Standard equation of a Linear Polynomial is y = mx + c, where m is the slope and ‘c’ is the constant or it is also known as ‘y-intercept’.
If the slope of the line i.e. ‘m’ is positive, then the line will be an Increasing Line.
While if the slope is negative, then the line will be decreasing.
Examples of Increasing line:
If we want to plot y = x/2 + 3. Many students find it difficult to plot the graph. It can be done in just two steps. First of all the line given is an increasing line since the slope ‘m’ is positive.
We just need to find the two points on X-axis and Y-axis respectively.
If x = 0, then y = 3, (3 is the y-intercept which we learnt in the equation y = mx + c).
And if y = 0, then x = –6.
We just need to plot these two points on the X-axis and Y-axis, join these two points.
(x = 0, y =3) will be point on positive Y-axis.
(x = –6, y =0) will be point on negative X-axis. On, joining the graph will look like this.
Similarly, if we want to plot the graph of y = x + 3. Again, this is an increasing line but with a slope of +1. We need just two points to plot the graph.
If x = 0, then y = 3, (again the y-intercept is same).
And if y = 0, then x = –3. Let us plot it on the same graph on which previous line was plotted.
After looking at the two graphs, we should be able to conclude that “if the slope increases, the line will be steeper”.
In the graph, the line y = x + 3 is more steep than y = x/2 + 3 because the slope of the first line is greater than the second line.
Now, if we want to plot y = 3x + 3.
If x = 0, then y = 3, (again the y-intercept is +3).
And if y = 0, then x = –1. Let us plot these two points and join them to get the graph.
After observing the third line, our conclusion should be justified that if the slope increases, the line will be steeper than the lines with the lesser slope.
Slope: Slope or Gradient of a line is the inclination or the angle of the line. A higher slope value indicates a steeper line. Slope can be calculated as the change in the y-coordinate divided by change in the x-coordinate.
So, if we know the co-ordinates of two points, we can find out the slope of the line joining them. For e.g. if P = (–5, 6) and Q = (1, –6).
So, the slope of the line is negative that means the line will be decreasing one.
Also, the relevance of the slope can be understood an example. Let us take the example of the line for which graph was drawn.
Y = x + 3;
| X | Y |
| 0 | 3 |
| 1 | 4 |
| 2 | 5 |
| 3 | 6 |
By, observing the table, we can say that when there is a change of 1 unit in ‘x’, y also increases by 1 unit. That’s why the slope of the line is 1.
We can conclude some properties of increasing line.
a) When x increases, then y will also increase depending on the slope.
b) And if x decreases, then y will also decrease depending on the slope.
Let us see another example of increasing line i.e. y = 3x + 3.
| X | Y |
| 0 | 3 |
| 1 | 6 |
| 2 | 9 |
| 3 | 12 |
Again in this line, for one unit increase in ‘x’, change in y-coordinate is of 3 units, thus the slope of the line is +3.
Conclusion: If an increase in ‘x-co-ordinate’ leads to increase in ‘y-co-ordinate’, the slope will be positive and the line will be INCREASING.
And If an increase in ‘x-co-ordinate’ leads to decrease in ‘y-co-ordinate’, the slope will be negative and the line will be DECREASING.
Examples of Decreasing Line: We have already seen how to identify whether a line is increasing or decreasing. If the slope is negative, line will always be decreasing.
For e.g. 2x + y = 4.
On seeing this line, many students will say that this is an increasing line with a slope of +2. But, actually, this is an example of decreasing line of slope –2. Whenever, we need to find the slope of a line, we will have to compare it with the standard equation of straight line y = mx + c.
If we rewrite the equation given in the question as y = –2x + 4, we get a decreasing line.
Properties of Decreasing Line: If ‘x’ increases, then ‘y’ will decrease depending on the slope and if ‘x’ decreases, then ‘y’ will increase depending on the slope.
| X | Y |
| 0 | 4 |
| 1 | 2 |
| 2 | 0 |
| 3 | –2 |
As there is a change of +1 unit in x-coordinate, y co-ordinate decreases by –2 units, thus a negative line with slope of –2.
Let us plot the graph of this line. Again we just need two points.
If x = 0, then y = 4 (4 is the y-intercept, so graph will cut the y-axis at +4) and when y = 0, then x = 2.
If we decrease the magnitude of the slope i.e. make it 1, and then try to plot it.
Let say the line y = –x + 4, a decreasing line with a slope of –1.
When x =0, then y = 4, and when y = 0, then x = 4.
| X | Y |
| 0 | 4 |
| 1 | 3 |
| 2 | 2 |
| 3 | –1 |
As ‘x’ increases by 1 unit, the y co-ordinate decreases by 1 unit, thus a slope of –1.
Again, we can see as the magnitude of slope of a line decreases, we get a flatter line. The line y = –x + 4 is a decreasing line with slope of (–1), while the line y = –2x + 4 is a decreasing line with a slope of –2. As the magnitude of the slope increases, we get a steeper line and vice-versa.
ROOTS & SOLUTIONS
Roots: Roots are the point where the graph of that equation cuts the x-axis. Or roots are the point where y-coordinate is equal to 0 which should be quite obvious. As roots are the points on X-axis and at any point on the X-axis, y-co-ordinate is equal to 0.
Also, a linear polynomial can have only one root. Root is dependent on the highest power of the polynomial.
So, y = –2x + 4, highest power of polynomial here is 1, that’s why it can have one root only.
Solutions: Solutions are the values which will satisfy the equation. But you will think that on plug-in value of the root also, the equation will be satisfied. Yes, its true. Roots are a part of Solutions. In other words, we can say that for every corresponding value of x, we will get a corresponding value of y and all of those will be solutions.
For e.g. if x =1, then y = 2. This is a solution.
If x = 2, then y = 0. This is also a solution and root.
If x = 3, then y = –2 , this is also a solution.
Like this, we can assume infinite values of x, and for all those infinite values of x, we will get corresponding values for y.
So, any linear equation has maximum of one root and infinite solutions.
If we want to interpret this logic graphically, then all the points on the line are solutions of that equation. And there are infinite points on a line.
In the graph shown above, the line cuts the X-axis at 2 which is the root and the line cuts the y-axis at 4 which is the y-intercept.
In the graph shown above, the point (x , y) as (1, 2) also lies on the graph and is a solution. That was just one point on the graph, like that we can get infinite points on the graph and all of them will be the solutions.
Parallel Lines: If the slopes are same, then the lines will be parallel.
For example, y = 2x + 4 and y = 2x – 4 are increasing lines with slope of (+2) and y-intercepts of +4 and –4 respectively. Let us plot them.
