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Here we will learn about reciprocal graphs, including how to recognise and sketch them. We will also look at plotting and interpreting graphs of reciprocal functions.

There are also reciprocal graphs worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

A **reciprocal graph** is of the form y = \frac{1}{x} .

E.g.

The graph is a smooth curve called a **hyperbola**.

We can see that there is a break in the graph when x = 0 . The curve gets very close to the x and y axes but never touches them.

This means that the x and y axes in this example are **asymptotes **to the curve.

In order to recognise a reciprocal graph:

**Identify linear or quadratic or any other functions.****Identify the reciprocal function**.**Identify your final answer**.

Get your free reciprocal graph worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free reciprocal graph worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONWhich is the correct equation for the graph?

y=x^2+3 \quad \quad \quad y=3x+1 \quad \quad \quad y=\frac{3}{x} \quad \quad \quad y=3^x**Identify linear or quadratic or any other functions**.

y=x^2+3 is a quadratic function; its graph would be a parabola.

y=3x+1 is a linear function; its graph would be a straight line.

y=3^x is an exponential function; its graph would be a growth curve.

2**Identify the reciprocal function**.

y=\frac{3}{x} is a function which has

3**Identify your final answer**.

The correct equation for the graph is:

y=\frac{3}{x}Which is the correct graph for the equation?

y=\frac{5}{x}**Identify linear or quadratic or any other functions**.

Graph A is a parabola; its function would be a quadratic function.

Graph B is a growth curve; its function would be an exponential function.

Graph C is a straight line; its function would be a linear function.

**Identify the reciprocal function**.

The original equation has x as a denominator so we know we are looking for a hyperbola. Graph D is a hyperbola.

**Identify your final answer**.

y=\frac{5}{x}

The correct graph for the equation is: Graph D.

In order to use a reciprocal graph to solve an equation:

**Find the given value on the y -axis.****Draw a straight horizontal line to the curve**.**Draw a straight vertical line from the curve to the x -axis**.**Read off the value on the x -axis**.

Use the graph of y=\frac{8}{x} to find an approximate solution of the equation.

\frac{8}{x}=6**Find the given value on the y-axis**.

We need to look on the y -axis for the value of 6 .

**Draw a straight horizontal line to the curve**.

Make sure you use a ruler and be as accurate as you can.

**Draw a straight vertical line from the curve to the x-axis**.

Continue to use a ruler and try to be as accurate as you can.

**Read off the value on the x-axis**.

The value on the x -axis is 1.4 .

Therefore the approximate solution to the equation is x = 1.4

Use the graph of y=\frac{7}{x-1}+2 to find an approximate solution of the equation.

\frac{7}{x-1}+2=5**Find the given value on the y-axis**.

We need to look on the y -axis for the value of 5 .

**Draw a straight horizontal line to the curve**.

Make sure you use a ruler and be as accurate as you can.

**Draw a straight vertical line from the curve to the x-axis**.

Continue to use a ruler and try to be as accurate as you can.

**Read off the value on the x-axis**.

The value on the x -axis is 3.4 .

Therefore the approximate solution to the equation is x = 3.4

In order to plot a reciprocal graph:

**Complete the table of values**.**Plot the coordinates**.**Draw a smooth curve through the points**.

Draw the curve for 1\leq{x}\leq12

y=\frac{12}{x}**Complete the table of values**.

Substitute the values of x into the equation. Write the values of y in the table.

\begin{aligned}
&x \quad \quad 1 \quad \quad 2 \quad \quad 3 \quad \quad 4 \quad \quad 6 \quad \quad 12 \\\\
&y \quad \quad 12 \quad \;\;6 \quad \quad 4 \quad \quad 3 \quad \quad 2 \quad \quad 1
\end{aligned}

**Plot the coordinates**.

It is best practice to use a pencil and plot the coordinates using small crosses.

The coordinates would be (1,12), (2,6) and so on.

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

Draw the curve for 0\leq{x}\leq9

y=\frac{10}{x+1}**Complete the table of values**.

Substitute the values of x into the equation. Write the values of y in the table.

When,

x=0, \quad \quad \quad y=\frac{10}{0+1}=\frac{10}{1}=10

\begin{aligned} &x \quad \quad 0 \quad \quad 1 \quad \quad 3 \quad \quad 4 \quad \quad 9 \\ &y \quad \quad 10 \quad \;\;5 \quad \quad 2.5 \quad \;5 \quad \quad 1 \end{aligned} .

**Plot the coordinates**.

It is best practice to use a pencil and plot the coordinates using small crosses.

