Function Families Math Example 2

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Example 2

medium
The family fk(x)=1x−kf_k(x)=\dfrac{1}{x-k} has a vertical asymptote at x=kx=k for each parameter kk. Analyze the graphs for k=−2,0,3k=-2, 0, 3 and describe the pattern.

Solution

  1. 1
    k=0k=0: f0(x)=1/xf_0(x)=1/x. Asymptote at x=0x=0 (the yy-axis). Standard hyperbola.
  2. 2
    k=−2k=-2: f−2(x)=1/(x+2)f_{-2}(x)=1/(x+2). Asymptote at x=−2x=-2. Same hyperbola shifted left 22.
  3. 3
    k=3k=3: f3(x)=1/(x−3)f_3(x)=1/(x-3). Asymptote at x=3x=3. Same hyperbola shifted right 33. Pattern: parameter kk shifts the vertical asymptote. All members have the same horizontal asymptote y=0y=0 and the same shape.

Answer

Asymptote moves with kk: at x=kx=k for each member of the family
A parameter family of rational functions 1/(x−k)1/(x-k) is really one function 1/x1/x shifted horizontally by kk. The parameter controls position; the shape is invariant across the family.

About Function Families

A function family is a group of functions sharing the same general form and behavior, differing only in the values of one or more parameters.

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