GCSE Maths Algebra Quadratic Graphs

Plotting Quadratic Graphs

Plotting Quadratic Graphs

Here we will learn about plotting quadratic graphs including how to substitute values to create a table and then draw the graph of a quadratic function from this table.

There are also quadratic 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 plotting quadratic graphs?

Plotting quadratic graphs is drawing up a table of values for the x and y coordinates, and then plotting these on a set of axes.

Quadratic graphs are graphs of quadratic functions – that is, any function which has x^2 as its highest power.

E.g.

y=x^{2}+2 x+5 is a quadratic function.

In order to complete a table of values, we substitute each x value into the quadratic function to obtain the matching y value. Each one of these is a coordinate pair. For the function y=x^{2}+2 x+5 , the table of values from x=-3 to x=2 would look like this:

\begin{aligned} &x \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 2 \\ &y \quad \quad \quad 8 \quad \quad\quad 5 \quad \quad \quad 4 \quad \quad \quad 5 \quad \quad \quad 8 \quad \quad \quad 13 \end{aligned}

and the graph would look like this:

Plotting quadratic graphs image 1 1

The shape made by the graph of a quadratic function is called a parabola, and is symmetric.

What is plotting quadratic graphs?

What is plotting quadratic graphs?

How to plot a quadratic graph

In order to plot a graph of a quadratic function:

  1. Draw a table of values, and substitute x values to find matching y values.
  2. Plot these coordinate pairs on a graph.
  3. Join the points with a smooth curve.

Explain how to plot a quadratic graph

Explain how to plot a quadratic graph

Plotting quadratic graphs worksheet

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

COMING SOON
x

Plotting quadratic graphs worksheet

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

COMING SOON

Related quadratic graphs lessons

Plotting quadratic graphs is part of a series of lessons to support the study of quadratic graphs. It may be useful to begin with the main quadratic graphs lesson as it presents an overview of the topic. You can also use the step-by-step guides below to focus on individual lessons in this topic:

Plotting quadratic graphs examples

Example 1: the standard quadratic graph

Draw the graph of

y=x^{2}

  1. Draw a table of values, and substitute x values to find matching y values.

You will usually be given the range of values to use for x . Sometimes this might be written as ‘from -3 to 3 ’, or you may see it written symbolically as -3 ≤ x ≤ 3 .

\begin{aligned} &x \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 2 \quad \quad \quad 3 \\ &y \quad \quad \quad 9 \quad \quad\quad 4 \quad \quad \quad 1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 4 \quad \quad \quad 9 \end{aligned}

When:

\begin{aligned} &x=-3, \;y=(-3)^{2}=9 \\\\ &x=-2,\; y=(-2)^{2}=4 \\\\ &x=-1, \;y=(-1)^{2}=1 \end{aligned}

And so on…

2Plot these coordinate pairs on a graph.

Plotting quadratic graphs example 1 step 2 1

3Join the points with a smooth curve.

Plotting quadratic graphs example 1 step 3 1

Example 2: a simple quadratic

Draw the graph of

y=x^{2}-4

\begin{aligned} &x \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad \; 0 \quad \quad \quad 1 \quad \quad \quad 2 \quad \quad \quad 3 \\ &y \quad \quad \quad 5 \quad \quad\quad 0 \quad \quad -3 \quad \quad \; -4 \quad \quad -3 \quad \quad \quad 0 \quad \quad \quad 5 \end{aligned}


When:


\begin{aligned} &x=-3, \;y=(-3)^{2}-4=9-4=5 \\\\ &x=-2, \;y=(-2)^{2}-4=4-4=0\\\\ &x=-1, \;y=(-1)^{2}-4=1-4=-3 \end{aligned}


And so on…

Plotting quadratic graphs example 2 step 2 1

Plotting quadratic graphs example 2 step 3 1

Example 3: another simple quadratic

Draw the graph of

y=x^{2}+3x

\begin{aligned} &x \quad \quad -5 \quad \quad -4 \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \\ &y \quad \quad \;\; 10 \quad \quad\quad 4 \quad \quad \quad 0 \quad \quad -2 \quad \quad -2 \quad \quad \quad 0 \quad \quad \quad 4 \end{aligned}


When:


\begin{aligned} &x=-5, \; y=(-5)^{2}+3(-5)=25-15=10 \\\\ &x=-4, \; y=(-4)^{2}+3(-4)=16-12=4\\\\ &x=-3, \; y=(-3)^{2}+3(-3)=9-9=0 \end{aligned}


And so on…


You might find it helpful to add extra rows into your table to work out these substitutions. Break the function down into its component terms, as in the shaded rows in the table. To get the y value, just sum the shaded portions.


