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GCSE Maths Algebra

Types of Graphs

Reciprocal Graph

Reciprocal Graph

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.

What is a reciprocal graph?

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

E.g.

Reciprocal Graph image 1 1

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.

What is a reciprocal graph?

What is a reciprocal graph?

How to recognise a reciprocal graph

In order to recognise a reciprocal graph:

  1. Identify linear or quadratic or any other functions.
  2. Identify the reciprocal function.
  3. Identify your final answer.

Reciprocal graph worksheet

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

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Reciprocal graph worksheet

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

COMING SOON

Recognising a reciprocal graph examples

Example 1: recognise a reciprocal graph

Which is the correct equation for the graph?

Reciprocal Graph example 1 1

y=x^2+3 \quad \quad \quad y=3x+1 \quad \quad \quad y=\frac{3}{x} \quad \quad \quad y=3^x

  1. 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.

2Identify the reciprocal function.

y=\frac{3}{x} is a function which has x as the denominator so its graph would be a hyperbola.

3Identify your final answer.

Reciprocal Graph example 1 2

The correct equation for the graph is:

y=\frac{3}{x}

Example 2: recognise a reciprocal graph

Which is the correct graph for the equation?

y=\frac{5}{x}

Reciprocal Graph example 2 1

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.

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

y=\frac{5}{x}


The correct graph for the equation is: Graph D.


Reciprocal Graph example 2 step 3 1

How to use a reciprocal graph

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

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

Explain how to recognise a reciprocal graph

Explain how to recognise a reciprocal graph

Using a reciprocal graph examples

Example 3: using a reciprocal graph

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

\frac{8}{x}=6

Reciprocal Graph example 3 1

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


Reciprocal Graph example 3 step 1 1

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


Reciprocal Graph example 3 step 2 1

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


Reciprocal Graph example 3 step 3 1

The value on the x -axis is 1.4 .


Therefore the approximate solution to the equation is x = 1.4

Example 4: using a reciprocal graph

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

\frac{7}{x-1}+2=5

Reciprocal Graph example 4 1

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


Reciprocal Graph example 4 step 1 1

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


Reciprocal Graph example 4 step 2 1

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

Reciprocal Graph example 4 step 3 1

The value on the x -axis is 3.4 .


Therefore the approximate solution to the equation is x = 3.4

How to plot a reciprocal graph

In order to plot a reciprocal graph:

  1. Complete the table of values.
  2. Plot the coordinates.
  3. Draw a smooth curve through the points.

Plotting a reciprocal graph examples

Example 5: plot a reciprocal graph

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

y=\frac{12}{x}

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}

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.


Reciprocal Graph example 5 step 2


Reciprocal Graph example 5 step 3

Example 6: plot a reciprocal graph

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

y=\frac{10}{x+1}

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} .

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.


Reciprocal Graph example 6 step 2 1


Reciprocal Graph example 6 step 3 1

Common misconceptions

  • Joining up curves

Curves should be smooth.
Do NOT join up the plotted points with straight lines like this:

Reciprocal Graph common misconceptions 1 1


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.

Reciprocal Graph common misconceptions 2 1

Practice reciprocal graph questions

1. Identify the correct equation for the graph:

 

Reciprocal Graph practice question 1 1

y=x^2+9
GCSE Quiz False

y=9x+2
GCSE Quiz False

y=\frac{9}{x}
GCSE Quiz True

y=9^x
GCSE Quiz False

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

Reciprocal Graph Practice Questions 2a

GCSE Quiz False

Reciprocal Graph Practice Questions 2b

GCSE Quiz True

Reciprocal Graph Practice Questions 2c

GCSE Quiz False

Reciprocal Graph Practice Questions 2d

GCSE Quiz False

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}

Reciprocal Graph practice question 3a

GCSE Quiz False

Reciprocal Graph practice question 3b

GCSE Quiz False

Reciprocal Graph practice question 3c

GCSE Quiz False

Reciprocal Graph practice question 3d

GCSE Quiz True

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

Reciprocal Graph practice question 4a

GCSE Quiz True

Reciprocal Graph practice question 4b

GCSE Quiz False

Reciprocal Graph practice question 4c

GCSE Quiz False

Reciprocal Graph practice question 4d

GCSE Quiz False

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

 

Reciprocal Graph practice question 5 1

3.7
GCSE Quiz True

33
GCSE Quiz False

4.1
GCSE Quiz False

5.2
GCSE Quiz False

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.

 

Reciprocal Graph practice question 5 (2) 1

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

 

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

 

Reciprocal Graph practice question 6 1

2.9
GCSE Quiz True

2.6
GCSE Quiz False

4.8
GCSE Quiz False

5.2
GCSE Quiz False

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.

 

Reciprocal Graph practice question 6 (2) 1

Reciprocal graph GCSE questions

1.  Match the correct graph to the equation.

 

y=\frac{10}{x}

 

Reciprocal Graph GCSE question 1 1

 

(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

 

Reciprocal Graph question 2b 1

 

(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)

 

Reciprocal Graph GCSE questions 2b(2) 1

 

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

 

Reciprocal Graph GCSE questions 3b 1

 

(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)

 

Reciprocal Graph GCSE questions final image 1

 

for plotting the 4 points correctly

(1)

for the smooth curve

(1)

 

(c)

 

The answer is x = 3.5

(1)

Learning checklist

You have now learned how to:

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

The next lessons are

  • Cubic graphs
  • Exponential graphs
  • Circle graphs
  • Trigonometry
  • Inverse proportion

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