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GCSE Maths Algebra

Plotting Graphs

Interpreting Graphs

Plotting Graphs

Here we will learn about plotting graphs, including how to plot straight-line graphs and the graphs of curves including quadratic graphs and cubic graphs

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

What is plotting graphs?

Plotting graphs allows us to accurately plot coordinates onto a grid to produce the graph of a function.

We can plot a variety of different graphs. Linear functions which produce straight-line graphs but we can also plot other polynomials such as quadratic functions or cubic functions and exponential functions.

To do this we need to find x -coordinates and their y -coordinates.

These are plotted on x-y axes and the points are joined up.

For example,

y = 3x + 2

Plotting Graphs image 1

Plotting Graphs image 2

What is plotting graphs?

What is plotting graphs?

How to plot a graph

In order to plot a graph:

  1. Find the coordinates.
  2. Plot the coordinates on the axes.
  3. Join up the coordinates.

Explain how to plot a graph

Explain how to plot a graph

Sketching graphs worksheet (includes plotting graphs)

Get your free plotting graphs worksheet of 20+ sketching surds questions and answers. Includes reasoning and applied questions.

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Sketching graphs worksheet (includes plotting graphs)

Get your free plotting graphs worksheet of 20+ sketching surds questions and answers. Includes reasoning and applied questions.

COMING SOON

Plotting graphs examples

Example 1: straight-line graph in the form y = mx + c

Complete the table of values for y=2x+3 and then on the grid draw the graph of y=2x+3:

Plotting Graphs example 1 image 1

Plotting Graphs example 1 image 2

  1. Find the coordinates.

We need to find the missing y -coordinates. We do this by substituting the x -values into the equation.

For example,

When x = -1

When y = 2(-1) + 3 = -2 + 3 = 1

Plotting Graphs example 1 step 1

2Plot the coordinates on the axes.

The x -values and y -values in the table give coordinates to plot on the grid.

(-2,-1), \ (-1, 1) and so on. The points should be plotted with small crosses.

Plotting Graphs example 1 step 2

3Join up the coordinates.

The coordinates need to be joined up with a straight-line. The line can be extended beyond the first and last points.

Plotting Graphs example 1 step 3

Example 2: straight-line graph in the form y = mx + c, no table given

On the grid draw the graph of y=3x-4 for values of x from -2 to 4:

Plotting Graphs example 2

We need to make our own table. We can choose at least 3 \ x -values. Simple x -values such as 0, 1, and 2 will be sensible. We can then find the corresponding y -coordinates. We do this by substituting the x -values into the equation.


For example,


When x=1


y=3(1)-4=3-4=-1


Plotting Graphs example 2 step 1

The x -values and y -values in the table give coordinates to plot on the grid.


(0,-4), \ (1, -1) and (2,2). The points should be plotted with small crosses.


Plotting Graphs example 2 step 2

The coordinates need to be joined up with a straight-line. The line can be extended beyond the first and last points to cover the whole of the grid.


Plotting Graphs example 2 step 3

Example 3: quadratic function

Complete the table of values for y=x^{2}-3x and then on the grid draw the graph of y=x^{2}-3x.

Plotting Graphs example 3 image 1

Plotting Graphs example 3 image 2

We need to find the missing y -coordinates. We do this by substituting the x -values into the equation.


For example,


When x=-1


y=(-1)^2-3(-1)=1+3=4


Plotting Graphs example 3 step 1

The x -values and y -values in the table give coordinates to plot on the grid.


(-1,4), \ (0, 0) and so on. The points should be plotted with small crosses.


Plotting Graphs example 3 step 2

The coordinates need to be joined up with a single smooth curve. The curve can be extended beyond the first and last points.


Plotting Graphs example 3 step 3

Example 4: cubic function

Complete the table of values for y=x^{3}+2x^{2}-3x and then on the grid draw the graph of y=x^{3}+2x^{2}-3x:

Plotting Graphs example 4 image 1

Plotting Graphs example 4 image 2

We need to find the missing y -coordinates. We do this by substituting the x -values into the equation.