\begin{aligned} &x \quad \quad -5 \quad \quad -4 \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \\ &x^2 \quad \quad 25 \quad \quad \;\;\;16 \quad \quad \;\;\; 9 \quad \quad \quad 4 \quad \quad \quad1 \quad \quad \quad 0 \quad \quad \quad 1 \\ + 3&x \quad \;\; -15 \quad \;\; -12 \quad \;\;\; -9 \quad \quad -6 \quad \quad -3 \quad \quad +0 \quad \quad+3 \\ &y \quad \quad \;\; 10 \quad \quad\quad 4 \quad \quad \quad 0 \quad \quad -2 \quad \quad -2 \quad \quad \quad 0 \quad \quad \quad 4 \end{aligned}

Plotting quadratic graphs example 3 step 2 1

Plotting quadratic graphs example 3 step 3 1


Note that you need to round out the bottom of the curve – don’t just join across the bottom with a straight line!

Example 4:  a quadratic trinomial (three parts)

Draw the graph of

y=x^{2}+3x+2

\begin{aligned} &x \quad \quad -5 \quad \quad -4 \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 2\\ &y \quad \quad \;\;12 \quad \quad\quad 6 \quad \quad \quad 2 \quad \quad \quad 0 \quad \quad \quad 0 \quad \quad \quad 2 \quad \quad \quad 6 \quad \quad \;\; 12 \end{aligned}


When:


\begin{aligned} &x=-5,\; y=(-5)^{2}+3(-5)+2=25-15+2=12 \\\\ &x=-4, \;y=(-4)^{2}+3(-4)+2=16-12+2=6\\\\ &x=-3, \;y=(-3)^{2}+3(-3)+2=9-9+2=2 \end{aligned}


And so on…


You might find it helpful to add extra rows into your table to work out these substitutions. Break the function down into its component terms, as in the shaded rows in the table. To get the y value, just sum the shaded portions.


\begin{aligned} &x \quad \quad -5 \quad \quad -4 \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 2 \\ &x^2 \quad \quad 25 \quad \quad \;\;16 \quad \quad \quad 9 \quad \quad \quad 4 \quad \quad \quad1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 4\\ + 3&x \quad \;\; -15 \quad \;\; -12 \quad \;\;\; -9 \quad \quad -6 \quad \quad -3 \quad \quad +0 \quad \quad+3 \quad \quad+6 \\ + &2 \quad \quad +2 \quad \quad +2 \quad \quad +2\quad \quad +2\quad \quad +2\quad \quad +2\quad \quad +2\quad \quad +2 \\ &y \quad \quad \;\;12 \quad \quad \quad 6 \quad \quad \quad 2 \quad \quad \quad 0 \quad \quad \quad 0 \quad \quad \quad 2 \quad \quad \quad 6 \quad \quad \;\;12 \end{aligned}

Plotting quadratic graphs example 4 step 2 1

Plotting quadratic graphs example 4 step 3 1


Note that you need to round out the bottom of the curve – don’t just join across the bottom with a straight line!

Example 5: dealing with negative terms

Draw the graph of

y=x^{2}-2x+4

\begin{aligned} &x \quad \quad -2 \quad \quad -1 \quad \quad \quad0 \quad \quad \quad 1 \quad \quad \quad 2 \quad \quad \quad 3 \quad \quad \quad 4 \\ &y \quad \quad \;\; 12 \quad \quad \quad 7 \quad \quad \quad 4 \quad \quad \quad 3 \quad \quad \quad 4 \quad \quad \quad 7 \quad \quad \quad 12 \end{aligned}


When:


\begin{aligned} &x=-2, \;y=(-2)^{2}-2(-2)+4=4+4+4=12 \\\\ &x=-1, \;y=(-1)^{2}-2(-1)+4=1+2+4=7\\\\ &x=0, \;y=(0)^{2}-2(0)+4=0+0+4=4 \end{aligned}


And so on…


Alternatively, using a table with the function broken down into rows:


\begin{aligned} &x \quad \quad -2 \quad \quad -1 \quad \quad \quad0 \quad \quad \quad 1 \quad \quad \quad 2 \quad \quad \quad 3 \quad \quad \quad 4 \\ &x^2 \quad \quad \;\; 4 \quad \quad \quad 1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad4 \quad \quad \quad 9 \quad \quad \;\; 16 \\ -2&x \quad \;\; +14 \quad \;\;\; +2 \quad \quad\: -0 \quad \quad -2 \quad \quad -4 \quad \quad -6 \quad \quad -8 \\ + &4 \quad \quad +4 \quad \quad +4 \quad \quad +4\quad \quad +4\quad \quad +4\quad \quad+4\quad \quad +4 \\ &y \;\; \quad \quad 12 \quad \quad \quad 7 \quad \quad \quad 4 \quad \quad \quad 3 \quad \quad \quad 4 \quad \quad \quad 7 \quad \quad \quad 12 \end{aligned}

