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Here you will learn about reciprocal graphs (reciprocal functions), including how to recognize them and sketch them on the coordinate plane.

Students first learn about what a reciprocal is when they work with real numbers in the 6 th and 7 th grades. However, the work with reciprocal graphs and functions occurs in high school when students study Algebra II and Precalculus.

A **reciprocal graph** or reciprocal function is mathematically defined to be f(x)=\cfrac{1}{f(x)} which is the multiplicative inverse of a function.

Consider the linear function, f(x)=x, the reciprocal of that function is f(x)=\cfrac{1}{x},

therefore, a reciprocal graph of f(x)=x is the graph of f(x)=\cfrac{1}{x} .

f(x)=\cfrac{1}{x} when sketched, it is a smooth curve called a hyperbola.

The general form of reciprocal functions is written like a fraction with a numerator and denominator:

f(x)=\cfrac{a}{x} or y=\cfrac{a}{x} where x and y are variables representing (x, y) points on the curve and a is the constant.

You may also see the equation rearranged to be xy=a .

Letβs take a look at the graph of a reciprocal function.

y=\cfrac{1}{x}Notice that the smooth curves approach the x- axis and y- axis but never touch them.

This means that the x- axis (y=0) and the y- axis (x=0) are the asymptotes to the reciprocal curve. Asymptotes are lines that the curve approaches but does not touch.

Note: The reciprocal function is NOT the inverse function!

For example, take a look at the linear function y=x and itβs reciprocal function y=\cfrac{1}{x}. Here you can see that the reciprocal function is not the inverse of y=x.

The constant in the numerator is what determines the steepness of the curve. The larger the number in the numerator, the less steep the curve is. You can see that in the three functions sketched below:

f(x)=\cfrac{1}{x}, g(x)=\cfrac{5}{x}, and h(x)=\cfrac{10}{x}

Notice how function f has the smallest numerator and function h has the largest. You can see on the graph how function f is more steep than function g and function h.

Like other functions, reciprocal functions can be translated vertically and horizontally.

For example, take a look at the functions sketched below, y=\cfrac{1}{x} and y=\cfrac{1}{x}+1.

Notice how the sketch of y=\cfrac{1}{x}+1 , the horizontal asymptote is the line y=1

instead of the x- axis (y=0). This is because the graph moved up vertically by one unit.

If you sketched the reciprocal graph, y=\cfrac{1}{x}-2, the horizontal asymptote would be the horizontal line, y=-2 because the graph would move down vertically by two units.

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DOWNLOAD FREELetβs take a look at the graphs y=\cfrac{1}{x} and y=\cfrac{1}{x-1} (where y=\cfrac{1}{x} is the parent function).

Notice how the sketch of y=\cfrac{1}{x-1} moved horizontally one unit to the right which means that the vertical asymptote is now at x=1 instead of x=0 (y- axis ).

If you were to sketch the reciprocal graph of y=\cfrac{1}{x-2} , the vertical asymptote would be x=2 because the graph would move two units horizontally to the right.

This is because the standard form of a reciprocal function is:

f(x)=\cfrac{a}{x-h}+k- Vertical asymptote: x=h
- Horizontal asymptote: y=k

Since the standard form of the equation is defined as x-h in the denominator, if the denominator is x+h that implies that the value of h is a negative number.

For example, determine the vertical asymptote and the horizontal asymptote of the reciprocal function:

y=\cfrac{3}{x+7}-2The vertical asymptote is x=-7 and the horizontal asymptote is y=-2.

This is because the reciprocal function is shifted two units vertically down which changes the horizontal asymptote and it is shifted 7 units horizontally to the left which changes the vertical asymptote.

How does this apply to high school math?

**High School Functions – Interpreting Functions (HSF-IF.C.7)**

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

In order to identify asymptotes of reciprocal graphs:

**Identify the values of \textbf{h} and \textbf{k}.****Write the equation(s) of the horizontal and vertical asymptotes.**

Determine the vertical and horizontal asymptotes of the given function:

f(x)=\cfrac{4}{x}-1**Identify the values of \textbf{h} and \textbf{k}.**

The equation of a reciprocal graph is y=\cfrac{a}{x-h}+k .

