Linear Functions: Overview, Questions, Preparation

When a function’s variables do not have an exponent, then it is said to be a linear function. There are either one or two variables in a linear function, but if there are more variables, they are either known variables or constant. 

Define Linear Function 

The graph of a linear function can be represented as a straight line. It is a polynomial function that has a degree of 0 or 1. The below-mentioned formula can give its expression:

y = f(x) = px + q

Here, y is the dependent variable, whereas x is the independent variable, and p is the term that remains constant. 

Graph of a Linear Function 

Follow these to draw a graph of a linear function:

Find 2 points that satisfy the linear function’s equation y = px + q and mark these points in an XY-plane.

Now, just join those points with a line. 

Check the below image for a better understanding. 

Slope of a Linear Function 

If a linear function has variables x and y, the change of the value of y as x changes remain the same. The rate at which y changes concerning x is called the slope of a linear function. 

For example, check the below table to examine the change in y with respect to x

By examining this table, you can see that the rate of change is 4. This can be represented by a linear function y = x + 4, and 4 will be its slope. 

How to draw a graph using the linear function and related concepts is taught in Class X, and it carries a weightage of 5 marks in the exam. 

Class XI comprises more difficult problems related to linear functions, and this topic carries a weightage of up to 4 marks in the exam.

1.Find the graph of linear function y = 2x + 1 

Solution.

For x = 1, y = 2 x 1 + 1 = 3 

For x = 2, y = 2 x 2 + 1 = 5

2. Ajay makes Rs. 500 for every watch he sells. It has a fixed cost of Rs. 400 and its variable cost is Rs. 50. How much profit will he make by selling 25 watches?

Solution.

R(x) = 500x 

C(x) = 400 + 50x 

Therefore, P(x) = 500x - (400 + 50x) = 450x - 400

When x = 25, the profit he will make is = 11250 - 400 = 10850.

3. A company makes a product at a fixed cost of Rs. 500 and the variable cost of Rs. 100. Calculate the total cost after manufacturing varying units. 

Solution.

Let x = manufactured units and A = total cost. 

Therefore, A = Fixed Cost + Variable Cost = 500 + 100x 

Suppose that the company manufactures 5 and 10 units respectively. The below table shows the total cost for each manufacturing batch or output. 

Therefore, the company spends Rs. 1000 and Rs. 1500 after manufacturing 5 and 10 units, respectively. 

Q: Who is the founder of linear equations?

Q: Define Nonlinear Function and give some examples. 

Q: Who can make use of linear functions?

Q:  What is the difference between exponential and linear functions?

Q: Is logarithmic functions linear?