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How to Graph Functions Using Vertical Tangent Lines in Calculus

Vertical tangent lines on a graph, called asymptotes, represent values on a graph with an infinite slope. The curve of a function f(x) never touches an asymptote but only approaches it as the function goes to infinity. This primarily occurs when graphing for logarithms, terms under radicals and rational expressions since there are values of "x" where the function doesn't exist. Determining the presence and placement of a vertical asymptote is a matter of finding values, if any, of f(x) where the function is undefined.

Instructions

- 1
  Set up an equation to find the value(s), if any, where the denominator of a rational expression is zero, or where a negative logarithm or root expression is being taken. For example, if f '(x) = 1 / (2 - x), then (2 - x) cannot equal zero.
- 2
  Solve for x. For example, solving for x in the equation (2 - x) = 0 finds: - x = (0 - 2) ---> x = -(0 - 2) = 2. So this function is undefined at x = 2, which is a point with an undefined, vertical tangent line.
- 3
  Draw a vertical dotted line on a Cartesian coordinate grid at the point(s) where x = 0. This line represents a vertical asymptote and the graph will approach, but never touch, the line.
- 4
  Draw a curve approaching the vertical asymptote from the right side. Consult the function to determine whether it is approaching either positive or negative infinity at the asymptote.
- 5
  Approach the asymptote as close as you possibly can but don't quite touch it with the curve. The graph approaches the asymptote for infinity coming arbitrarily close to, but never touching, the line.
- 6
  Jump to the left of the asymptote. Consult the function again to determine whether the graph is approaching positive or negative infinity. The general shape of the graph of the right and left sides may differ once the curve reaches a certain distance from the asymptote but both sides approach the line in the same way, though possibly increasing in opposite directions (positive or negative infinity).

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