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A linear function is defined as a function with one or two variables and no exponents in mathematics. It's a graphing function for the straight line. This entire topic is based on a single equation that shows the increase and decrease of a variable when all other variables remain constant. It is also important to learn the graphical representation of the equations. Here, we will learn about linear functions along with their formulae, graph, characteristics and a few important questions.
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What is a Linear Function?
Linear functions are algebraic equations with straight lines as graphs and unique slope and y-intercept values. A straight line with a slope is used to represent it on the graph. Although Linear functions can be expressed in terms of both calculus and linear algebra, the function notation is the only change. It's also required to know how to write an ordered pair in function notation. A function is defined as f(a) where an is an independent variable on which the function depends. The linear function graph has a straight line with the following equation or formula:
y= f(x)= ax+b
Here, the Y-axis in the graph is represented by ‘y,' which is a dependent variable because its value is reliant on the value of ‘x.' In this case, on the other hand, ‘x' is an independent variable that represents the X-axis. ‘a' is a constant in this equation, and it is also the value of ‘y' for ‘x=0.' In this equation, ‘a' is also known as the y-intercept. The other constant in this equation is b', which has a value of 0 at all times.
What is a Nonlinear Function?
The term "nonlinear function" refers to a function that is not linear. In other words, a function in a graph that does not form a straight line. Exponential functions, parabolic functions, inverse functions, quadratic functions, and so on are examples of such functions. The linear equation y = m x + c is violated by all of these functions. All of these functions have different expressions.
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Linear Function Graph
We must first determine the x and y-intercepts before graphing a linear function. The x-intercept is the point where the graph crosses the x-axis, while the y-intercept is the point where the graph crosses the y-axis.
To find the x-intercept:
- In the equation, write y = 0.
- Find the value of x. The x-coordinate of the x-intercept is the value obtained.
- The point (x, 0) is the x-intercept, with x being the value calculated in step 2.
To find the y-intercept:
- In the equation, make x = 0.
- Find the value of y. The y-coordinate of the y-intercept is the value obtained.
- The point (0, y) is the y-intercept, with y being the value calculated in step 2.
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Linear Function Formula
Standard form and slope-intercept form are two approaches to express a linear function. The standard form of a linear equation is a more formal way of formulating it, although the slope-intercept form makes it easier to graph.
| Form | Equation | Note |
|---|---|---|
| Standard | Ax+ By= c | A and B are not 0. |
| Slope- intercept | y= mx+c | The slope of the line is m, and the y-intercept is b. |
Formulation of a Linear Function through Table
The notation of the ordered pair is generalised in normal form and function form in the table below:
| A normal ordered pair | A function notation ordered pair |
|---|---|
| (a,b) = (1,3) | f(a) = y coordinate, a=1 and y = 3, f(1) = 3 |
We may check the linear function by looking at the values of x and y in the table. The rate of change of y with respect to the variable x remains constant for the linear function. The slope is the rate of change after that.
Check Important Notes for Surface Area of a Hemisphere
Take a look at the table below.
| x | y |
|---|---|
| 0 | 2 |
| 1 | 4 |
| 2 | 6 |
| 3 | 8 |
It can be seen from the table that the rate of change between x and y is 2. The linear function y=x+2 can be used to express this.
Linear Function Characteristics
- Relation: It's a group of pairs that are arranged in a certain way.
- Functions: A function is a relationship between a set of allowed inputs and outputs. It has the property of having exactly one output for each input.
- Variable: In a math expression, this symbol represents a quantity.
- Linear Function: An algebraic equation is defined as a set of terms in which each term is either a constant or the product of a constant and (the first power of) a single variable.
- Direction: Increasing, decreasing, horizontal, or vertical movements are all possible.
- Steepness: The rate at which a function departs from a reference value.
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Points to Remember
- A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
- A function is a relationship in which each input corresponds to only one output.
- A linear function's graph is a straight line.
- We must first determine the x and y-intercepts before graphing a linear function. The x-intercept is the point where the graph crosses the x-axis, while the y-intercept is the point where the graph crosses the y-axis.
- The slope of a line is a number that describes the line's direction as well as its steepness; the sign denotes the direction, while the magnitude indicates the steepness.
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Sample Questions
Ques. Calculate the area bounded by the line x + y = 10 and both the coordinate axes. [CBSE 2012]
Ans.
Area of triangle = ½ × base × corresponding altitude
= ½ × 10cm × 10 cm
= 50 cm sq.
Ques. Represent the following pair of equations graphically and write the coordinates of points where the lines intersect y-axis. [3 marks]
Ans.
x + 3y = 6
⇒ x = 6 - 3y
| x | 6 | 3 | 0 |
|---|---|---|---|
| y | 0 | 1 | 2 |
Therefore the coordinates are (6, 0), (3,1), (0,2)
Now,
2x - 3y = 12
⇒ 2x = 12 - 3y
⇒ x = 12 - 3y / 2
| x | 0 | 6 | 3 |
|---|---|---|---|
| y | -4 | 0 | -2 |
Therefore the coordinates are (0, -4), (6 , 0), (3 , 2)
By plotting the points and joining them, the lines intersect at A (6, 0).
