There are several forms of reciprocal functions. One of them has the form y = , where k is a real number and x ≠ 0.
What are the characteristics of the Graph of a Reciprocal Function?The graph of y = gets closer to the xaxis as the value of x increases, but it never meets the x axis. This is called the horizontal asymptote of the graph.
Each piece of the graph also gets closer to the yaxis as x gets closer to 0 but it never meets the yaxis because there is no value for y when x = 0. This is called the vertical asymptote of the graph.
This type of curve is called a rectangular hyperbola.
Note that this type of curve, the graphs of y = where k is a real number and x ≠ 0, has two lines of symmetry: y = x and y = x.
Example:
Draw the graph of y = for values between 4 and 4, except for x = 0.
Solution:
x 
0.2 
0.5 
1 
2 
3 
4 
5 
y 
25 
10 
5 
2.5 
1.25 
1 
x 
0.2 
0.5 
1 
2 
3 
4 
5 
y 
25 
10 
5 
2.5 

1.25 
1 
Example:
Draw the graph of y = for 4 ≤ x ≤4 and x ≠ 0.
Solution:
x 
4 
3 
2 
1 
0.8 
0.8 
1 
2 
3 
4 
y 
0.19 
0.33 
0.75 
3 
4.69 
4.69 
3 
0.75 
0.33 
0.19 
Notice that graphs of y = , where k is a real number and x ≠ 0, has an axis of symmetry on the yaxis (i.e. x = 0)