What Are Linear Graphs?
The graph of an equation that is in the shape of a straight line is called a linear graph. In other words, a linear graph is a pictorial representation of a linear equation.
Here, graph 1 is a linear graph as it is a straight line unlike what is seen in graph 2.
What Is a Linear Equation?
An equation that can be represented as a linear graph is called a linear equation. Every point on the graph of a linear equation is a solution to the equation.
A linear equation is usually represented as:
y = mx + b
Here, m is the slope of the line and b is the y-intercept.
The equation here is also known as slope-intercept form.
How to Draw a Linear Graph?
We can draw the graph of a linear equation by following these steps:
Step 1: Construct a table of values for the variables using the linear equation
Step 2: Plot and mark the ordered pairs
Step 3: Join the points to obtain a straight line
Solved Examples
Example 1: Show that the graph of y = -x + 1 is a linear graph.
Solution:
Let’s draw the graph of the given equation, y = -x + 1.
Step 1: Construct a table for the variable values
x | y = -x + 1 | y | (x,y) |
0 | y = -(0) + 1 | 1 | (0,1) |
1 | y = -(1) + 1 | 0 | (1,0) |
2 | y = -(2) + 1 | -1 | (2,-1) |
Step 2: Plot the points.
Step 3: Join the points.
The graph obtained is a straight line.
Hence, the graph of y = -x + 1 is a linear graph.
Example 2: Which equation does not belong with the other two?
- y = x – 2
- y = x + 5
- y = x2 + 6
Solution:
All three equations are in terms of the variables ‘x’ and ‘y’. Let’s draw the graph of each equation to find the odd one among the three.
a. y = x – 2
x | y = x – 2 | y | (x,y) |
0 | y = (0) – 2 | -2 | (0,-2) |
1 | y = (1) – 2 | -1 | (1,-1) |
2 | y = (2) – 2 | 0 | (2,0) |
b. y = x + 5
x | y = x + 5 | y | (x,y) |
0 | y = (0) + 5 | 5 | (0,5) |
1 | y = (1) + 5 | 6 | (1,6) |
2 | y = (2) + 5 | 7 | (2,7) |
c.\(y = x^{2} + 6 \)
x | y = x2+6 | y | (x,y) |
0 | y = (0)2+6 | 6 | (0,6) |
1 | y = (1)2+6 | 7 | (1,7) |
2 | y = (2)2+6 | 10 | (2,10) |
Looking at graphs of the given three equations we can say that the graphs of the first two equations, which are equation a and equation b, are straight lines. However, the graph of the equation c is not a straight line.
Hence, the equation \(y = x^{2} + 6 \) does not belong with the other two options.
An ordered pair is used to locate a point on the coordinate plane. It comprises the values of the x-coordinate and the y-coordinate of a point. For example, (0, 1), (-2, 3) and (9, -2) are ordered pairs.
The ‘y-intercept’ of a line is the value of the y-coordinate corresponding to ‘x = 0’.
The slope of a line is the ‘ratio’ of rise to run, where rise is the change in the y-coordinate values, and run is the change in the corresponding x-coordinate values of a line.
The steepness of a road on a hill or the steepness of a staircase are real life examples where the slope of a line is important.
The slope of a horizontal line is ‘0’.
The slope of a vertical line is ‘undefined’.