Graphs of Polynomials (Edexcel IGCSE Further Pure Maths)

Revision Note

Dan Finlay

Written by: Dan Finlay

Reviewed by: Lucy Kirkham

Quadratic Functions & Graphs

What are the key features of quadratic graphs?

  • A quadratic graph can be written in the form y equals a x squared plus b x plus c where a not equal to 0

  • The shape of the graph is known as a parabola

  • The value of a affects the shape of the curve

    • If a is positive the shape is 'u-shaped'

    • If a is negative the shape is 'upside down u-shaped'

  • The y-intercept is at the point (0, c)

  • The roots are the solutions to a x squared plus b x plus c equals 0

    • These are also known as the x-intercepts or zeroes

    • They can be found by

      • Factorising

      • Quadratic formula

      • Completing the square

      • Your calculator may also be able to find these for you

    • There can be 0, 1 or 2 x-intercepts

      • This is determined by the value of the discriminant

  • There is an axis of symmetry at x equals negative fraction numerator b over denominator 2 a end fraction

    • If there are two x-intercepts then the axis of symmetry goes through the midpoint between them

  • The vertex lies on the axis of symmetry

    • It can be found by completing the square

    • The x-coordinate is x equals negative fraction numerator b over denominator 2 a end fraction

    • The y-coordinate can be found by calculating y when x equals negative fraction numerator b over denominator 2 a end fraction

    • If a is positive then the vertex is the minimum point

    • If a is negative then the vertex is the maximum point

Graphs of a quadratic
Key features of a quadratic

What are the equations of a quadratic function?

  • straight f left parenthesis x right parenthesis equals a x squared plus b x plus c

    • This is the general form

    • It clearly shows the y-intercept (0, c)

    • You can find the axis of symmetry by x equals negative fraction numerator b over denominator 2 a end fraction

  • straight f left parenthesis x right parenthesis equals a left parenthesis x minus p right parenthesis left parenthesis x minus q right parenthesis

    • This is the factorised form

    • It clearly shows the roots (p, 0) & (q, 0)

    • You can find the axis of symmetry by x equals fraction numerator p plus q over denominator 2 end fraction

  • straight f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k

    • This is the vertex form (or completed square form)

    • It clearly shows the vertex (h, k)

    • The axis of symmetry is therefore x equals h

    • It clearly shows how the function can be transformed from the graph of y equals x squared

      • Vertical stretch by scale factor ­a

      • Translation by vector stretchy left parenthesis table row h row k end table stretchy right parenthesis

How do I find an equation of a quadratic?

  • If you have the roots x = p and x = q...

    • Write in factorised form space y equals a left parenthesis x minus p right parenthesis left parenthesis x minus q right parenthesis

    • You will need a third point to find the value of a

  • If you have the vertex (h, k) then...

    • Write in vertex form y equals a left parenthesis x minus h right parenthesis squared plus k

    • You will need a second point to find the value of a

  • If you have three random points (x1, y1), (x2, y2) & (x3, y3) then...

    • Write in the general form y equals a x squared plus b x plus c

    • Substitute the three points into the equation

    • Form and solve a system of three linear equations to find the values of a, b & c

Examiner Tips and Tricks

  • Your calculator may be able to find the roots and turning point of a quadratic function

    • Even on a 'show that' question this can be used to check your answers

Worked Example

The diagram below shows the graph of space y equals straight f left parenthesis x right parenthesis, where space straight f left parenthesis x right parenthesis is a quadratic function.

The vertex and the intercept with the y-axis have been labelled.

2-2-1-ib-aa-sl-we-image


Find an expression for space y equals f left parenthesis x right parenthesis.

