Unsure about Sketching Polynomials?

Unsure about Sketching Polynomials?

Let the fantastic wealth of resources below teach you all about the Sketching of Polynomials ...

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Covers the whole of the AH Maths course

Created by an experienced maths teacher

Resources used with students in Scottish Secondary Schools

Sketching of Polynomials

Welcome to advancedhighermaths.co.uk

A sound understanding of how to sketch polynomials is essential to ensure exam success.

Please find resources for all other Maths courses HERE.

For students looking for a ‘good’ pass at AH Maths in 2019 you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack.

To return to the Home Page please click HERE.

We hope you find this website useful and wish you the very best with your studies.

Please find below:

1. Links and Exam Formulae

2. Differentiation Resources

3. Further Differentiation Resources

4. MIA Text Book Questions – Differentiation

5. MIA Text Book Questions – Further Differentiation 

6. About Sketching Polynomials

7. AH Maths Past and Practice Papers

8. AH Maths Past Paper Questions by Topic

9. Recommended AH Book

10. Exam Focused Online Study Pack – For students looking for a ‘good’ Pass

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Advanced Higher Maths Resources

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1. Links and Exam Formulae

AH Maths - Home Page LinkHERE
AH Maths - Whole Course LinkHERE
AH Maths - Exam Formula ListHERE
Higher Maths - Exam Formulae ListHERE
AH Maths - Complete Formula ListHERE
SQA Full Course Content and OutlineHERE

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2. Differentiation Resources

Thanks to the author(s) for making these available.

Differentiation Theory Guide 1HERE
Differentiation Theory Guide 2HERE
Differentiation HSN Summary NotesHERE

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3. Further Differentiation Resources

Thanks to the author(s) for making these available.

Further Differentiation Theory Guide 1HERE
Further Differentiation Theory Guide 2HERE
Further Differentiation HSN Summary NotesHERE

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4. MIA Text Book Questions – Differentiation

Recommended questions from the new MIA Text Book are shown below.

Worked solutions to all questions below are available in the Online Study Pack.

Subtopic
___________________________
Page Number
____________
Exercise
___________
Questions
____________________________
Worked Solutions
_______________
Derivative from First PrinciplesPage 45Exercise 4.1Q1,3,5,7HERE
The Chain RulePage 48Exercise 4.3Q1a,d,2a,c,3b,4a,5aHERE
The Product RulePage 51Exercise 4.5Q1a-h,Q2b,Q3a-lHERE
The Quotient RulePage 52Exercise 4.6Q1,2,3,4HERE
Differentiation - A Mixture!Page 53Exercise 4.7Q1,2,3,4,5HERE
Sec, Cosec & CotPage 55Exercise 4.8Q1a,b,2a,c,d,3a,c,e,gHERE
Exponential FunctionsPage 58Exercise 4.9Q1a,c,e,2a,3e,4a,b,5a,eHERE
Logarithmic FunctionsPage 58Exercise 4.9Q1k,m,o,q,s,2f,g,3a,b,c,4d,e,5dHERE
Nature & Sketching PolynomialsPage 70Exercise 5.5Q1a,b,c,2a,bHERE
ConcavityPage 73Exercise 5.7Q5a,b,c,Q1a,bHERE
ApplicationsPage 187Ex 11.1Q1a,b,e,f,2a,c,3a,cHERE

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5. MIA Text Book Questions – Further Differentiation 

The new MIA Text Book worked solutions below are courtesy of Robert Milton, Teacher of Mathematics.

Subtopic
_______________________________________
Page Number
_____________
Exercise
___________
Questions
___________________
Worked Solutions
_______________
Inverse Trig Functions & Chain RulePage 85Exercise 6.2Q1a,b,c,Q2b,c,dQ3a,dHERE
Inverse Trig Fns & Product/Quotient RulesPage 86Exercise 6.3Q2,Q3HERE
Implicit & Explicit Functions - 1Page 89Exercise 6.4Q1,Q2HERE
Implicit & Explicit Functions - 2Page 89Exercise 6.4Q5,Q9,Q4HERE
Second Derivatives of Implicit FunctionsPage 90Exercise 6.5Q1a,d,f,k(i),6HERE
Logarithmic DifferentiationPage 92Exercise 6.6Q1,Q2HERE
Parametric EquationsPage 95Exercise 6.7Q1a,b,cHERE
Parametric Eqns - Differentiation Page 96Exercise 6.8Q1,2,3HERE
Parametric Eqns - Differentiation (Alternative)Page 96Exercise 6.8Q1(i)HERE
Parametric Eqns - Differentiation (Alternative)Page 96Exercise 6.8Q1(ii),Q2,Q3HERE
Applications of Further DifferentiationPage 193Exercise 11.2Q1,Q2,Q3HERE

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6. About Sketching of Polynomials

For a more detailed explanation, please first read the Theory Guides above.

When sketching Polynomials we determine:

  • Where the graph cuts the x and y-axis
  • Where the stationary points occur (eg set f ‘ (x) = 0 and solve)
  • How f(x) behaves as x approaches ± ∞
  • The nature of the stationary points.

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At AH Maths to find the nature of the stationary points the second derivative is used.

