There are several forms of reciprocal functions. One of them has the form y = , where k is a real number and x ≠ 0.
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 |
The curve consists of two separate pieces, but they should be regarded as one graph.
The graph of y = gets closer to the x-axis 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 y-axis as x gets closer to 0 but it never meets the y-axis 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.
Another form of reciprocal functions is y = , where k is a real number and x ≠ 0.
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
y-axis (i.e. x = 0)
The following video shows how to use transformation to graph reciprocal functions.
This video shows how to get the equation of a reciprocal function when given its graph
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