Method 1

Since we know the vertex (turning point), it will be easiest to start with the completed square version of the equation 
This is  straight f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k,  where the vertex is at open parentheses h comma space k close parentheses

table row cell straight f open parentheses x close parentheses end cell equals cell a open parentheses x minus open parentheses negative 1 close parentheses close parentheses squared plus 8 end cell row blank equals cell a open parentheses x plus 1 close parentheses squared plus 8 end cell end table

We also know the curve y equals straight f open parentheses x close parentheses goes through (0, 6)
Put those coordinates into the equation and solve for a

table row 6 equals cell a open parentheses 0 plus 1 close parentheses squared plus 8 end cell row 6 equals cell a plus 8 end cell row a equals cell negative 2 end cell end table

Substitute a equals negative 2 into the expression for straight f open parentheses x close parentheses
Expand the brackets and rearrange into the form required

table row cell straight f open parentheses x close parentheses end cell equals cell negative 2 open parentheses x plus 1 close parentheses squared plus 8 end cell row blank equals cell negative 2 open parentheses x squared plus 2 x plus 1 close parentheses plus 8 end cell row blank equals cell negative 2 x squared minus 4 x minus 2 plus 8 end cell end table

bold f begin bold style stretchy left parenthesis x stretchy right parenthesis end style bold equals bold minus bold 2 bold italic x to the power of bold 2 bold minus bold 4 bold italic x bold plus bold 6

Method 2

It is also possible to start with the y equals a x squared plus b x plus c form
Because the y-intercept is (0, 6) we know that c equals 6

Goes through open parentheses 0 comma space 6 close parentheses means  straight f open parentheses x close parentheses equals a x squared plus b x plus 6

It also goes through (-1, 8)
Substitute those coordinates into the equation of y equals f open parentheses x close parentheses

Goes through open parentheses negative 1 comma space 8 close parentheses means

table row 8 equals cell a open parentheses negative 1 close parentheses squared plus b open parentheses negative 1 close parentheses plus 6 end cell row 8 equals cell a minus b plus 6 end cell row cell a minus b end cell equals 2 end table

We need one more piece of information
You may remember that the turning point lies on the line  x equals negative fraction numerator b over denominator 2 a end fraction
If not, then use the fact that the x-coordinate of the turning point satisfies  straight f to the power of apostrophe open parentheses x close parentheses equals 0

Turning point at open parentheses negative 1 comma space 8 close parentheses means  straight f to the power of apostrophe open parentheses negative 1 close parentheses equals 0

Differentiate to find straight f to the power of apostrophe open parentheses x close parentheses
Then solve straight f to the power of apostrophe open parentheses negative 1 close parentheses equals 0 to find another equation with aand b

straight f to the power of apostrophe open parentheses x close parentheses equals 2 a x plus b

table row cell 2 a open parentheses negative 1 close parentheses plus b end cell equals 0 row cell negative 2 a plus b end cell equals 0 end table

We now have two simultaneous equations that we can solve to find a and b

table row cell a minus b end cell equals cell 2 space space space space open square brackets 1 close square brackets end cell row cell negative 2 a plus b end cell equals cell 0 space space space space open square brackets 2 close square brackets end cell end table

Add [1] and [2] together to eliminate b

table row cell negative a end cell equals 2 row a equals cell negative 2 end cell end table

Substitute into [2] and solve to find b

table row cell negative 2 open parentheses negative 2 close parentheses plus b end cell equals 0 row cell 4 plus b end cell equals 0 row b equals cell negative 4 end cell end table

Write final answer in form requested

bold f stretchy left parenthesis x stretchy right parenthesis bold equals bold minus bold 2 bold italic x to the power of bold 2 bold minus bold 4 bold italic x bold plus bold 6

Cubic Functions & Graphs

What are the key features of cubic graphs?

  • cubic graph can be written in the form  y equals a x cubed plus b x squared plus c x plus d  where a not equal to 0

  • When asked to consider the equation of a cubic and its graph, think about the following

    • y-axis intercept

      • This occurs at  open parentheses 0 comma space d close parentheses

    • x-axis intercepts (roots)

      • x-coordinates are the solutions to  a x cubed plus b x squared plus c x plus d equals 0

    • turning points (maximum and/or minimum)

      • A cubic can have 0, 1 or 2 turning points

      • x-coordinates are the solutions to  3 a x squared plus 2 b x plus c equals 0

      • these are when the derivative of a x cubed plus b x squared plus c x plus d equals 0

    • The value of a affects the shape of the curve

      • If a is positive the ends of the curve will go 'down on the left and up on the right'

      • If a is negative the ends of the curve will go 'up on the left and down on the right'

Key features of a cubic

 

How do I sketch the graph of a cubic?

