Functions and graphs > Characteristics


Many functions have so-called asymptotes - straight lines that the graph approaches with increasing distance to the origin of the coordinate system.
Especially vertical asymptotes are often immediately obvious when looking at the function rule: any values for `x` that would require you to divide by 0 would result in such an asymptote.
A horizontal asymptote is seen if the function value approaches a certain `y` -value when input values get either very large or very small (large negative numbers).

The function f with rule f ( x ) = 1 x is the prototype of a function with asymptotes. The graph of this function is shown here. This graph has:

  • a horizontal asymptote given by the line y = 0 , because the larger or smaller (more negative) values of x become, the closer the function value gets to 0;

  • a vertical asymptote that is given by the line x = 0 , because there is no function value for this value of `x` (you cannot divide by 0 ), and the closer the line gets to the origin (to 0 ) the larger the function values get (in positive or negative direction).

The domain of f is D f = , 0 ( 0 , .
And the range is B f = , 0 ( 0 , .
The sign indicates the combined set of all numbers in the two intervals.

When you choose a proper window to plot the graph of funtion f , then all its characteristics will be visible. These characteristics are:

  • the points of intersection with the axes, which are the zeros (or roots) of the function and the y -intercept;

  • the asymptotes;

  • the vertices, the points where the function has (local) maxima and minima, which you can find using your graphing calculator.

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