In this case, h=0 and k=-1 .

2**Write the equation(s) of the horizontal and vertical asymptotes.**

The equations of both asymptotes are:

Vertical asymptote: x=0 (y- axis )

Horizontal asymptote: y=-1

Determine the vertical and horizontal asymptotes of the reciprocal graph, f(x)=\cfrac{2}{x-3}

**Identify the values of \textbf{h} and \textbf{k}. **

The equation of a reciprocal graph is y=\cfrac{a}{x-h}+k .

In this case, h=3 and k=0 .

**Write the equation(s) of the horizontal and vertical asymptotes.**

The equations of both asymptotes are:

Vertical asymptote: x=3

Horizontal asymptote: y=0 (x- axis )

Determine the vertical and horizontal asymptotes of the reciprocal graph, f(x)=\cfrac{3}{x+4}+5 .

**Identify the values of \textbf{h} and \textbf{k}. **

The equation of a reciprocal graph is y=\cfrac{a}{x-h}+k .

In this case, h=-4 and k=5 .

**Write the equation(s) of the horizontal and vertical asymptotes.**

The equations of both asymptotes are:

Vertical asymptote: x=-4

Horizontal asymptote: y=5

In order to plot a reciprocal graph:

**Identify the horizontal and vertical asymptotes.****Create a table of values with some points to the right and left of the vertical asymptote.****Plot the coordinates and graph the asymptotes.****Draw a smooth curve through the points.**

Plot the reciprocal function:

f(x)=\cfrac{2}{x}-1**Identify the horizontal and vertical asymptotes.**

h=0 so the vertical asymptote is x=0 (y- axis )

k=-1 so the horizontal asymptote is y=-1

**Create a table of values with some points to the right and left of the vertical asymptote.**

Remember you cannot select an x value of 0 because the denominator cannot be equal to 0.

**Plot the coordinates and graph the asymptotes.**

From the table of values, plot points on the coordinate plane.

**Draw a smooth curve through the points.**

Plot the reciprocal function:

f(x)=\cfrac{1}{x+4}**Identify the horizontal and vertical asymptotes.**

h=-4 because there is a plus sign meaning that the function is really, f(x)=\cfrac{1}{x-(-4)} so the vertical asymptote is x=-4

k=0 so the horizontal asymptote is y=0 (x- axis )

**Create a table of values with some points to the right and left of the vertical asymptote.**

Remember, you cannot select an x value of -4 because the denominator cannot be equal to 0.

**Plot the coordinates and graph the asymptotes.**

From the table of values, plot points on the coordinate plane.

**Draw a smooth curve through the points.**

Plot the reciprocal function:

f(x)=\cfrac{8}{x-1}+3**Identify the horizontal and vertical asymptotes.**

h=1 so the vertical asymptote is x=1

k=3 so the horizontal asymptote is y=3

**Create a table of values with some points to the right and left of the vertical asymptote.**

Remember, you cannot select an x value of 1 because the denominator cannot be equal to 0.

**Plot the coordinates and graph the asymptotes.**

From the table of values, plot points on the coordinate plane.

**Draw a smooth curve through the points.**

- Have students sketch the graphs on large poster paper or graph paper and conduct gallery walks so that students can collaborate, critique work, and generate math discussion on the reciprocal graphs.

- Instead of worksheets, have students practice skills through game-playing, classroom scavenger hunts, or using digital platforms such as Khan Academy so that they can watch tutorial videos as well as practice skills.

**Mixing up the vertical and horizontal asymptotes**For example, in the reciprocal function, y=\cfrac{1}{x-6}+4, writing the horizontal asymptote as x=6 and the vertical asymptote as y=4 instead of the vertical asymptote as x=6 and the horizontal asymptote as y=4.

**Forgetting to sketch both parts of the graph**For example, sketching y=\cfrac{2}{x} as:

Instead of:

**Having points off the curve**

When plotting the points, if there is a point that is not on the curves, go back and check the coordinate. For example, in the graph below there is a point that does not lie on the smooth curve, which means the calculation of the coordinates is incorrect. Go back and recheck your arithmetic.

