in a patient’s body gradually decreases over time $t$t (in hours) according to the equation $M=1050-15t$M=1050−15t.
After $61$61 hours, how many milligrams of medication are left in the body?
How many hours will it take for the medication to be completely removed from the body?
Question 2
Valentina left for a road trip at midday. The following graph shows the total distance travelled (in kilometres) $t$t hours after midday.
Find the slope of the straight line.
What does the slope of the line represent?
the total distance travelled
Athe car's acceleration
Bthe car’s speed
Cthe slope of the road
D