Mr Smiley and Mr Frowny can help us remember the above:

  • Happy ⇒ Double Differentiate  ⇒ Eyes Positive (f ”(x) > 0) and a Smile  (Minimum TP)
  • Frowny ⇒ Double Differentiate  ⇒ Eyes Negative (f ”(x) < 0) and a Frown (Maximum TP)

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When the second derivative is zero there is no information about the nature of the stationary point and a nature table must be used to investigate,

Points of inflexion occur when the second derivative is zero.

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7. AH Maths Past & Practice Papers

Thanks to the SQA for making these available.

Worked solutions to SQA Exam Questions available in the Online Study Pack.

Year
_____
Paper Type
_________________
Exam Paper
______________
Marking Scheme
_________________
2018Advanced HigherExam PaperMarking Scheme
2017Advanced HigherExam PaperMarking Scheme
2016Advanced HigherExam PaperMarking Scheme
2016AH SpecimenSpecimen
2016AH ExemplarExemplarExemplar Guidance
2015Advanced HigherExam PaperMarking Scheme
2014Advanced HigherExam PaperMarking Scheme
2013Advanced HigherExam PaperMarking Scheme
2012Advanced HigherExam PaperMarking Scheme
2011Advanced HigherExam PaperMarking Scheme
2010Advanced HigherExam PaperMarking Scheme
2009Advanced HigherExam PaperMarking Scheme
2008Advanced HigherExam PaperMarking Scheme
2007Advanced HigherExam PaperMarking Scheme
2006Advanced HigherExam PaperMarking Scheme
2005Advanced HigherExam PaperMarking Scheme
2004Advanced HigherExam PaperMarking Scheme
2003Advanced HigherExam PaperMarking Scheme
2002Advanced HigherExam PaperMarking Scheme
2001Advanced HigherExam Paper

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8. AH Maths Past Paper Questions by Topic

Thanks to the SQA for making these available.

Questions and answers have been split up by topic for ease of reference.

Worked solutions to all SQA questions below available in the Online Study Pack.

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Paper
______
.
Marking
______
Binomial
Theorem
________
Partial
Fractions
________
.
Differentiation
___________
Further Differentiation
___________
.
Integration
___________
Further
Integration
____________
Functions
& Graphs
___________
Systems of
Equations
____________
Complex
Numbers
__________
Seq &
Series
_________
Further Seq
& Series
____________
.
Matrices
_________
.
Vectors
__________
Methods
of Proof
__________
Further No
Theory
___________
Differential
Equations
____________
Further
Differential Eqns
_________________
2018MarkingQ3Q2Q1bQ1a,c,6,13Q8Q15aQ16aQ4,10Q14Q17Q7,11Q16Q9,12Q5Q15b
2017MarkingQ1Q2Q3Q11,18Q16Q6Q12Q5Q17Q4,10Q7Q15Q13Q8Q9Q14
2016MarkingQ3Q13Q1a,bQ1c,11Q13Q9Q12Q4Q8Q2Q6Q7Q14Q5,10Q16Q15
2015MarkingQ1,9Q2Q4,6,8Q17Q10Q14Q13Q3Q5,11Q15Q12Q7Q18Q16
2014MarkingQ214bQ1,13Q1,4,6Q10,12Q15Q11Q3Q16Q14Q9Q7Q5Q7Q8
2013MarkingQ1Q2Q11Q4,6Q8Q13Q7,10Q17Q3Q15Q9,12Q5Q16Q14
2012MarkingQ415aQ1Q12,13Q8Q11Q7Q14Q3,16bQ2Q6Q9Q516aQ10Q15
2011MarkingQ2Q13b,73aQ1,11aQ1,11,16Q6Q10Q8,13Q5Q4Q15Q12Q9Q14
2010MarkingQ5Q1Q13Q15Q3,7Q10Q16Q2Q9Q4,14Q6Q8,12Q11
2009MarkingQ8Q14Q1aQ1b,11Q5,7Q9Q1316aQ6Q12Q14Q2Q16Q4Q10Q3Q15
2008MarkingQ8Q4Q10,15Q2,5Q4,9,10Q7Q3Q16Q1Q12Q6Q14Q11Q13
2007MarkingQ1Q4Q2Q13Q4,10Q4Q16Q3,11Q9Q6Q5Q15Q12Q7Q14Q8
MixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixed

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9.   AH Maths Text Book – Maths In Action (2nd Edition) by Edward Mullan 

A fully revised course for the new Curriculum for Excellence examination that is designed to fully support the course’s new structure and unit assessment. A part of the highly regarded Maths in Action series, it provides students with a familiar, clear and carefully structured learning experience that encourages them to build confidence and understanding.
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10. Exam focused Online Study Pack 

Through step-by-step worked solutions to exam questions available in the Online Study Pack we cover everything you need to know about how to sketch polynomials to pass your final exam.

As well as students studying Advanced Higher Mathematics, the resources will benefit young adults studying A-Level Mathematics and undergraduates who need a little extra help. Particular benefit will be to students who have gained a ‘Conditional’ University place and are therefore required to pass in order to gain entry onto the course of their choice.

Subscribing to the Online Study Pack may therefore be one of your best ever investments. Please click Online Study Pack to view screenshots, examples and instructions on how to subscribe.

Please speak with your mum/dad and show them our fantastic website.

We hope the resources on this website prove useful and wish you the very best of success with your AH Maths course in 2018/19.

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