For example,


When x=-2


y=(-2)^3+2(-2)^2-3(-2)=-8+8+6=6


Plotting Graphs example 4 step 1

The x -values and y -values in the table give coordinates to plot on the grid.


(-3,0), \ (-2, 6) and so on. The points should be plotted with small crosses.


Plotting Graphs example 4 step 2

The coordinates need to be joined up with a single smooth curve. The curve can be extended beyond the first and last points.


Plotting Graphs example 4 step 3

Example 5: exponential function

Complete the table of values for y=2^x and then on the grid draw the graph of y=2^x:

Plotting Graphs example 5 image 1

Plotting Graphs example 5 image 2

We need to find the missing y -coordinates. We do this by substituting the x -values into the equation.


For example,


When x=-1


y=2^{-1} = 0.5


Plotting Graphs example 5 step 1

The x -values and y -values in the table give coordinates to plot on the grid.


(-1,0.5), \ (0, 1) and so on. The points should be plotted with small crosses.


Plotting Graphs example 5 step 2

The coordinates need to be joined up with a single smooth curve. The curve can be extended beyond the first and last points.


Plotting Graphs example 5 step 3

How to plot a graph using the intercept method

In order to plot a graph using the intercept method:

  1. Substitute \bf{x=0} to find the \textbf{y} -intercept.
  2. Substitute \bf{y=0} to find the \textbf{x} -intercept.
  3. Join the two points, then substitute the values of a third point to check your answer.

Explain how to plot a graph using the intercept method

Explain how to plot a graph using the intercept method

Plotting graphs using the intercept method examples

Example 6: straight-line graph in the form ax + by = c

On the grid draw the graph of 2x+3y=12.

Plotting Graphs example 6 image 1

Because of the way the equation of the straight-line is written, it is simple to substitute x=0 to find where the line crosses the y -axis.


\begin{aligned} 2x+3y &= 12 \\\\ 2(0)+3y &=12\\\\ 3y&=12\\\\ y&=4 \end{aligned}


This means the line crosses the y -axis at 4. The y -intercept is 4.


Plotting Graphs example 6 step 1

Similarly it is simple to substitute y=0, to find where the line crosses the x -axis.


\begin{aligned} 2x+3y &= 12 \\\\ 2x+3(0) &=12\\\\ 2x&=12\\\\ x&=6 \end{aligned}


This means the line crosses the x -axis at 6. The x -intercept is 6.


Plotting Graphs example 6 step 2

Join the two points with a straight line.


Plotting Graphs example 6 step 3


Always check your answer by substituting the values of a third point into the original equation.


The point (3,2) lies on the line we have drawn. We substitute x=3, \ y=2 into the equation 2x+3y=12 to check.


\begin{aligned} 2x+3y &= 12 \\\\ 2(3)+3(2) &=12\\\\ 6+6&=12 \end{aligned}


The statement is correct, so we know that our line is drawn correctly.

Example 7: straight-line graph in the form ax + by = c

Draw the graph of 2x+3y=18.

Plotting Graphs example 7 image 1

Substitute x=0 into 2x+3y=18.


\begin{aligned} 2x+3y &= 18 \\\\ 2(0)+3y &=18\\\\ 3y&=18\\\\ y&=6 \end{aligned}


This means the line crosses the y -axis at 4. The y -intercept is 4.


Plotting Graphs example 7 step 1

Substitute y=0 into 2x+3y=18.


\begin{aligned} 2x+3y &= 18 \\\\ 2x+3(0) &=18\\\\ 2x&=18\\\\ x&=9 \end{aligned}


This means the line crosses the x -axis at 9. The x -intercept is 9.


Plotting Graphs example 7 step 2

Join the two points with a straight line:

Plotting Graphs example 7 step 3


Always check your answer by substituting the values of a third point into the original equation.


The point (3,4) lies on the line we have drawn. We substitute x=3, \ y=4 into the equation 2x+3y=18 to check.


\begin{aligned} 2x+3y &= 18 \\\\ 2(3)+3(4) &=18\\\\ 6+12&=18 \end{aligned}


The statement is correct, so we know that our line is drawn correctly.

Example 8: straight-line graph in the form ax + by = c

Draw the graph of 4x-2y=20.