Plotting quadratic graphs example 5 step 2 1

Plotting quadratic graphs example 5 step 3 1

Example 6: coefficient of x2 ≠ 1

Draw the graph of

y=2x^{2}+3x-5

\begin{aligned} &x \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad \; 0 \quad \quad \quad 1 \quad \quad \quad 2 \quad \quad \quad 3 \\ &y \quad \quad \quad 4 \quad \quad -3 \quad \quad -6 \quad \quad \; -5 \quad \quad \quad0 \quad \quad \quad 9 \quad \quad \;\; 22 \end{aligned}


When:


\begin{aligned} &x=-3, \;y=2(-3)^{2}+3(-3)-5=18-9-5=4 \\\\ &x=-2, \;y=2(-2)^{2}+3(-2)-5=8-6-5=-3\\\\ &x=-1, \;y=2(-1)^{2}+3(-1)-5=2-3-5=-6 \end{aligned}


And so on…


Alternatively, using a table with the function broken down into rows:


\begin{aligned} &x \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 2 \quad \quad \quad 3 \\ 2&x^2 \quad \quad 18 \quad \quad \quad 8 \quad \quad \quad 2 \quad \quad \, \quad 0 \quad \quad \quad 2 \quad \quad \quad 8 \quad \quad \;\; 18 \\ +3&x \quad \quad-9 \quad \quad -6 \quad \quad -3 \quad \quad +0 \quad \quad +3 \quad \quad +6 \quad \quad+9 \\ - &5 \quad \quad -5 \quad \quad -5 \quad \quad-5\quad \quad -5\quad \quad -5\quad \quad-5 \quad \quad -5 \\ &y \quad \quad \quad 4 \quad \quad -3 \quad \quad -6 \quad \quad -5 \quad \quad \quad0 \quad \quad \quad 9 \quad \quad \;\; 22 \end{aligned}

Plotting quadratic graphs example 6 step 2 1

Plotting quadratic graphs example 6 step 3 1

Example 7: coefficient of x2 ≠ 1 , negative x2 term

Draw the graph of

y=-2x^{2}-4x+3

\begin{aligned} &x \quad \quad -4 \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 2\\ &y \quad \;\; -13 \quad \quad -3 \quad \quad \quad 3 \quad \quad \quad 5 \quad \quad \quad3 \quad \;\;\, -3 \quad \;\; -13 \end{aligned}


When:


\begin{aligned} &x=-4,\; y=-2(-4)^{2}-4(-4)+3=-32+16+3=-13 \\\\ &x=-3,\; y=-2(-3)^{2}-4(-3)+3=-18+12+3=-3\\\\ &x=-2, \;y=-2(-2)^{2}-4(-2)+3=-8+8+3=3 \end{aligned}


And so on…


Alternatively, using a table with the function broken down into rows:


\begin{aligned} &x \quad \quad -4 \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 2\\ -2&x^2 \quad \;\; -32 \quad \;\; -18 \quad \;\; -8 \quad \;\;\;-2 \quad \quad \quad \,0 \quad \quad -2 \quad \;\;\;-8 \\ -4&x \quad \quad +16 \quad \;\; +12 \quad \;\; +8 \quad \;\;\, +4 \quad \quad \; -0 \quad \quad -4 \quad \quad -8 \\ + &3 \quad \quad \, +3 \quad \quad +3 \quad \quad +3\quad \;\;\; +3 \quad \quad \; +3 \quad \quad+3 \quad \quad +3 \\ &y \quad \quad-13 \quad \;\; -3 \quad \quad \quad \, 3 \quad \quad \;\;\; 5 \quad \quad \quad\: 3 \quad \quad -3 \quad \quad -13 \end{aligned}

Plotting quadratic graphs example 7 step 2 1

Plotting quadratic graphs example 7 step 3 1

Common misconceptions

  • Drawing a pointy vertex

Make sure that the vertex of the graph is a smooth curve, not pointed:

Plotting quadratic graphs Common Misconceptions 1 1

  • Making errors when dealing with negative x values, particularly when squaring

E.g.

(-3)2=9 , not -9 .

If you’re using your calculator, make sure you include brackets around the x value that you are squaring.

Practice plotting quadratic graphs questions

1. Draw the graph of:

 

y=x^{2}+4

Plotting quadratic graphs Practice questions 1a 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 1b 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 1c 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 1d 1

GCSE Quiz True

The y -intercept is (0,4) and the graph is a u shape because the x^2 coefficient is positive.

2. Draw the graph of:

 

y=x^{2}+4x

Plotting quadratic graphs Practice questions 2a 1

GCSE Quiz True

Plotting quadratic graphs Practice questions 2b 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 2c 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 2d 1

GCSE Quiz False

The quadratic factorises to give x(x+4) so the roots are x=0 and x=-4.

