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## Using reciprocal graphs

Reciprocal Graphs are useful to visually represent relationships that are inversely proportional, which means that they behave in opposite ways – if one decreases, the other one increases, and vice versa. For example, if the Number of workers in a shop increases, the amount of time that the customers spend waiting to be served will be reduced.

### 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 $y=0$.

See the graph below for $y=\frac{1}{x}$

$x=0$ is a vertical asymptote because you cannot divide by zero; therefore, x cannot be zero. $y=0$ is a horizontal asymptote because there are no values of x that make $y=0$, so y cannot be zero either.

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):

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.

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 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 $x\ne 0$.

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.

## 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:

**Identify the vertical and horizontal asymptotes.**

For $y=\frac{1}{x}$, $x=0$ and $y=0$ are asymptotes.

**Identify the type of reciprocal function $y=\frac{a}{x}$ or $y=\frac{a}{{x}^{2}}$, and if a is positive or negative. T**his information will give you an idea of where the graphs will be drawn on the coordinate plane. This step is optional.

In our example $y=\frac{1}{x}$, the reciprocal function is of type y = $\frac{a}{x}$ and a> 0; therefore, the graphs will be drawn on quadrants I and III.

**Plot points strategically reveal the graph's behaviour as it approaches the asymptotes from each side.**

**Negative side:**

$f(-1)=\frac{1}{-1}=-1$

$f(-2)=\frac{1}{-2}=\frac{-1}{2}$

$f(-3)=\frac{1}{-3}=\frac{-1}{3}$

Notice that the further we go to the left, the closer we get to zero.

Now let's try some fractions of negative 1:

$f\left(\frac{-1}{2}\right)=\frac{1}{\frac{-1}{2}}=-2$

$f\left(\frac{-1}{3}\right)=\frac{1}{\frac{-1}{3}}=-3$

**Positive side:**

$f\left(1\right)=\frac{1}{1}=1$

$f\left(2\right)=\frac{1}{2}$

$f\left(3\right)=\frac{1}{3}$

Notice that the further we go to the right, the closer we get to zero.

Now let's try some fractions of positive 1:

$f\left(\frac{1}{2}\right)=\frac{1}{\frac{1}{2}}=2$

$f\left(\frac{1}{3}\right)=\frac{1}{\frac{1}{3}}=3$

x | $-3$ | $-2$ | $-1$ | $\frac{-1}{2}$ | $\frac{-1}{3}$ | $\frac{1}{3}$ | $\frac{1}{2}$ | $1$ | $2$ | $3$ |

y | $\frac{-1}{3}$ | $\frac{-1}{2}$ | $-1$ | $-2$ | $-3$ | $3$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{1}{3}$ |

**Draw the graph using the table of values obtained:**

A reciprocal function $y=\frac{a}{x}$ has been transformed if its equation is written in the Standard Form $y=\frac{a}{x+h}+k$, where a, h and k are real constants, the vertical asymptote of the function is $x=-h$, and the horizontal one is $y=k$.

For the reciprocal function $y=\frac{1}{x+2}+1$, the asymptotes are $x=-2$ and $y=1$.

You might be asked to find the interceptions of the reciprocal function graph with the x and y axes. You can proceed as follows:

**x-intercept:**Substitute y = 0 in the equation and solve for x.

$0=\frac{1}{x+2}+1$

$-1=\frac{1}{x+2}$

$-(x+2)=1$

$-x-2=1$

$x=-2-1$

$x=-3$

The point where the graph of the function crosses the x-axis is (-3, 0)

**y-intercept:**Substitute x = 0 in the equation and solve for y.

$y=\frac{1}{0+2}+1$

$y=\frac{1}{2}+1$

$y=\frac{3}{2}$

The point where the graph of the function crosses the y-axis is $(0,\frac{3}{2})$

## 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:

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.

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

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

Find the equation for the reciprocal graph below:

- Vertical asymptote $x=-3$, therefore$h=3$
- Horizontal asymptote $y=1$, therefore$k=1$
- Substitute the point A (-5, 0) in the reciprocal function $y=\frac{a}{x+h}+k$ to find the value of $a$:

$0=\frac{a}{-5+3}+1$

$0=\frac{a}{-2}+1$

$-1=\frac{a}{-2}$

$a=2$

The equation of the reciprocal function is $y=\frac{2}{x+3}+1$

## 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 $\frac{1}{f\left(x\right)}$.

Find the reciprocal of the function $y=x-5$

The reciprocal of $y=x-5$ is$y=\frac{1}{x-5}$

## Reciprocal Graphs - Key takeaways

- Reciprocal graphs are graphical representations of reciprocal functions, where the numerator is a real constant, and the denominator contains an algebraic expression with a variable x.
- To sketch reciprocal graphs, you must take into account their asymptotes.
- An asymptote is a line that the curve gets very close to, never touching it.
- The domain of reciprocal functions will be all real numbers apart from the vertical asymptote.
- The range of reciprocal functions will be all real numbers apart from the horizontal asymptote.

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##### Frequently Asked Questions about Reciprocal Graphs

What is a reciprocal graph?

Reciprocal graphs are graphical representations of reciprocal functions, where the numerator is a real constant, and the denominator contains an algebraic expression with a variable x.

How do you draw a reciprocal graph?

To graph this function you need to follow these steps:

- Identify the vertical and horizontal asymptotes.
- Identify the type of reciprocal function y = a/x or y = a/x², and if a is positive or negative. This information will give you an idea of where the graphs will be drawn on the coordinate plane. (Optional)
- Plot points strategically to reveal the behaviour of the graph as it approaches the asymptotes from each side.
- Draw the graph using the table of values obtained.

How do you find the equation of a reciprocal graph?

To find the equation of a reciprocal function y = a/(x+h) + k follow these steps:

- Find the vertical asymptote. This is the value that you need to add or subtract from the variable in the denominator (h). h will have the opposite sign of the vertical asymptote.
- Find the horizontal asymptote. This will be the value of k, which is added or subtracted from the fraction depending on its sign.
- Find the value of a by substituting the values of x and y corresponding to a given point on the curve in the equation.

How do you find the reciprocal of a function?

To find the reciprocal of a function f(x) you can find the expression 1/f(x).

Why do we use reciprocal graphs?

Reciprocal graphs are useful to visually represent relationships that are inversely proportional, which means that they behave in opposite ways. If one decreases the other one increases, and vice versa.

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