The coordinates would be (0,10), (1,5) and so on.

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

**Joining up curves**

Curves should be smooth.

Do NOT join up the plotted points with straight lines like this:

The points should be joined by a single, smooth curve.

**Points on the curve**

All points should be on the curve. If there is a point that does not join up with the other points when you draw a smooth curve – go back and check the coordinate.

In the example below there is a point that does not lie on the smooth curve so this needs checking.

1. Identify the correct equation for the graph:

y=x^2+9

y=9x+2

y=\frac{9}{x}

y=9^x

The graph is a hyperbola. We are looking for a function which has x as the denominator.

y=\frac{9}{x} has x as the denominator so it is a reciprocal function and its graph will be a hyperbola.

y=x^2+9 is a quadratic function; its graph will be a parabola.

y=9x+2 is a linear function; its graph will be a straight line.

y=9^x is an exponential function; its graph will be a growth curve.

2. Identify the correct graph for the equation:

y=\frac{5}{x}+1

The function which has x as the denominator. We need the graph of a reciprocal function. We are looking for a hyperbola.

Graph B is the hyperbola.

Graph A is a growth curve; its function will be an exponential function.

Graph C is a straight line; its function will be a linear function.

Graph D is a parabola; its function will be a quadratic function.

3. Identify the correct graph for the equation:

y=\frac{10}{x}

A correct table of values would be:

\begin{aligned} x \quad \quad 1 \quad \quad 2 \quad \quad 4 \quad \quad 5 \quad \quad 8 \quad \quad 10\\ y \quad \quad 10 \quad \;\; 5 \quad \quad 2.5 \quad \;\, 2 \quad \;\; 1.25 \quad \;\, 1 \end{aligned}

These values give coordinates such as (1,10), (2,5) and so on.

The coordinates are plotted on the grid.

A smooth curve should be drawn through the points.

4. Identify the correct graph for the equation:

y=\frac{8}{x-1}-2

A correct table of values would be:

\begin{aligned} &x \quad \quad 2 \quad \quad 3 \quad \quad 5 \quad \quad 9 \\ &y \quad \quad 6 \quad \quad 2 \quad \quad 0 \quad \; -1 \end{aligned}

These values give coordinates such as (2,6), (3,2) and so on.

The coordinates are plotted on the grid.

A smooth curve should be drawn through the points.

5. Use the graph of y=\frac{11}{x} to solve the following equation:

\frac{11}{x}=3

3.7

33

4.1

5.2

Find 3 on the y-axis . Draw a straight line across to the curve.

Draw a straight line down to the x-axis and read off the value.

6. Use the graph of y=\frac{15}{x-1}+3 to solve the following equation:

\frac{15}{x-1}+3=11

2.9

2.6

4.8

5.2

Find 11 on the y-axis . Draw a straight line across to the curve.

Draw a straight line down to the x-axis and read off the value.

1. Match the correct graph to the equation.

y=\frac{10}{x}

**(1 mark)**

Show answer

The correct graph is Graph B.

**(1)**

2. (a) Complete the table of values for y=\frac{6}{x}

\begin{aligned} &x \quad \quad 1 \quad \quad 2 \quad \quad 3 \quad \quad 6\\ &y\end{aligned}

(b) On the grid, draw the graph of y=\frac{6}{x} for 1\leq{x}\leq6

**(4 marks)**

Show answer

(a)

Correct Values are: 6, 3, 2 and 1

For 2 correct y -values

**(1)**

For all correct y -values

**(1)**

(b)

for plotting the 4 points correctly

**(1)**

for the smooth curve

**(1)**

3. (a) Complete the table of values for y=\frac{9}{x+1}

\begin{aligned} &x \quad \quad 0 \quad \quad 1 \quad \quad 2 \quad \quad 8\\ &y\end{aligned}

(b) On the grid, draw the graph of y=\frac{9}{x+1} for 0\leq{x}\leq8

(c) Use your graph to solve:

\frac{9}{x+1}=2

**(5 marks)**

Show answer

(a)

Correct Values are: 9, 4.5, 3 and 1

For 2 correct y -values

**(1)**

For all correct y -values

**(1)**

(b)

for plotting the 4 points correctly

**(1)**

for the smooth curve

**(1)**

(c)

The answer is x = 3.5

**(1)**

You have now learned how to:

- Recognise a graph of a reciprocal function
- Plot reciprocal graphs
- Use reciprocal graphs

- Cubic graphs
- Exponential graphs
- Circle graphs
- Trigonometry
- Inverse proportion

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