Reciprocal Graphs

Asymptotes

To sketch this type of graph, you need to take into account its asymptotes. An asymptote is a line that the curve gets very close to, but never touches. The graph of reciprocal Functions $y=\frac{a}{x}$ and $y=\frac{a}{x^2}$ have asymptotes at $x=0$ and $y=0$.

In general, the domain of reciprocal functions will be all Real Numbers apart from the vertical asymptote, and the range will be all Real Numbers apart from the horizontal asymptote.

Types of reciprocal graphs

The Graphs article discusses that the coordinate plane is divided into four quadrants named using roman numbers (I, II, III and IV):

Coordinate plane, Marilú García De Taylor - StudySmarter Originals

The possible types of reciprocal graphs include:

• Reciprocal functions of the type $y=\frac{a}{x}$

a) If a> 0:

For example, if $a=1$, $y=\frac{1}{x}$, the shape of the graph is shown below. Notice that the graph is drawn on quadrants I and III of the coordinate plane.

Reciprocal function, Marilú García De Taylor - StudySmarter Originals

b) If a <0:

For example, if $a=-1$, $y=\frac{-1}{x}$, the shape of the reciprocal function is shown below. In this case, the graph is drawn on quadrants II and IV. This graph is the reflection of the previous one because the negative sign in the function means that all positive values of $x\ne 0$ will now have negative values of y, and all negative values of x will now have positive values of y.

Reciprocal function with negative numerator, Marilú García De Taylor - StudySmarter Originals

• Reciprocal functions of the type $y=\frac{a}{x^2}$

a) If a> 0:

For example, if $a=1$, $y=\frac{1}{x^2}$, the shape of the graph is shown below. Notice that the graph is drawn on quadrants I and II of the coordinate plane. The shape of the graph of $y=\frac{1}{x^2}$ changes in comparison to the previous graph of $y=\frac{1}{x}$, because having $x^2$ in the denominator means that all values of y will be positive for all values of $x\ne 0$.

Reciprocal squared function, Marilú García De Taylor - StudySmarter Originals

b) If a <0:

For example, if $a=-1$, $y=\frac{-1}{x^2}$, the shape of the reciprocal function is shown below. In this case, the graph is drawn on quadrants III and IV. This graph is also the reflection of the previous one due to the negative sign in the numerator of the function.

Reciprocal squared function with negative numerator, Marilú García De Taylor - StudySmarter Originals

Drawing reciprocal graphs

To show you how to draw the graph of a reciprocal function, we will use the example of $y=\frac{1}{x}$. To graph this function you need to follow these steps:

How do you find the equation of a reciprocal graph?

If you are given a reciprocal graph, you can find its equation $y=\frac{a}{x+h}+k$ by following these steps:

1. Find the vertical asymptote. This is the value you need to add or subtract from the variable in the denominator $\left(h\right)$. It will have the opposite sign of the vertical asymptote.

2. Find the horizontal asymptote. This will be the value of $k$, which is added or subtracted from the fraction depending on its sign.

3. Find the value of $a$ by substituting the x and y corresponding to a given point on the curve in the equation.

Finding the reciprocal of a function

We know from Algebra that you can calculate the reciprocal of a Number by swapping the numerator and the denominator. The same applies to functions. To find the reciprocal of a function $f\left(x\right)$ you can find the expression $\frac{1}{f\left(x\right)}$.