Question about a graph.....?

Question

what does the tangent line to a graph with asymptotes look like? for example, what would a tangent line to the graph (-2x)/(x-1) look like?

I mean, the line that is tangent to a point on a graph with asymptotes. could it bisect the two graphs?

Answer 1

One asymptote would be vertical line x = 1. The other would be a horizontal line at y = -2.

Let me draw a graph for you.

Okay, now you may wonder how you figure this out. First look at the formula:
y = (-2x) / (x - 1)

When will this result in an undefined value (e.g. division by zero)? It will happen when the denominator is zero.
x - 1 = 0
x = 1

So that is one asymptote.

Now solve the equation in terms of x:
y = (-2x) / (x - 1)
y(x - 1) = -2x
xy - y = -2x
xy + 2x = y
x(y + 2) = y
x = y / (y + 2)

So when will this have an undefined result? Right, when the denominator is zero.
y + 2 = 0
y = -2

So the aymptotes are x = 1 and y = -2 as shown in the attached link.

To get a tangent to the graph, take the first derivative to get an equation for slope. Now plug in a point that you know to be on the graph. Use the point/slope form and get the equation for the line. You'll see in all cases, the tangent line will fail to cross the other half of the graph... check the picture below to confirm. You'll see that it can never get steeper or flatter than either of the asymptotes, so it will never cross over to the other half.

Answer 2

Do you mean the tangent graph or the tangent line?

The graph would look like a funnel reaching toward -infinity at the asymptote.

If you want a tangent line, you'd need a specific point or x value first to find it.

Answer 3

Here is what they look like:

http://mathforum.org/library/drmath/view/53941.html

Guido