Line x + 3y = 6 intersects y-axis at B(0, 2) and line 2x – 3y = 12 intersects y-axis at
C(0, -4).
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Ques. Draw the graph of 2y = 4x – 6; 2x = y + 3 and determine whether this system of linear equations has a unique solution or not. [ 5 marks]
Ans:
2y = 4x - 6
= > y = 2x - 3
| x | 1 | 2 | 3 |
|---|---|---|---|
| y | -1 | 1 | 3 |
Therefore the coordinates are (-1 , 1), (2, 1), (3 , 3)
Now,
2x = y + 3
=> y = 2x - 3
| x | 1 | 2 | 3 |
|---|---|---|---|
| y | -1 | 1 | 3 |
Therefore the coordinates are (-1 , -1), (2, 1), (3 , 3)
Since both the lines coincide there are infinitely many solutions.
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Ques. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden. [3 marks]
Ans.
Given, half the perimeter of a rectangular garden = 36 m
So, 2 (l + b)/2 = 36
(l + b) = 36 ……….(1)
Given, the length is 4 m more than its width.
Let width = x
And length = x + 4
Substituting this in eq(1), we get;
x + x + 4 = 36
2x + 4 = 36
2x = 32
x = 16
Therefore, the width is 16 m and the length is 16 + 4 = 20 m.
Ques. Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be”. Isn’t this interesting? Represent this situation algebraically and graphically. [5 marks]
Ans.
Let present age of Aftab = x years and
The present age of Aftab’s daughter = y years
1st Condition:
Seven years ago
x - 7 = 7 (y - 7)
=> x - 7 = 7y - 49
=> x - 7y = -42
Table:
| x | 0 | -42 | -35 |
|---|---|---|---|
| y | 6 | 0 | 1 |
2 nd Condition:
Three years later,
x + 3 = 3 (y + 3)
x + 3 = 3y + 9
x - 3y = 6
Table:
| x | 6 | 0 | 9 |
|---|---|---|---|
| y | 0 | -2 | 1 |
Thus, the algebraic equations are:
x - 7y + 42 - 0 AND x - 3y - 6 = 0
Graphical Representation:
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Ques. The coach of a cricket team buys 3 bats and 6 balls for Rs.3900. Later, she buys another bat and 3 more balls of the same kind for Rs.1300. Represent this situation algebraically and geometrically. [5 marks]
Ans.
Let the cost of 1 bat be Rs x and the cost of 1 ball be Rs y
Then, according to the question, we have:
3x + 6y = 3900 ….. (i)
x + 3y = 1300……(ii)
For geometrical representation,
From equation (i), we have : y = 3900 - 3x / 6
When x = 100, then y = 3900 - 300/6 = 600
When x = 300, then y = 3900 - 900/6 = 500
When x = 700, then y = 3900 - 2100/6 = 300
Thus, we have the following table of points:
| x | 100 | 300 | 700 |
|---|---|---|---|
| y | 600 | 500 | 300 |
From equation (ii), we have : y = 1300 - x / 3
When x = 100, then y = 1300 - 100 / 3 = 400
When x = 400, then y = 1300 - 400 / 3 = 300
When x = 700, then y = 1300 - 700 / 3 = 200
Thus, we have the following table of points:
| x | 100 | 400 | 700 |
|---|---|---|---|
| y | 400 | 300 | 200 |
Plotting the points of each table, we obtain the required graph of two intersecting lines.
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Ques. A family of linear functions is given by[5 marks]
f(x) = mx + (3 - 2 m)
where x is the independent variable and m is a constant.
- Graph f for m = 0, 1, 2, -3 and -5
- What do all the graphs in part a) have in common?
- Justify your answer to part b) analytically.
- Write the equation of the family of functions whose graphs pass by the same point (- 2, - 4).
Ans.
- All the graphs pass by the same point (2 , 3)
- To prove that all lines described by the equation f(x) = mx + (3 - 2 m) pass by the point (2 , 3), show that f(2) = 3
f(2) = 2 m + (3 - 2m) = 3
- The point-slope form of the equation of a line is used to find equation of the family of lines that pass by the point (-2,-4) is found as follows
y - (-4) = m (x - (-2))
y = mx + (2m - 4)
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Ques. The cost of producing x tools by a company is given by: [6 marks]
C(x) = 1200 x + 5500 (in Rupees)
- What is the cost of 100 tools?
- What is the cost of 101 tools?
- Find the difference between the cost of 101 and 100 tools.
- Find the slope of the graph of C?
- Interpret the slope.
Ans.
- C(100) = 1200*100 + 5500
= Rs. 1,25,500
- C(101) = 1200*101 + 5500
= Rs. 1,26,700
- C(101)-C(100) = Rs 1200
- Slope m is given by
m = Rs 1200 / (1 tool)
= Rs 1200
- Slope is the increase in the total cost C when the number of tools produced increases by 1 unit.
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