Types of Graphs (AQA GCSE Maths)

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Daniel I

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Daniel I

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Types of Graphs

Why do we need to know what graphs look like?

  • Graphs are used in various aspects of mathematics – but in the real world they can take on specific meanings
  • For example a linear (straight line) graph could be the path a ship needs to sail along to get from one port to another
  • An exponential graph (y equals k to the power of x) can be used to model population growth – for instance to monitor wildlife conservation projects

What are the shapes of graphs that we need to know?

  • Recalling facts alone won’t do much for boosting your GCSE Mathematics grade!
  • But being familiar with the general shapes of graphs will help you quickly recognise the sort of maths you are dealing with and features of the graph a question may refer to
  • Below the basic form of the five types of function (other than trig graphs) you need to recognise;
    • linear (y equals plus-or-minus x)
    • quadratic (y equals plus-or-minus x squared)
    • cubic (y equals plus-or-minus x cubed)
    • reciprocal (y equals plus-or-minus 1 over x)
    • exponential (y equals k to the power of plus-or-minus x end exponent)

w2DxgV4S_edexcel-igcse-3-graphs-types-of-graphs

  • In addition, you need to recognise the three basic trigonometric graphs- but these are dealt with in the next section

Worked example

Match the graphs to the equations.

Graphs:

A

screen-shot-2022-11-28-at-9-55-22-am

B

screen-shot-2022-11-28-at-9-55-27-am

C

screen-shot-2022-11-28-at-9-55-34-am

D

screen-shot-2022-11-28-at-9-55-39-am

E

screen-shot-2022-11-28-at-9-55-44-am

Equations:

(1) y equals 0.6 x plus 2,    (2) y equals 3 to the power of x,    (3) y equals negative 0.7 x cubed,    (4) y equals 4 over x,   (5) y equals negative x squared plus 3 x plus 2

Starting with the equations,
(1) is a linear equation (ymx c) so matches the only straight line, graph (D)
(2) is an exponential equation with a positive coefficient so matches graph (A)
(3) 
is a cubic equation with a negative coefficient so matches graph (E)
(4)
is a reciprocal equation (notice that it takes the same form as inverse proportion) with a positive coefficient so matches graph (B)
(5) is a quadratic equation with a negative coefficient so matches graph (C)

Graph (A) → Equation (2)

Graph (B) → Equation (4)

Graph (C) → Equation (5)

Graph (D) → Equation (1)

Graph (E) → Equation (3)

Quadratic Graphs

A quadratic is a function of the form y equals a x squared plus b x plus c where a is not zero
They are a very common type of function in mathematics, so it is important to know their key features

What does a quadratic graph look like?

  • The shape made by a quadratic graph is known as a parabola
  • The parabola shape of a quadratic graph can either look like a “u-shape” or an “n-shape”
    • A quadratic with a positive coefficient of x squared will be a u-shape
    • A quadratic with a negative coefficient of x squared will be an n-shape
  • A quadratic will always cross the y-axis
  • A quadratic may cross the x-axis twice, once, or not at all
    • The points where the graph crosses the x-axis are called the roots
  • If the quadratic is a u-shape, it has a minimum point (the bottom of the u)
  • If the quadratic is an n-shape, it has a maximum point (the top of the n)
  • Minimum and maximum points are both examples of turning points

Quadratic Graphs Notes Diagram 1

How do I sketch a quadratic graph?

  • We could create a table of values for the function and then plot it accurately, however we often only require a sketch to be drawn, showing just the key features
  • The most important features of a quadratic are
    • Its overall shape; a u-shape or an n-shape
    • Its y-intercept
    • Its x-intercept(s), these are also known as the roots
    • Its minimum or maximum point (turning point)
  • If it is a positive quadratic (a in a x squared plus b x plus c is positive) it will be a u-shape
  • If it is a negative quadratic (a in a x squared plus b x plus c is negative) it will be an n-shape
  • The y-intercept of y equals a x squared plus b x plus c will be open parentheses 0 comma space c close parentheses
  • The roots, or the x-intercepts will be the solutions to y equals 0a x squared plus b x plus c equals 0
    • You can solve a quadratic by factorising, completing the square, or using the quadratic formula
    • There may be 2, 1, or 0 solutions and therefore 2, 1, or 0 roots
  • The minimum or maximum point of a quadratic can be found by;
    • Completing the square 
      • Once the quadratic has been written in the form y equals p open parentheses x minus q close parentheses squared plus r, the minimum or maximum point is given by open parentheses q comma space r close parentheses
      • Be careful with the sign of the x-coordinate. E.g. if the equation is y equals open parentheses x minus 3 close parentheses squared plus 2 then the minimum point is open parentheses 3 comma space 2 close parentheses but if the equation is y equals open parentheses x plus 3 close parentheses squared plus 2 then the minimum point is open parentheses negative 3 comma space 2 close parentheses

Worked example

a)

Sketch the graph of y equals x squared minus 5 x plus 6 showing the x and y intercepts

