Undefined Slope as the name suggests is the slope of any curve or line where the change in vertical direction became exponentially too large compared to horizontal direction. Undefined Slope of any line or curve becomes increasingly steep, and its slope cannot be expressed as a finite numerical value.
In this article, we will discuss about undefined slope in detail along with the equation for undefined slope and how we can identify the undefined slope in graphs. We will also see some solved examples and practice problems on undefined slope equations.
What is Undefined Slope?
Undefined slope refers to a situation in which the slope of a line or curve cannot be determined or expressed as a finite numerical value. Undefined Slope typically occurs when the change in the vertical direction becomes infinitely large compared to the horizontal direction. For example, if we have any vertical line, slope of this line is undefined as with no change in horizontal direction there is infinitely large change in vertical direction.
In mathematics, the slope of a line is typically calculated as the ratio of the change in the vertical direction (the “rise”) to the change in the horizontal direction (the “run”). If the run is zero, which is the case for a vertical line, you cannot calculate a finite slope because division by zero is undefined in mathematics.
Undefined Slope Definition
Slope of a line is defined as the ratio of the change in the vertical direction (y-coordinates) to the change in the horizontal direction (x-coordinates) between two points on the line.
When change is y-coordinate becomes infinitely large compare to change in x-coordinate, thus the slope is undefined for that curve or line.
How to Find Slope?
Slope is calculated by dividing the difference in vertical (y) values by the difference in horizontal (x) values i.e., Δy/Δx. When there’s no change in the horizontal values (x) along the line, the slope becomes undefined. Slope is determined by the difference in vertical (y) values divided by the difference in horizontal (x) values. It becomes undefined when there’s no change in the horizontal values (x) along the line.
Lets take an example of an undefined slope: A line that goes through the points (1, 0) and (1, 1). If we use these values in the slope formula: (1-0)/(1-1) = 1/0, here we get an undefined result.
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Undefined Slope Equation
A line with an undefined slope is parallel to the y-axis going straight up and down. This corresponds to a 90° angle where the tangent is undefined. The equation for an undefined slope is x = a, where ‘a’ represents the x-coordinate of the intercept point on the x-axis.
The slope of a straight line can be described as the “rise” (the vertical change) over the “run” (the horizontal change) when you move along the line. When a line is parallel to the y-axis, it means it goes straight up and down and it’s perpendicular to the x-axis which goes side to side. This perpendicular angle is 90 degrees. In this case, the tangent of 90 degrees is undefined. An undefined slope corresponds to a vertical line, and its equation is x = a, where ‘a’ is a constant that represents the x-coordinate of the intercept point on the x-axis.
Let’s understand this better with an example:
As we can clearly see in this figure that the slope by the points (3, 2)0 and (3, -3) have an undefined slope.
This is how we can graphically represent any question and find that whether the slope is undefined or not.
Undefined Slope Formula
To identify an undefined slope, you can look at the linear equation’s form. If it’s in the form ‘x = a,’ where ‘a’ is a constant then the slope is undefined indicating a vertical line.
The formula for identifying an undefined slope is straightforward: if you have a linear equation in the form ‘x = a,’ where ‘a’ is a constant, the slope is undefined. This implies that the line is vertical and is parallel to the y-axis.
Undefined Slope Examples
Examples of undefined slope include vertical lines such as x = 7 where ‘x’ is a constant. In these cases, the slope is undefined because the line goes straight up and down, making it impossible to quantify its steepness with a single number. The slope is undefined because the line runs straight up and down.
Undefined Slope Graph
Graphically representing an undefined slope involves plotting points that reveal a perfectly vertical line indicating an undefined slope. Undefined slope occurs when the slope of a line is not defined and is represented by vertical lines in the form ‘x = a.’ The undefined slope is parallel to y-axis and perpendicular to x-axis forming an angle of 90 degree with the x-axis. Here, we have graphically represent the undefined slope at x = 5.
Read More about Graphing Linear Equation.
How To Find The Undefined Slope?
Undefined slope doesn’t require calculation because it’s inherent in the form of the equation. Below are the steps to Find The Undefined Slope:
- To find undefined slope, any equation in the form ‘x = a’ where ‘a’ is a constant can be represented as a vertical line with an undefined slope.
- When the slope is undefined, you simply recognise that the line is vertical and is parallel to the y-axis.
- Undefined slope is represented as being a line perpendicular to the x-axis forming an angle of 90 degree with the x-axis.
- Undefined slope has another distinguishing factor that its steepness cannot be quantified by a single number.
Zero Slope vs Undefined Slope
It’s important to distinguish between zero and undefined slopes. Zero slope represents a perfectly horizontal line, while undefined slope signifies a perfectly vertical line. In the case of zero slope, the line is flat and its incline is quantified as 0 while an undefined slope indicates a vertical line with no defined incline.
Below are the differences between zero slope and undefined slope in tabular form for better understanding:
| Symbolically |
m = 0 |
Not applicable (no defined slope value) |
| Geometric Interpretation |
A line with zero slope is horizontal and parallel to the x-axis. |
There is no line with an undefined slope; this situation typically arises in vertical lines. |
| Angle with the x-axis |
Forms a 0-degree angle with the x-axis. |
Does not form an angle with the x-axis. |
| Equation of Line |
y = constant (horizontal line) |
x = constant (vertical line) |
| Graph |
A horizontal line. |
A vertical line. |
| Slope Calculation |
Δy / Δx = 0 |
Not applicable (division by zero error) |
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Solved Examples on Undefined Slope
Example 1: Represent the equation x = 2 and find it’s slope.
Solution:
Representing the mentioned equation x = 2, the line is perfectly vertical running along the x-coordinate 2 making its slope undefined.
Example 2: Draw the equation x = 4 and find it’s slope.
Solution:
Representing the mentioned equation x = 4, the line is perfectly vertical running along the x-coordinate 4 making its slope undefined.
Example 3: Represent the equation x = -4 on cartesian plane and find it’s slope.
Solutoin:
Representing the mentioned equation x = -4, the line is perfectly vertical running along the x-coordinate -4 making its slope undefined.
Example 4: For the given figure, write all the equations shown in the graph and also mention the slope represented by each.
Solution:
The above figure contains the equation x = -4, x= 1 and x=4. Each line is perfectly vertical running along the x-coordinate and has an undefined slope.
Practice Problems on Undefined Slope
Problem 1: Draw the equation x = 1 and find it’s slope.
Problem 2: Draw the equation x = – 1, x = 1 and find it’s slope.
Problem 3: Draw the equation y= 4 and find it’s slope.
Problem 4: Draw the equation x = -5 and find it’s slope.
Problem 5: Draw the equation y = -6 and find the slope of both.
FAQs on Undefined Slope
1. What is the Definition of Undefined Slope?
Undefined slope occurs when the slope of a line is not defined and is represented by vertical lines in the form ‘x = a.’
2. What is the Equation of Undefined Slope?
The equation of an undefined slope is ‘x = a,’ where ‘a’ is a constant representing the x-coordinate of the intercept on the x-axis.
3. How do You Calculate the Undefined Slope?
Undefined slope doesn’t require calculation because it’s inherent in the form of the equation.
4. Is 0 an Undefined Slope?
No, 0 is not an undefined slope. It represents a perfectly horizontal line indicating zero incline.
5. Is 0/0 Undefined or Zero?
0/0 is an indeterminate form in mathematics and does not represent either an undefined or zero slope.
6. How do you solve if the Slope is Undefined?
When the slope is undefined, you simply recognise that the line is vertical and its steepness cannot be quantified by a single number.
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