How Is A Slope Undefined

In the xy-coordinate aeroplane, the slope of a line is a way of measuring its "steepness". A large positive gradient, for example, ways that the line rises from left to right very quickly. But considering this, how tin we take an undefined gradient? The respond is based on how we calculate slope with the gradient formula. In this lesson, we will await at what kinds of lines have an undefined gradient and at why this is the instance.

What kinds of lines have an undefined slope?

Any vertical line, like the i shown beneath, volition have an undefined slope. These lines are always of the form \(x = c\), where \(c\) is some number.

example of a line with undefined slope: x = 4

To understand the give-and-take beneath, yous should exist familiar with finding the slope using the slope formula.

Why is the slope undefined for vertical lines?

Let's use the example of the line \(x = 4\), which is graphed higher up. 2 points on this line would exist \((iv, 1)\) and \((4, two)\). Since you can use any two points to calculate the slope of the line, we can so apply the slope formula using \((x_1,y_1) = (4,one)\) and \((x_2, y_2) = (4,2)\).

\(\begin{align}\text{slope} = chiliad &= \dfrac{y_2 – y_1}{x_2 – x_1} \\&= \dfrac{2 – i}{4 – 4}\\ &= \dfrac{1}{0}\cease{marshal}\)

At present, nosotros can see the upshot. Since the two x-values were the same, the denominator of the slope ends up being 0. Division by zero is always undefined. Every signal on this line has an x-coordinate of four, then this will happen regardless of the points picked.

Generalizing this idea a bit, for a vertical line \(ten = c\), the x-coordinates will ever be the number \(c\). Therefore, the slope formula will always consequence in sectionalisation by cypher and therefore the slope will be undefined.

Summary

We have seen how the slope of a line may be undefined. The other possibilities when calculating the gradient are:

Negative slope – the line falls from left to right.
Positive slope – the line rises from left to correct.
Cypher – the line is horizontal (of the form \(y = c\)). (See lesson: zero slope)