What Is Slope?
As mentioned, slope measures the steepness of a line. In general, if we consider ramps or staircases, then we can say that some ramps are steeper than others. When moving up a staircase or a ramp we can either be moving backwards or forwards. Similarly, when moving down a ramp we will either move forward or backwards. It should be noted that the forward or backward movement is a horizontal movement.
Mathematically, slope is measured as the ratio of rise to run. Here, rise refers to the vertical distance and run refers to the horizontal distance between two points. Therefore, slope is also a measure of the horizontal and vertical distances between the two points.
Formula for Slope Calculation
The slope of a line or a graph is the ratio of rise to run between any two points on the line.
\(slop, m = \frac{Rise}{Run} = \frac{Change\text{ } in\text{ }y\text{ coordinates}}{Change\text{ } in\text{ }x\text{ coordinates}} = \frac{y_{2}-y_{1}}{x_{2} – x_{1}}\)
What Are the Types of Slopes?
In mathematics, slopes can be of four types:
- Positive Slope
If the graph of a line rises up when moving left to right on a coordinate plane, then it is said that the line has a positive slope.
2. Negative Slope
If the graph of a line moves downwards when moving left to right, then it is said that the line has a negative slope.
3. Zero Slope
If the graph of a line is a horizontal line, then we can say that it has a zero slope. For such a case, the graph does not have any
vertical fall or rise.
Let’s calculate the slope of a horizontal line.
\(slop, m =\frac{Rise}{Run}\)
For a horizontal line, Rise = 0, so
\(m =\frac{0}{Run}=0\)
4. Undefined Slope
If the graph of a line is a vertical line, then we can say that it has an undefined slope. In such a case, the graph does not have
any horizontal run.
Let’s calculate the slope of a vertical line.
\(slop, m = \frac{Rise}{Run}\)
For a vertical line, Run = 0, so
\(m = \frac{Rise}{0} = undefined\)
Rapid Recall
Solved Examples
Example 1: Find the slope of the graph. Verify your answer.
The graph is a horizontal line. So, its slope is zero.
Let’s verify the answer using the slope formula. Here, we can see that the two points that lie on the graph are (-2, 5) and (6, 5), so we will use these points to calculate the slope.
\(Slop,\text{ }m = \frac{y_{2} – y_{1}}{x_{2} – x_{1}}\) [Formula for slope]
\(m = \frac{5 – 5}{6-\left( -2 \right)}\) [Substitute values]
\(m = \frac{0}{8}\) [Simplify]
\(m = 0\) [Simplify further]
Example 2: John calculates the slope of the graph shown as 2. Is John’s calculation correct?
Solution:
The graph is a vertical line. So, its slope is undefined.
Let’s check our answer by calculating the slope using the slope formula.
The two points that lie on the graph are (1, 5) and (1, -3). So,
\(Slop, m = \frac{y_{2} – y_{1}}{x_{2} – x_{1}}\) [Formula for slope]
\(m = \frac{\left( -3 \right) -5}{1 – 1}\) [Substitute values]
\(m = \frac{-8}{0}\) [Simplify]
\(m = undefined\) [Simplify further]
Therefore, the slope of the graph is undefined and hence John’s calculation is not correct.
Example 3: Identify the lines with zero slope and undefined slope.
Solution:
The graph of ‘line a’ and ‘line y = 2’ is a horizontal line. Hence, ‘lines a and y = 2’ have a zero slope.
The graph of ‘line b’ and ‘line x = -5’ is a vertical line. Hence, ‘lines b and x = -5’ have an undefined slope.
When the graph of an equation or a function is a vertical line, that is, it has zero horizontal distance between two points then the slope of such a graph is undefined.
The ratio of rise to run for two or more parallel lines is equivalent. So, parallel lines have the same slope.
The x-axis is a horizontal line, hence, it has a zero slope.
The slope of the y-axis is undefined, hence the slope of any line parallel to it is also undefined.