Plotting Graphs example 8 image 1

Substitute x=0 into 4x-2y=20.


\begin{aligned}4x-2y &= 20 \\\\ 4(0)-2y &=20\\\\ -2y&=20\\\\ y&=-10 \end{aligned}


This means the line crosses the y -axis at -10. The y -intercept is -10.


Plotting Graphs example 8 step 1

Substitute y=0 into 4x-2y=20.


\begin{aligned} 4x-2y &= 20 \\\\ 4x-2(0) &=20\\\\ 4x&=20\\\\ x&=-5 \end{aligned}


This means the line crosses the x -axis at 5. The x -intercept is 5.


Plotting Graphs example 8 step 2

Join the two points with a straight line:


Plotting Graphs example 8 step 3


Always check your answer by substituting the values of a third point into the original equation.


The point (4,-2) lies on the line we have drawn. We substitute x=4, \ y=-2 into the equation 4x-2y=20 to check.


\begin{aligned} 4x-2y &= 20 \\\\ 4(4)-2(-2) &=20\\\\ 16+4&=20 \end{aligned}


The statement is correct, so we know that our line is drawn correctly.

Common misconceptions

  • Take care with negative values

It is easy to make mistakes when substituting negative values into algebraic expressions.

For example, substituting x=-2 into y=x^{2}+3 will be:

y=(-2)^2+3=4+3=7 \ \color{green} \textbf{✔}

NOT

y=-2^2+3=-4+3=-1 \ \color{red} \textbf{✘}

  • Graphs of quadratic functions have symmetry

Graphs of quadratic functions have a line of symmetry. This can be useful as a check on whether you have plotted the points correctly.

Plotting Graphs example 8 common misconceptions image 1

  • Turning points should be drawn carefully

Quadratic graphs and cubic graphs have turning points. They should be drawn with care so they are not very flat, nor very pointy.

Plotting Graphs example 8 common misconceptions image 2

Practice plotting graphs questions

1. Complete the table of values for y=5x-4.

 

Plotting Graphs practice questions 1 image 1

Plotting Graphs practice questions 1 image 2

GCSE Quiz False

Plotting Graphs practice questions 1 correct answer 1

GCSE Quiz True

Plotting Graphs practice questions 1 correct answer 2

GCSE Quiz False

Plotting Graphs practice questions 1 correct answer 3

GCSE Quiz False

The missing y -coordinates can be found by substituting the x -values into y=5x-4.

 

When x=-1, \ y=5\times -1-4=-5-4=-9

 

When x=0, \ y=5\times 0-4=0-4=-4

 

When x=2, \ y=5\times 2-4=10-4=6

2. Draw the graph of y=4-2x.

Plotting Graphs practice questions 2 image 1

GCSE Quiz False

Plotting Graphs practice questions 2 image 2

GCSE Quiz False

Plotting Graphs practice questions 2 image 3

GCSE Quiz False

Plotting Graphs practice questions 2 correct answer

GCSE Quiz True

The graph of y=4-2x is a straight-line going through these points.

 

Plotting Graphs practice questions 2 explanation image

 

The points should be joined with a straight-line.

3. Draw the graph of 4x+3y=24.

Plotting Graphs practice questions 3 correct answer 1

GCSE Quiz True

Plotting Graphs practice questions 3 correct answer 2

GCSE Quiz False

Plotting Graphs practice questions 3 correct answer 3

GCSE Quiz False

Plotting Graphs practice questions 3 correct answer 4

GCSE Quiz False

We can work out where the line crosses the axes.

 

To find where the line crosses the y -axis, substitute x=0.

 
\begin{aligned} 4x+3y &= 24 \\\\ 4(0)+3y &=24\\\\ 3y&=24\\\\ y&=8 \end{aligned}
 

This means the line crosses the y -axis at 8.

 

To find where the line crosses the x -axis, substitute y=0.
 
\begin{aligned} 4x+3y &= 24 \\\\ 4x+3(0) &=24\\\\ 4x&=24\\\\ x&=6 \end{aligned}
 
This means the line crosses the x -axis at 6.