  • STEP 1       
    Find the y-axis intercept by setting x = 0

  • STEP 2
    Find the x-axis intercepts (roots) by setting y = 0 

    • This may require factorising the cubic

  • STEP 3
    Consider the shape and “start”/”end” of the graph

    • a positive cubic graph starts in third quadrant (“bottom left”) and ends in first quadrant (“top right”)

    • negative cubic graph starts in second quadrant (“top left”) and ends in fourth quadrant (“bottom right”)

  • STEP 4
    Consider where any turning points should go

    • Differentiate the equation of the curve and set equal to zero

  • STEP 5
    Draw with a smooth curve (this takes practice!) 

Steps to sketch a cubic

  

Worked Example

Consider the function  straight f open parentheses x close parentheses equals negative x cubed plus b x squared plus c x,  where b and c are constants.

The graph of  y equals straight f open parentheses x close parentheses  has two turning points, the x-coordinates of which are 1 and 3.

(a) Find the values of b and c.

The turning points occur when straight f to the power of apostrophe open parentheses x close parentheses equals 0

Start by differentiating to find straight f to the power of apostrophe open parentheses x close parentheses

straight f to the power of apostrophe open parentheses x close parentheses equals negative 3 x squared plus 2 b x plus c

That is equal to 0 when x equals 1 and when x equals 3

Substitute those values in to find two equations in b and c

table row cell negative 3 open parentheses 1 close parentheses squared plus 2 b open parentheses 1 close parentheses plus c end cell equals 0 row cell negative 3 plus 2 b plus c end cell equals 0 row cell 2 b plus c end cell equals 3 end table

table row cell negative 3 open parentheses 3 close parentheses squared plus 2 b open parentheses 3 close parentheses plus c end cell equals 0 row cell negative 27 plus 6 b plus c end cell equals 0 row cell 6 b plus c end cell equals 27 end table

Those are simultaneous equations in b and c
Subtract the first from the second to eliminate c and find b
Then substitute that value into either equation to find c

bold italic b bold equals bold 6 bold space bold space bold space bold space bold space bold space bold italic c bold equals bold minus bold 9

(b) Sketch the graph of y equals straight f open parentheses x close parentheses.

There is no constant term in straight f open parentheses x close parentheses, so the y-intercept will be 0

Substitute the x-coordinates of the turning points into straight f open parentheses x close parentheses to find the corresponding y-coordinates

 

negative open parentheses 1 close parentheses cubed plus 6 open parentheses 1 close parentheses squared minus 9 open parentheses 1 close parentheses equals negative 1 plus 6 minus 9 equals negative 4

minus open parentheses 3 close parentheses cubed plus 6 open parentheses 3 close parentheses squared minus 9 open parentheses 3 close parentheses equals negative 27 plus 54 minus 27 equals 0

So the turning points are at  open parentheses 1 comma space minus 4 close parentheses and open parentheses 3 comma space 0 close parentheses

Find the x-intercepts by solving straight f open parentheses x close parentheses equals 0

table row cell negative x cubed plus 6 x squared minus 9 x end cell equals 0 row cell negative x cubed open parentheses x squared minus 6 x plus 9 close parentheses end cell equals 0 row cell negative x cubed open parentheses x minus 3 close parentheses squared end cell equals 0 end table


x equals 0 comma space space space space x equals 3

So the x-intercepts are open parentheses 0 comma space 0 close parentheses and open parentheses 3 comma space 0 close parentheses
Note that the first is also the y-intercept. and the second is also a turning point

Finally consider the shape of the curve
It's a 'negative cubic', so it's going to be 'up on the left and down on the right'
Draw a smooth curve incorporating all these features
Label the turning points and intercepts

Graph of cubic function

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Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

Lucy Kirkham

Author: Lucy Kirkham

Expertise: Head of STEM

Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels.Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all.