Undefined slope

In the xy-coordinate plane, the slope of a line is a way of measuring its “steepness”. A large positive slope, for example, means that the line rises from left to right very quickly. But considering this, how can we have an undefined slope? The answer is based on how we calculate slope with the slope formula. In this lesson, we will look at what kinds of lines have an undefined slope and at why this is the case.

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What kinds of lines have an undefined slope?

Any vertical line, like the one shown below, will 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 discussion below, you should be 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 above. Two points on this line would be \((4, 1)\) and \((4, 2)\). Since you can use any two points to calculate the slope of the line, we can then apply the slope formula using \((x_1,y_1) = (4,1)\) and \((x_2, y_2) = (4,2)\).

\(\begin{align}\text{slope} = m &= \dfrac{y_2 – y_1}{x_2 – x_1} \\&= \dfrac{2 – 1}{4 – 4}\\ &= \dfrac{1}{0}\end{align}\)

Now, we can see the issue. Since the two x-values were the same, the denominator of the slope ends up being 0. Division by zero is always undefined. Every point on this line has an x-coordinate of 4, so this will happen regardless of the points picked.

Generalizing this idea a bit, for a vertical line \(x = c\), the x-coordinates will always be the number \(c\). Therefore, the slope formula will always result in division by zero 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 slope are:

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

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