- Types of graphs
- Linear graph
- Cubic function graph
- Exponential function graph
- Equation of a circle

1. Identify the equation that matches the graph.

Β y=x^2+9

y=\cfrac{1}{x}

y=\cfrac{9}{x}

y=9^x

To determine which equation matches the graph, you know that it is a reciprocal function.

The curves of the reciprocal graph are wider than they would be with the function y=\cfrac{1}{x}. But to be sure, check some of the points.

For example, when x=9\text{:}

\begin{aligned} & y=\cfrac{9}{9} \\\\ & y=1 \end{aligned}

So the point (9, 1) should be on the graph.

Also check x=-9\text{:}

\begin{aligned} & y=\cfrac{9}{-9} \\\\ & y=-1 \end{aligned}

So the point (-9, -1) should be on the graph.

Since both points are on the graph, y=\cfrac{9}{x} is the correct equation.

2. Identify the vertical asymptote of the reciprocal function:

y=\cfrac{3}{x}+4

Β x = 0

x = 4

y = 0

y = 4

For the reciprocal function, y=\cfrac{a}{x-h}+k, h represents the vertical asymptote.

So, for the function y=\cfrac{3}{x}+4, h=0Β so the vertical asymptote is x=0.

3. Which graph represents the reciprocal function, y=\cfrac{1}{x-3}+2?

The reciprocal function is represented by y=\cfrac{a}{x-h}+k so the function,

y=\cfrac{1}{x-3}+2, \, h=3 and k=2.

This means that the vertical asymptote is the line x = 3 and the horizontal asymptote is the line y=2.

4. If the vertical asymptote of a reciprocal function is x=-5, the horizontal asymptote of the same function is y=2, and the constant is 1, which of the following functions could represent the function?

Β y=\cfrac{-5}{x}+2

y=\cfrac{1}{x+5}+2

y=\cfrac{1}{x-5}-2

y=\cfrac{1}{x+5}-2

If a reciprocal function has a vertical asymptote at x=-5 that means that the value of h=-5.

If that same function has a horizontal asymptote at y=2 that means the k=2.

The constant value was given to be 1 which means that a=1.Β

So, using the standard equation of the reciprocal function, y=\cfrac{a}{x-h}+k, substitute values into the equation to write the reciprocal function:

\begin{aligned} & y=\cfrac{1}{x-(-5)}+2 \\\\ & y=\cfrac{1}{x+5}+2 \end{aligned}

5. Which reciprocal function represents the graph below?

Β y=\cfrac{1}{x-4}+1

y=\cfrac{1}{x-4}-1

y=\cfrac{1}{x+4}-1

y=\cfrac{1}{x+4}+1

From the graph, you can see that the vertical asymptote is x=-4 and the horizontal asymptote is y=1.

So, using the values h=-4 and k=1, substitute into the reciprocal function where a=1,

\begin{aligned} & y=\cfrac{1}{x-(-4)}+1 \\\\ & y=\cfrac{1}{x+4}+1 \end{aligned}

6. Using the reciprocal function, y=\cfrac{-4}{x+3}-1, if x=-1 find the value of y.

Β -1

-3

-6

3

Using the function, y=\cfrac{-4}{x+3}-1 substitute the value of -1 into the function for x and calculate the value of y.

This is similar to creating a chart of values because, technically, you are looking for the y- coordinate of a point that exists on the reciprocal graph.

\begin{aligned} & y=\cfrac{-4}{-1+3}-1 \\\\ & y=\cfrac{-4}{2}-1 \\\\ & y=-2-1 \\\\ & y=-3 \end{aligned}

Trigonometric functions, logarithmic functions, polynomial functions, and quadratic functions can be part of a rational function and rational functions are a type of reciprocal function.

End behavior describes what the graph of the function looks like as it approaches positive and negative infinity.

Rational functions can be complicated to graph. Itβs always beneficial to know non-digital strategies to sketch a rough graph of a rational function so that you can understand how the function behaves.

- Graphing linear equations
- Rate of change
- Systems of equations

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