Question

What is the slope for x = 4?

Answer 1

The slope is not defined for points with the same `x` coordinate.

Explanation:

The definition of slope is for the slope of a line through the points `(x_1, y_1)` and `(x_2, y_2)` with `x_1 != x_2`.

The case `x_1 = x_2`. is not defined.

(You may often hear people say that the slope is infinity. This is the result of a confusion of two or more ideas.)

Answer 2

A vertical line has an infinitely steep slope because it's straight up and down!

Explanation:

Remember that the a typical equation of a line can be expressed as

`y=mx+b`

where `m` is the slope of the line. The slope of a line describes ratio of rise (the difference in vertical distance, or `y`-values), divided by the run (the difference in horizontal distance, or `x`-values). In other words, slope can be defined as:

`m=(x_2-x_1)/(y_2-y_1)`

What this means is that as the top part of the fraction gets big (compared to the denominator), the slope gets steeper and steeper, ever creeping closer to a vertical line. Here you have a slope of just 5:

graph{5x+1 [-11.25, 11.26, -5.63, 5.62]}

And here's a slope of 50:

graph{50x+1 [-11.25, 11.26, -5.63, 5.62]}

So the line becomes vertical as `m` gets large. But the equation `x=1` is simply a vertical line at `x=1`. So the slope is `oo`.