It is a positive quadratic, so will be a u-shape

The plus c at the end is the y-intercept, so this graph crosses the y-axis at (0,6)

Factorise

y equals open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses

Solve y equals 0

open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses equals 0

x equals 2 space or space x equals 3

So the roots of the graph are

(2,0)  and (3,0)

cie-igcse-quadratic-graphs-we-1

 

b)

Sketch the graph of y equals x squared minus 6 x plus 13 showing the y-intercept and the turning point

It is a positive quadratic, so will be a u-shape

The plus c at the end is the y-intercept, so this graph crosses the y-axis at

(0,13)

We can find the minimum point (it will be a minimum as it is a positive quadratic) by completing the square: 

x squared minus 6 x plus 13 equals open parentheses x minus 3 close parentheses squared minus 9 plus 13 equals open parentheses x minus 3 close parentheses squared plus 4

This shows that the minimum point will be

(3,4)

As the minimum point is above the x-axis, this means the graph will not cross the x-axis i.e. it has no roots

We could also show that there are no roots by trying to solve x squared minus 6 x plus 13 equals 0

If we use the quadratic formula, we will find that x is the square root of a negative number, which is not a real number, which means there are no real solutions, and hence no roots

cie-igcse-quadratic-graphs-we-2

 

c)

Sketch the graph of y equals negative x squared minus 4 x minus 4 showing the root(s), y-intercept, and turning point

It is a negative quadratic, so will be an n-shape

The plus c at the end is the y-intercept, so this graph crosses the y-axis at (0, -4)

We can find the maximum point (it will be a maximum as it is a negative quadratic) by completing the square: 

negative x squared minus 4 x minus 4 equals negative 1 open parentheses x squared plus 4 x plus 4 close parentheses equals negative 1 open parentheses open parentheses x plus 2 close parentheses squared minus 4 plus 4 close parentheses equals negative open parentheses x plus 2 close parentheses squared

This shows that the maximum point will be

(-2, 0)

As the maximum is on the x-axis, there is only one root

We could also show that there is only one root by solving negative x squared minus 4 x minus 4 equals 0

If you use the quadratic formula, you will find that the two solutions for x are the same number; in this case -2

cie-igcse-quadratic-graphs-we-3

Drawing Graphs Using a Table

How do we draw a graph using a table of values in a non-calculator exam?

  • Before you start, think what the graph might look like- see the previous notes on being familiar with shapes of graphs
  • Using the rules of BIDMAS/ order of operations, substitute each x- value into the given function
  • PLOT POINTS and join with a SMOOTH CURVE
  • If there are any points that don't seem to fit with the shape of the rest of the curve, check your calculations for them again!

How do we draw a graph using a table of values in a calculator exam?

  • Before you start, think what the graph might look like – see the previous notes on being familiar with shapes of graphs
  • Find the TABLE function on your CALCULATOR
  • Enter the FUNCTION – f(x)

    (use ALPHA button and x or X, depending on make/model)

    (Press = when finished)

    (If you are asked for another function, g(x), just press enter again)
  • Enter Start, End and Step (gap between x values)
  • Press = and scroll up and down to see y values
  • PLOT POINTS and join with a SMOOTH CURVE
  • To avoid errors always put negative numbers in brackets and use the (-) key rather than the subtraction key
  • If your calculator does not have a TABLE function, then you will have to work out each y value separately using the normal mode on your calculator

Exam Tip

  • When using the TABLE function of your calculator, double-check that your calculator's y-values are the same as any that are given in the question

Worked example

Calculator Allowed

(a)
Complete the table of values for the function y equals x cubed minus 5 x plus 2.

x negative 3 negative 2 negative 1 0 1 2 3
y   4         14

Use the TABLE function on your calculator for
f open parentheses x close parentheses equals x squared minus x minus 6, starting at -3, ending at 3 and with steps of 1
If your calculator does not have a TABLE function then substitute the values of x into the function one by one for the missing values, being careful to put negative numbers in brackets, e.g.
x equals negative 3 comma space y equals open parentheses negative 3 close parentheses cubed minus 5 open parentheses negative 3 close parentheses plus 2 equals negative 10

x negative 3 negative 2 negative 1 0 1 2 3
y bold minus bold 10 4 bold 6 bold 2 bold minus bold 2 bold 0 14

(b)
On the grid provided, draw the graph of y equals x cubed minus 5 x plus 2 for values of x from negative 3 to 3.


Carefully plot the points from your table of values in (a) on the grid, noting the different scales on the and  axes

For example, the first column represents the point open parentheses negative 3 comma negative 10 close parentheses

After plotting the points, join them with a smooth curve- do not use a ruler!

2-14-drawing-graphs

It is best practice to label the curve with its equation

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Daniel I

Author: Daniel I

Daniel has taught maths for over 10 years in a variety of settings, covering GCSE, IGCSE, A-level and IB. The more he taught maths, the more he appreciated its beauty. He loves breaking tricky topics down into a way they can be easily understood by students, and creating resources that help to do this.