Syllabus Edition

First teaching 2023

First exams 2025

|

Using Graphs (CIE IGCSE Maths: Core)

Revision Note

Test yourself
Mark

Author

Mark

Last updated

Drawing Graphs Using a Table

How do I draw a graph using a table of values?

  • To create a table of values
    • substitute different x-values into the equation
    • This gives the y-values
  • To plot the points
    • Use the and y-values to mark crosses on the grid at the coordinates (x , y )
    • Each point is expected to be plotted to an accuracy within half of the smallest square on the grid
  • Drawsingle smooth freehand curve 
    • Go through all the plotted points
    • Make it the shape you would expect
      • For example, quadratic curves have a vertical line of symmetry
    • Do not use a ruler for curves!

Which numbers should I be careful with?

  • For quadratic graphs, be careful substituting in negative numbers 
  • Always put brackets around them and use BIDMAS
    • For example, x  = -3 in y  = -x2  + 8x 
      • becomes y  = -(-3)2 + 8(-3)
      • which simplifies to -9 - 24
      • so = - 33
  • For reciprocal graphs like y equals 1 over x, do not include = 0
    • You cannot divide by zero
      • You get an error on your calculator
    • There is no value at x  = 0 
      • The L-shaped branches can't cross the y-axis
    • An example is given below with y equals 1 over x
x -3 -2 -1 0 1 2 3
y negative 1 third negative 1 half negative 1 No value 1 1 half 1 third

How do I use the table function on my calculator?

  • Calculators can create tables of values for you
  • Find the table function
    • Type in the graph equation (called the function, f(x)) 
      • Use the alpha button then X or x
      • Press = when finished
    • If you are asked for another function, g(x), press = to ignore it
  • Enter the start value
    • The first x-value in the table
    • Press =
  • Enter the end value
    • The last x-value in the table
  • Enter the step size
    • How big the steps (gaps) are from one x-value to the next
    • Press =
  • Then scroll up and down to see all the y-values

Examiner Tip

  • If you find a point that doesn't seem to fit the shape of the curve, check your working!
  • If any y-values are given in the question, check that your calculations agrees with them.

Worked example

(a)
Complete the table of values for the graph of y equals 10 minus 8 x squared.

x negative 1.5 negative 1 negative 0.5 0 0.5 1 1.5
y   2         negative 8
 
Use the table function on your calculator for straight f open parentheses x close parentheses equals 10 minus 8 x squared
Start at -1.5, end at 1.5 and use steps of 0.5
On a non-calculator paper, substitute the x-values into the equation, for example x = -1.5

 
table row y equals cell 10 minus 8 open parentheses negative 1.5 close parentheses squared end cell row blank equals cell 10 minus 8 cross times 2.25 end cell row blank equals cell 10 minus 18 end cell row blank equals cell negative 8 end cell end table
 

x negative 1.5 negative 1 negative 0.5 0 0.5 1 1.5
y -8 2 8 10 8 2 negative 8
 

(b)
Plot the graph of y equals 10 minus 8 x squared on the axes below, for values of x from negative 1.5 to 1.5.
 
Carefully plot the points from your table on to the grid
Note the different scales on the axes
Join the points with a smooth curve (do not use a ruler)
 

cie-igcse-2018-may-jun-1-7

(c)
Write down the equation of the line of symmetry of the curve.
 
There is a vertical line of symmetry about the y-axis
The equation of the y-axis is x = 0
x = 0

Solving Equations Using Graphs

How do I find the coordinates of points of intersection?

  • Plot two graphs on the same set of axes
    • The points of intersection are where the two lines meet
  • For example, plot y  = x2  + 3x  + 1 and y  = 2x  + 1 on the same axes
    • They meet twice, as shown
    • The coordinates of intersection are (-1, -1) and (0, 1)
       

Points of intersection between a curve and a line

How do I solve simultaneous equations graphically?

  • The and solutions to simultaneous equations are the and coordinates of the point of intersection
  • For example, to solve 2- = 3 and 3 + y  = 7 simultaneously
    • Rearrange them into the form y  = mx  + c
      • y  = 2x  - 3 and y  = -3x  + 7
    • Use a table of values to plot each line
    • Find the point of intersection, (2, 1)
    • The solutions are therefore x  = 2 and = 1

 

Solving simultaneous equations graphically

How do I use graphs to solve equations?

  • This is easiest explained through an example
  • You can use the graph of y equals x squared minus 4 x minus 2 to solve the following equations
    • x squared minus 4 x minus 2 equals 0
      • The solutions are the two x-intercepts
      • Where the curve cuts the x-axis (also called roots)
    • x squared minus 4 x minus 2 equals 5
      • The solutions are the two x-coordinates where the curve intersects the horizontal line y equals 5 
    • x squared minus 4 x minus 2 equals x plus 1,
      • the solutions are the two x-coordinates where the curve intersects the straight line y equals x plus 1
      • The straight line must be plotted on the same axes first
  • To solve a different equation like x squared minus 4 x plus 3 equals 1
    • add / subtract terms to both sides to get "graph = ..."
      • For example, subtract 5 from both sides
        • x squared minus 4 x minus 2 equals negative 4
        • This can now be done as above

Examiner Tip

  • When solving equations in x, only give x-coordinates as final answers
    • Include the y-coordinates if solving simultaneous equations

Worked example

Use the graph of y equals 10 minus 8 x squared shown to estimate the solutions of each equation given below.

The graph of y equals ten minus eight x squared
 

(a)
10 minus 8 x squared equals 0
  
This equals zero, so the x-intercepts are the solutions
Read off the values where the curve cuts the x-axis

Use a suitable level of accuracy (no more than 2 decimal places from the scale of this graph)
  
-1.12 and 1.12
 
These are the two solutions to the equation

= -1.12 and = 1.12

A range of solutions are accepted, such as "between 1.1 and 1.2"
Solutions must be ± of each other (due to the symmetry of quadratics)

 
(b)
10 minus 8 x squared equals 8
  
This equals 8, so draw the horizontal line y = 8
Find the x-coordinates where this cuts the graph
  
-0.5 and 0.5
 
These are the two solutions to the original equation

x  = -0.5 and = 0.5

The solutions here are exact

You've read 0 of your 10 free revision notes

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

Did this page help you?

Mark

Author: Mark

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.