4. Complete the table of values for y=x^{2}+2x.

 

Plotting Graphs practice questions 4 image 1

Plotting Graphs practice questions 4 image 2

GCSE Quiz False

Plotting Graphs practice questions 4 image 3

GCSE Quiz False

Plotting Graphs practice questions 4 image 4

GCSE Quiz False

Plotting Graphs practice questions 4 correct answer

GCSE Quiz True

The missing y -coordinates can be found by substituting the x -values into y=x^{2}+2x.

 

When x=-2, \ y=(-2)^2 +2(-2)=4-4=0

 

When x=1, \ y=(1)^2 +2(1)=1+2=3

 

When x=2, \ y=(2)^2 +2(2)=4+4=8

5. Draw the graph of y=x^{2}-3.

Plotting Graphs practice questions 5 image 1

GCSE Quiz False

Plotting Graphs practice questions 5 image 2

GCSE Quiz False

Plotting Graphs practice questions 5 correct answer 1

GCSE Quiz True

Plotting Graphs practice questions 5 image 4

GCSE Quiz False

The graph of y=x^{2}-3 is a curve going through these points.

 

Plotting Graphs practice questions 5 explanation answer

 

The points should be joined with a smooth curve.

6. Complete the table of values for y=4^{x}.

 

Plotting Graphs practice questions 6 image 1

Plotting Graphs practice questions 6 correct answer 1

GCSE Quiz True

Plotting Graphs practice questions 6 correct answer 2

GCSE Quiz False

Plotting Graphs practice questions 6 correct answer 3

GCSE Quiz False

Plotting Graphs practice questions 6 correct answer 4

GCSE Quiz False

The missing y -coordinates can be found by substituting the x -values into y=4^{x}.

 

When x=-1, \ y=4^{-1}=0.25

 

When x=1, \ y=4^1=4

 

When x=2, \ y=4^2=16

Plotting graphs GCSE questions

1. (a) Complete the table of values for y=3x-5.

 

Plotting Graphs GCSE question 1a image 1

 

(b) On the grid draw the graph of y=3x-5, for the values of x from -1 to 3.

 

Plotting Graphs GCSE question 1b image 1

 

(4 marks)

Show answer

(a)

 

Plotting Graphs GCSE question 1a image 2

 

For two correct y -values.

(1)

For all four correct y -values.

(1)

 

(b)

 

Plotting Graphs GCSE question 1b image 2

 

For at least 4 points correctly plotted.

(1)

For graph fully drawn correctly.

(1)

2. (a) Complete the table of values for y=x^{2}+3x-1.

 

Plotting Graphs GCSE question 2a image 1

 

(b) On the grid draw the graph of y=3x-5, for the values of x from -3 to 3.

 

Plotting Graphs GCSE question 2b image 1

 

(4 marks)

Show answer

(a)

 

Plotting Graphs GCSE question 2a image 2

 

For two correct y -values.

(1)

For all four correct y -values.

(1)

 

(b)

 

Plotting Graphs GCSE question 2b image 2

 

For at least 6 points correctly plotted.

(1)

For graph fully drawn correctly.

(1)

3. (a) Complete the table of values for y=\frac{12}{x}.

 

Plotting Graphs GCSE question 3a image 1

 

(b) On the grid draw the graph of y=\frac{12}{x}, for the values of x from 1 to 12.

 

Plotting Graphs GCSE question 3b image 2

 

(4 marks)

Show answer

(a)

 

Plotting Graphs GCSE question 3a image 1

 

For two correct y -values.

(1)

For all four correct y -values.

(1)

 

(b)

 

Plotting Graphs GCSE question 3b image 3

 

For at least 4 points correctly plotted.

(1)

For graph fully drawn correctly.

(1)

Learning checklist

You have now learned how to:

  • Plot straight-line graphs
  • Plot graphs of functions including quadratic functions, cubic functions and exponential functions

The next lessons are

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FREE GCSE Maths Practice Papers - 2022 Topics

Practice paper packs based on the advanced information for the Summer 2022 exam series from Edexcel, AQA and OCR. 



Designed to help your GCSE students revise some of the topics that will come up in the Summer exams.

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