In the previous article we studied the definition of mathematical function and discussed about essential components of functions in ‘mathematics for the biologist’ series. In this article, we are going one step further and discuss linear function in mathematics and its representation and use in biological sciences.
In our daily lives, we are deeply connected to mathematical knowledge, same way mathematics in science make up the other side of the coin of the biological studies. comprehensive subject knowledge of biology of the living organism needs to be supplemented with basic understanding of the science math.
If you have paid an attention to the meter of the auto-rickshaw or our electricity consumption measuring meter, you will realise that people use the values of readings from digital meters to calculate the amount of cost incurred on distance traveled or the amount of electricity consumed. As discussed in the last article the reading from the meter is an independent variable (it can be any number thus independent) whereas charges incurred after availing of services are a dependent variable (it changes with reading thus dependent).
When you are at the store to buy a burger, you pay required amount of money for a particular number of burgers that you buy. We can easily calculate the amount of money we need to pay simply by knowing the rate of one burger. Unknowingly here we are using the linear function to calculate the value of a dependent variable (bill) using the independent variable (number of burgers) and the slope most commonly known as rate/item.
use of linear functions in real life
Y= mX+C.
In the case of the above example, Y values are the total price of the burgers purchased and X values are the quantity of the burgers in nos.
When we plot the graph of a number of the burger (X-axis) bought against the corresponding bill (Y-axis), we obtain the straight-line plot. And hence the function is called a linear function. For this function, the letter m represents the slope i.e. fixed price for one burger in that store. The graph can be a straight line only if the rate or price or slope is constant, which is Rs. 50 for blue line, Rs. 75 for green line and Rs. 100 for grey line. When we know the values of X and the slope we can figure out the bill. Blue, Grey, and Green lines represent the prices of the burgers from different sellers.
Now, look at the following graph. What’s the difference between the earlier graph and this graph? The line is not passing through the origin. Means though you are buying zero burgers you are paying a certain amount of money.
Let’s understand this way, suppose you like the burger from a particular store and unfortunately that store is located in the park which charges INR 50 for the entry in the park. To buy a Burger you will have to pay entry fees of the park. This way one burger will cost you INR 50 more than its original price.
Here, the cost per Burger is constant i.e. INR 100 and the graph is also a straight line but the line is not passing through the origin. This also is an example of a linear function where some extra but CONSTANT value is added to the bill which is represented as C in the equation Y=mX+C.
Let’s understand this with another example. Nowadays we can hire cabs online by using some mobile apps. We have to pay some basic charge for just for availing the service and then based upon the distance traveled total charges are calculated.
Let’s say we have two different companies A&B, providing online cabs services. Both companies operate with their own pricing policy. Company A is charges INR 35 as basic fare and company B charges INR 60 as a base fare.
The equation for company A can be written as,
Y (total fare) = m(charge per km)X (distance travelled in km) + 35
and same for company B is,
Y (total fare) = m(charge per km)X (distance travelled in km) + 60.
One might think company A seems to be offering cheaper rates by looking at the basic fare. But wait, there’s more to the story. Observe the following graphs. For the initial few kilometers, the charges incurred by company A are lesser than company B. But as the distance travelled increases, the scenario reverses and company B seems to be charging lesser bill than company A for longer distances.
If you are planning to travel longer distances, you just cannot go ahead with hiring a cab by looking at the base fares charged by the company. Using a linear function you are able to calculate the possible amount of the bill for certain kilometers. By looking at the graph it is now clear if you want to travel for longer distance company B should be your choice.
These are the few examples where we routinely use linear functions. In the similar manner, the knowledge of math and science are truly revolutionary in academic and research areas. In the next article, we will see applications of linear functions in biological studies as a part of bio math exercises.