Interpreting linear functions :
Let us consider the linear expression Ax + B.
Mostly, this linear expression will be equated to some other variable, say "y".
Then, we will have y = Ax + B.
"Y = Ax + B" is called as linear function or linear cost function"
We can interpret this linear function "y = Ax + B" using many real life examples.
In our real life, this linear function is most commonly used in business.
Let us interpret this linear function with business.
In business, this linear function is used to find the total cost of "x" units of a product produced.
For any product, if the cost curve is linear, then the linear cost function of the product will be in the form of
y = Ax + B
Here, "y" stands for total cost
"x" stands for number of units.
"A" stands for cost of one unit of the product
"B" stands for fixed cost.
Linear cost function is called as bi parametric function. Here the two parameters are "A" and "B".
Once the two parameters "A" and "B" are known, the complete function can be known.
In co-ordinate geometry, the same linear cost function is called as slope intercept form equation of a straight line.
The following steps are involved in interpreting linear functions for the given situation.
Step 1 :
First we have to go through the information carefully and understand them completely.
After having gone through the information, we have to conclude whether the information given in the question fits linear function.
If the information fits the linear function, we have to follow step 2
Step 2 :
Target : We have to know what has to be found.
In linear function, mostly the target would be to find either the value of "y" (total cost) or "x" (number of units).
Step 3 :
In step 3, we have to calculate the two constants "A" and "B" from the given information. It has been shown clearly in the example problem given below.
Step 4 :
Once the values of "A" and "B" in y = Ax + B are found, the linear function would be completely known.
Step 5 :
After step 4, based on the target of the question, we have to find either the value of "y" or "x" for the given input.
For example, if the value of "x" (number of units) is given, we can find the value of "y" (total cost).
If the value of "y" (total cost) is given, we can find the value of "x" (number of units).
To have better understanding on "Interpreting linear functions", let us go through the following example.
A manufacturer produces 80 units of a particular product at a cost of $ 220000 and 125 units at a cost of $ 287500. Assuming the cost curve to be linear, find the cost of 95 units.
Step 1 :
When we go through the question, it is very clear that the cost curve is linear.
And the function which best fits the given information will be a linear-cost function.
That is, y = Ax + B
Here "y" --------> Total cost
"x" --------> Number of units
Step 2 :
Target : We have to find the value of "y" for "x = 95"
Step 3 :
From the question, we have
x = 80 and y = 220000
x = 75 and y = 287500
Step 4 :
When we plug the above values of "x" and "y" in y = Ax + B, we get
220000 = 80A + B
287500 = 75A + B
Step 5 :
When we solve the above two linear equations for A and B, we get
A = 1500 and B = 100000
Step 6 :
From A = 1500 and B = 100000, the linear-cost function for the given information is
y = 1500x + 100000
Step 7 :
To estimate the value of "y" for "x = 95", we have to plug "x = 95" in "y = 1500x + 100000"
y = 1500x95 + 100000
y = 142500 + 100000
y = 242500
Hence, the cost of 95 units is $ 242500
After having gone through the stuff given above, we hope that the students would have understood "Interpreting linear functions".
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