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

Covers the whole of the AH Maths course

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Resources used with students in Scottish Secondary Schools

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A sound understanding of Functions and Graphs is essential to ensure exam success.

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Please find below:

2. Functions and Graphs Resources

3. MIA Text Book Questions – Functions and Graphs

5. AH Maths Past & Practice Papers

6. AH Maths Past Paper Questions by Topic

7. AH Maths Recommended Text Book

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

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**1. Links and Exam Formulae
**

AH Maths - Home Page Link | HERE |

AH Maths - Whole Course Link | HERE |

AH Maths - Exam Formula List | HERE |

Higher Maths - Exam Formulae List | HERE |

AH Maths - Complete Formula List | HERE |

SQA Full Course Content and Outline | HERE |

**2. Functions and Graphs Resources**

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

Functions & Graphs Theory Guide | HERE |

Functions & Graphs HSN Summary Notes | HERE |

**3. MIA Text Book Questions – Functions and Graphs**

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 _______________ |

Sketching Modulus Function y = |x| | Page 66 | Exercise 5.2 | Q1-9 | HERE |

Inverse Functions | Page 67 | Exercise 5.3 | Q1a,c,e,g,i,2a,c,e,3 | HERE |

Odd & Even Functions | Page 74 | Exercise 5.8 | Q1a-l | HERE |

Vertical Asymptotes & Behaviour | Page 75 | Exercise 5.9 | Q1a-f | HERE |

Horizontal & Oblique Asymptotes | Page 76 | Exercise 5.10 | Q1a,b,f,g,k,l | HERE |

Sketching Graphs | Page 77 | Exercise 5.11 | Q1a,c,e,i,k | HERE |

**4. About Sketching Graphs
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For a more detailed explanation, please first read the Theory Guides above.

**Steps**

- Find where the curve cuts the x and y-axis
- Find the vertical asymptote
- Find the nature of the vertical asymptote (go either side of the VA)
- Find the non-vertical asymptote
- Find the nature of the non-vertical asymptote (Note : For f(x) = m(x)/n(x), if degree of m(x) ≥ n(x) long division is required)
- Sketch the graph

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**Example**

Sketch y = 1/x

Cuts x and y-axis

Since we cannot divide by zero, the curve does not cut the x or y-axis

Vertical asymptotes of y = 1/x

Look at the denominator

Since x cannot be zero then y is undefined

Therefore there is a vertical asymptote at x = 0

Behaviour either side of the V.A.

If y is +’ve (x > 0) then there is positive infinity (approaches from right)

If y is -‘ve (x < 0) then there is negative infinity (approaches from left)

or, use a table below and go either side of the asymptote:

**y = 1/x**

x | - 0.1 (left of asymptote) ____________________________ | 0 (asymptote) ______________________ | + 0.1 (right of asymptote) ____________________________ |

y | -'ve value | +'ve value | |

Curve | Approaches asymptote from left | Approaches asymptote from right |

.Non-vertical asymptotes of y = 1/x

Ask, is synthetic division required? (no – since degree of numerator **is not** ≥ degree of denominator)

As x ⇒ +/- ∞, y ⇒ 0

Therefore y = 0 is a horizontal asymptotes

Behaviour either side of the horizontal asymptote

Use the table below and go either side of the asymptote:

**y = 1/x**

x | + infinity | - infinity |

y | +'ve value | -'ve infinity |

Curve | Approaches asymptote from above | Approaches asymptote from below |

Sketch

Analysis of both the vertical and horizontal asymptotes above, the sketch below can be drawn:

**Note 2**

Given y = (x + 3)/((x + 2)

Synthetic division required since the degree of numerator ≥ degree of denominator)

Here we divide to get:

y = 1 + 1/(x + 2)

Horizontal asymptote y = 1

Vertical asymptote x = -2

**Exam Question**

**Source: SQA AH Maths Paper 2001 Question A8**

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**5. 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 _________________ |

2018 | Advanced Higher | Exam Paper | Marking Scheme |

2017 | Advanced Higher | Exam Paper | Marking Scheme |

2016 | Advanced Higher | Exam Paper | Marking Scheme |

2016 | AH Specimen | Specimen | |

2016 | AH Exemplar | Exemplar | Exemplar Guidance |

2015 | Advanced Higher | Exam Paper | Marking Scheme |

2014 | Advanced Higher | Exam Paper | Marking Scheme |

2013 | Advanced Higher | Exam Paper | Marking Scheme |

2012 | Advanced Higher | Exam Paper | Marking Scheme |

2011 | Advanced Higher | Exam Paper | Marking Scheme |

2010 | Advanced Higher | Exam Paper | Marking Scheme |

2009 | Advanced Higher | Exam Paper | Marking Scheme |

2008 | Advanced Higher | Exam Paper | Marking Scheme |

2007 | Advanced Higher | Exam Paper | Marking Scheme |

2006 | Advanced Higher | Exam Paper | Marking Scheme |

2005 | Advanced Higher | Exam Paper | Marking Scheme |

2004 | Advanced Higher | Exam Paper | Marking Scheme |

2003 | Advanced Higher | Exam Paper | Marking Scheme |

2002 | Advanced Higher | Exam Paper | Marking Scheme |

2001 | Advanced Higher | Exam Paper |

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**6. 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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**7. ****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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**8. 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 Sketching Graphs 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.

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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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Topic Links

Free resources to dozens of AH Maths topics are available by clicking on any of the links to the right