Math Lesson 15.4.3 - Asymptotes of Reciprocal Graphs

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Welcome to our Math lesson on Asymptotes of Reciprocal Graphs, this is the third lesson of our suite of math lessons covering the topic of Reciprocal Graphs, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Asymptotes of Reciprocal Graphs

Given that a reciprocal graph may be displaced from the position of the basic graph y = 1/x, it is advised to identify the position where the symmetry takes place first so that the graph is easier to plot because you have the possibility to choose values around the intercept with the symmetry line. For this, we need to have some milestones in the coordinates system on which to be based when plotting the graph. Hence, we introduce the concept of asymptotes - the borderlines of the graph, i.e. the two lines (one horizontal and the other vertical) beyond which the reciprocal graph cannot go. Thus, in the graph of the simplest form of a reciprocal function y = 1/x, the two axes also act as asymptotes, as the graph approaches these two axes but it cannot intercept them or go further. For this function and for similar functions having the general form

the X-axis acts as a horizontal asymptote while the Y-axis as a vertical one.

What about reciprocal functions of the general form

Well, in such cases the graph is displaced in respect to the basic form y = a/x by h units in the horizontal direction and k units in the vertical one. Therefore, the lines

are the vertical and horizontal asymptotes of the reciprocal graph respectively, as shown in the figure.

Example 6

Find the asymptotes of the reciprocal graph given by the equation

Solution 6

First, let's write the equation (function) in the form

Thus, we have

Thus, comparing the last expression with the general form of a reciprocal function

we obtain the following asymptotes given that the horizontal asymptote (in short H.A.) is y = k while and vertical asymptote (in short V.A.) is x = h:

The graph below confirms the above findings.

You have reached the end of Math lesson 15.4.3 Asymptotes of Reciprocal Graphs. There are 5 lessons in this physics tutorial covering Reciprocal Graphs, you can access all the lessons from this tutorial below.

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4. Types of Graphs Practice Questions: Reciprocal Graphs. Test and improve your knowledge of Reciprocal Graphs with example questins and answers
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6. Continuing learning types of graphs - read our next math tutorial: Exponential Graphs

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