Math Revision 15.4 - Reciprocal Graphs

In these revision notes for Reciprocal Graphs, we cover the following key points:

• What does the term "reciprocal" mean in math?
• How to find the reciprocal of a number or expression?
• What is a reciprocal function?
• What is the simplest reciprocal function that exists?
• What is the general form of a reciprocal function?
• How to plot the graph of a reciprocal function?
• What are the asymptotes of a reciprocal graph? How to find them?
• How to find the equation of the symmetry line of a reciprocal graph?

Reciprocal Graphs Revision Notes

In mathematics, the reciprocal of a given number a is 1/a. From this definition, it is clear that multiplying a number a and its reciprocal 1/a gives always 1.

There are the following things to consider when dealing with reciprocals:

1. The reciprocal of a number a is 1/a.
2. The reciprocal of a variable x is 1/x.
3. The reciprocal of an expression E(x) is 1/E(x).
4. The reciprocal of 1 is 1 because 1/1 = 1.
5. The number 0 has no reciprocal as 1/0 is undefined. However, in the proximity of 0 we can assume the reciprocal of 0 to be + ∞.
6. The reciprocal of a negative number -n is -1/n.
7. The reciprocal of a fraction a/b is b/a.
8. The reciprocal of a mixed number is found by converting the mixed number into an improper fraction first, and then applying the 7th point.
9. The reciprocal of a decimal n is 1/n. However, this is not sufficient, as it is not appropriate to write the denominator of a fraction as a decimal. Therefore, the next step involves multiplying both the numerator and denominator of the reciprocal obtained by a suitable number that allows getting rid of the decimal point. Then, we make any further simplification if necessary.

The basic (simplest) form of a reciprocal function occurs when the product of the two variables contained in it is equal to 1. In this way, we say the two variables x and y are reciprocal to each other.

In symbols, we write

However, given that in functions the variable y is written on the left side and the rest of the terms on the right side, we transform the above formula as

In wider terms, a reciprocal function represents the relationship between two variables in inverse proportion to each other.

A reciprocal function may contain its independent variable not only in the denominator but also in the numerator in certain cases, (when either the denominator or the numerator contains a constant term besides the first-degree monomial). Therefore, despite the basic form of a reciprocal function does not contain variables in the numerator, don't hurry up to exclude a function from the set of reciprocal ones, as perhaps you can remove the variable from the numerator by doing some transformations in the original function.

In more specific terms, a reciprocal function is a function that after doing all the necessary operations can be written in the form.

where a, h and k are numbers and x is the independent variable.

In simpler forms as those we have seen earlier, one or more elements of the above formula may not be present. We think about them as being zero. For example, in the simplest form of a reciprocal function

we have a = 1, h = 0 and k = 0.

The shape of the graph obtained by expressing all values of an indirect proportion in a coordinates system is a curve that gets closer to the axes when the values of the variables represented through those axes increase more and more. In mathematical terms, this curve is called a hyperbola. The shape of a reciprocal function is also a hyperbola.

The features of a hyperbola showing a reciprocal function are as follows:

A hyperbola extends towards infinity on one side of a given direction but it approaches a fixed value on the other side of the same direction.

Lines of the form y = x + c act as lines of symmetry for the hyperbolas showing reciprocal functions. For example, the line y = x acts as a symmetry line for the hyperbola y = 1/x,

Sometimes we have the graph of a reciprocal function given but not the function's formula. We can find it by following the procedure below:

Step 1: Assume the function's formula to be in the form

Step 2: Check whether you are able to determine the equations of asymptotes by sight. This is possible if the units are clearly defined (as in all our examples where the units are shown by the squares). If you can find the asymptotes, then you have found the values of h and k in the above formula.

Step 3: After having identified the position of asymptotes, substitute the coordinates of a known point on the graph to find the coefficient a. If the asymptotes cannot be determined by sight, then solve a system of three equations obtained when you substitute the coordinates of three known points of the graph in the general formula above. It is impossible to solve a system of three equations with the concepts explained so far, but you will learn how to solve them in the upcoming topics of this course when dealing with matrices and determinants.

The equation of the symmetry line will have the form

where c is a constant.

If there is a negative sign preceding the coefficient a in the reciprocal function, the graphs lie in the second and the fourth quadrant. Therefore, the equation of the symmetry line will be

To determine the value of c (and therefore the equation of the symmetry line), we find the intercept of the two asymptotes, as it is a point of the symmetry line graph. It is clear that this intercept point is A(h, k), where h and k are the coefficients of the original reciprocal function that determine the asymptotes. Then, after substituting the values of h and k in the equation of the symmetry line, we find the value of c.