# Graphs of Reciprocal Functions

In these lessons, we will learn
• how to graph reciprocal functions by plotting points.
• the characteristics of graphs of reciprocal functions.
• how to use transformations to graph a reciprocal function.
• how to graph a reciprocal function when given its equation
• how to get the equation of a reciprocal function when given its graph.

Related Topics: More Algebra Lessons,

How to graph Reciprocal Functions?

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 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.

How to graph reciprocal functions by plotting points?

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.

How to use Transformation to Graph Reciprocal Functions?
Introduction to reciprocal functions, identifying asymptotes and graphs of reciprocal functions, stretching, shrinking, and translating reciprocal functions, and graphing reciprocal functions. y = 1/x and y = a/(xh) + k. Stretch when a > 1 and shrink when 0 < a < 1. For positive and negative a values. h translates horizontally and k translates vertically.
The following video describes the characteristics of the reciprocal function f(x) = 1/x. It explains the domain, range, vertical and horizontal asymptotes.

Transformation of Reciprocal Functions
The following video shows how to use transformation to graph reciprocal functions.
$$f(x) = \frac{a}{{x - h}} + k$$
h is the horizontal translation
if h is positive, shifts left
if h is negative, shifts right
h also shifts the vertical asymptote
k is the vertical translation
if k is positive, shifts up
if k is negative, shifts down
k also shifts the horizontal asymptote
a is orientation and shape
if a is negative, reflection across x-axis if a > 1, stretched vertically
if 0 < a < 1, compressed vertically
Example: Graph. State the domain, range and asymptotes
a) $$f(x) = \frac{2}{{x - 4}} + 2$$
b) $$f(x) = \frac{{ - 4}}{{x - 2}} - 1$$
Sketch the Reciprocal Function using Transformations
How to graph a reciprocal function with a horizontal translation. y = 1/(x - 3).

Graphing Transformations of Reciprocal Function
Given the function $$y = \frac{{ - 2}}{{3(x - 4)}} + 1$$, determine the parent function, argument, and the values of a, k, d and c to determine transformations. Then, state the transformations verbally in the correct order and graph using three different (but obviously related) methods.
How to graph Reciprocal Functions when given the equation
How to graph functions that are 1/x and variations of that. Find asymptotes, intercepts, and use a couple test values. $$y = \frac{{ - 17}}{{18 - 8x}} - 4$$.

How to find the equation of a Reciprocal function when given its graph
This video shows how to get the equation of a reciprocal function when given its graph
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)

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

[?] Subscribe To This Site