3. Draw the graph of:

 

y=x^{2}-6x

Plotting quadratic graphs Practice questions 3a 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 3b 1

GCSE Quiz True

Plotting quadratic graphs Practice questions 3c 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 3d 1

GCSE Quiz False

The quadratic factorises to give x(x-6) so the roots are x=0 and x=6 .
 
The graph is a u shape because the x^2 coefficient is positive.

4. Draw the graph of:

 

y=x^{2}+2x+3

Plotting quadratic graphs Practice questions 4a 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 4b 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 4c 1

GCSE Quiz True

Plotting quadratic graphs Practice questions 4d 1

GCSE Quiz False

The quadratic has no real roots so it doesn’t intersect the x axis.
 
The constant term is 3 so the y intercept is (0,3) . The graph is a u shape because the x^2 coefficient is positive.

5. Draw the graph of:

 

y=x^{2}-5x-3

Plotting quadratic graphs Practice questions 5A 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 5B 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 5C 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 5D 1

GCSE Quiz True

The quadratic has two roots, which could be found by completing the square or using the quadratic formula.
 
The constant term is -3 so the y intercept is (0,-3) .
 
The graph is a u shape because the x^2 coefficient is positive.

6. Draw the graph of:

 

y=2x^{2}+4x-7

Plotting quadratic graphs Practice questions 6A 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 6B 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 6C 1

GCSE Quiz True

Plotting quadratic graphs Practice questions 6D 1

GCSE Quiz False

The quadratic has two roots, which could be found by completing the square or using the quadratic formula.
 
The constant term is -7 so the y intercept is (0,-7) .
 
The graph is a u shape because the x^2 coefficient is positive.

7. Draw the graph of:

 

y=4x-3x^{2}

Plotting quadratic graphs Practice questions 7A 1

GCSE Quiz True

Plotting quadratic graphs Practice questions 7B 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 7C 1

GCSE Quiz False

Plotting quadratic graphs Practice questions 7D 1

GCSE Quiz False

The quadratic factorises to give x(4-3x) so the roots are x=0 and x=\frac{4}{3} .
 
The graph is a n shape because the x^2 coefficient is negative.

Plotting quadratic graphs GCSE questions

1. (a) Complete the table of values for y=x^{2}+4x-2

 

\begin{aligned} &x \quad \quad -6 \quad \quad -5 \quad \quad -4 \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1\\\\ &y \quad \quad\;\; 10 \quad \quad \quad \quad \quad\quad\quad\quad\quad-5 \end{aligned}

 

(b) On the grid draw the graph of y=x^{2}+4x-2 for values of x from -6 to 1 .

 

Plotting quadratic graphs GCSE questions 1b 1

 

(4 marks)

Show answer

(a)

 

\begin{aligned} &x \quad \quad -6 \quad \quad -5 \quad \quad -4 \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1\\\\ &y \quad \quad\;\; 10 \quad \quad \quad 3 \quad \quad -2 \quad \quad -5 \quad \quad -6 \quad \quad -5 \quad \quad -2 \quad \quad 3 \end{aligned}

(2)

 

(b)

 

Plotting quadratic graphs GCSE questions 1b ans 1

(2)

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

 

\begin{aligned} &x \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 2 \quad \quad \quad 3 \quad \quad \quad 4 \quad \quad \quad 5\\\\ &y \quad \quad \quad \quad \quad\;\;-2 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 2 \end{aligned}

 

(b) Draw the graph of y=2+3x-x^{2} for values of x from -2 to 5 .

 

Plotting quadratic graphs GCSE question 2b 1

 

(4 marks)

Show answer

(a)

 

\begin{aligned} &x \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 2 \quad \quad \quad 3 \quad \quad \quad 4 \quad \quad \quad 5\\\\ &y \quad \quad -8 \quad \quad -2 \quad \quad \quad 2 \quad \quad \quad 4 \quad \quad \quad 4 \quad \quad \quad 2 \quad \quad -2 \quad \quad -8 \end{aligned}

(2)

 

(b)

 

Plotting quadratic graphs GCSE questions 2b ans 1

(2)

3. (a) Complete the table of values for y=2x^{2}+x-5

 

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

 

(b) On the grid draw the graph of y=2x^{2}+x-5 for values of x from -3 to 2 .

 

Plotting quadratic graphs GCSE questions 3b 1

 

(4 marks)

Show answer

(a)

 

\begin{aligned} &x \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 2 \\\\ &y \quad \quad \;\; 10 \quad \quad \quad 1 \quad \quad -4 \quad \quad -5 \quad \quad -2 \quad \quad \quad 5 \end{aligned}

(2)

 

(b)

 

Plotting quadratic graphs GCSE questions 3b ans 1

(2)

The next lessons are

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths revision programme.