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Curve Sketching is an incredibly useful tool in mathematical problem solving, as well as an opportunity to improve and test your algebraic understanding.
As you study mathematics through school and college to degree level, your algebraic skills will be increasingly tested. In order to become a strong mathematician, you need to understand what the algebra is telling you. Curve Sketching is often examinable and can be a challenging topic to master due to the multitude of techniques that need to be learnt. This course is here to help.
Computer programs like Autograph, Desmos, Maple and Matlab can all plot curves, but understanding why a curve behaves the way it does relies on your understanding of algebra and calculus. Using techniques that we will learn on this course, you will be able to successfully sketch complicated functions and learn about the behaviour of different graphs.
The course is structured so that you will learn about Graph Transformations and Differentiation and its uses in the initial sections. You will not need to have met these concepts before. I go through Differentiation from its basics, through the derivatives of different functions, and up to the Chain Rule, Product Rule and Quotient Rule.
We then start Sketching, and within this we will learn many different techniques along the way.
Linear Graphs:
Quadratic Graphs:
Cubics and Higher Polynomials:
Rational Functions:
Trigonometric Functions:
Exponential and Logarithmic Functions:
Modulus Functions:
Each of these sections is introduced from scratch and involves several worked examples and exercises for you to complete. There are several quizzes to try along the way to test your understanding, and if there are any problems please do not hesitate to start a discussion and ask for help.
With over 100 lectures and 13 hours of content, this course is perfect for anybody studying a Calculus course or ALevel Mathematics, or for those wanting to test and improve their mathematical ability before tackling a Mathematicsrelated undergraduate degree course at university.
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Section 1: Introductions  

Lecture 1 
Who is this course for?
Preview

01:16  
Lecture 2 
What will be covered on this course and what do I need to know?
Preview

02:57  
Lecture 3 
How will my progress be assessed?
Preview

01:15  
Lecture 4 
Sketching vs Plotting  what is the difference?
Preview

04:01  
Lecture 5 
Introducing the "Algebra Skills Practice" Quiz
Preview

01:03  
Quiz 1  8 questions  
This quiz will go through several different algebra skills that you will want to be good at before continuing with this course. If you find the quiz too challenging, this course will likely not be at the right level for you at the moment. 

Section 2: Transformations  
Lecture 6 
Introducing Function Notation

03:35  
Lecture 7 
EXERCISE: Using Function Notation

03:58  
Lecture 8 
Introducing the "Using Function Notation for Substitution" Quiz

00:17  
Quiz 2 
Using Function Notation for Substitution

5 questions  
Lecture 9 
Introducing Transformations

00:36  
Lecture 10 
Introducing Translations

03:41  
Lecture 11 
Introducing Stretches in the x and ydirection

04:28  
Lecture 12 
Introducing Reflections in the x and yaxes

04:10  
Lecture 13 
EXERCISE: Describing Transformations

05:43  
Lecture 14 
Introducing the "Transformations" Quiz

00:18  
Quiz 3 
Transformations

8 questions  
Section 3: Introducing Differentiation  
Lecture 15 
Differentiation: before we begin...

01:51  
Lecture 16 
Introducing Differentiation

04:58  
Lecture 17 
Differentiating Linear and Constant Terms

02:01  
Lecture 18 
Differentiating ax^n

03:07  
Lecture 19 
Differentiating Polynomials

01:28  
Lecture 20 
EXERCISE: Differentiating Polynomials

03:26  
Lecture 21 
Introducing the "Differentiating Polynomials" Quiz

00:23  
Quiz 4 
Differentiating Polynomials

8 questions  
Lecture 22 
Differentiating sin(x) and cos(x)

02:15  
Lecture 23 
EXERCISE: Differentiating sin(x) and cos(x)

02:22  
Lecture 24 
Differentiating exp(x) and ln(x)

01:54  
Lecture 25 
EXERCISE: Differentiating exp(x) and ln(x)

02:55  
Lecture 26 
Introducing the "Differentiation so far" Quiz

00:23  
Quiz 5 
Differentiation so far

5 questions  
Lecture 27 
Introducing the Chain Rule

05:10  
Lecture 28 
Basic Examples of using The Chain Rule

10:14  
Lecture 29 
More Examples of using the Chain Rule

06:03  
Lecture 30 
EXERCISE: The Chain Rule

07:05  
Lecture 31 
Introducing The Product Rule

02:19  
Lecture 32 
Examples of using the Product Rule

04:53  
Lecture 33 
Examples of using the Product Rule with the Chain Rule

04:34  
Lecture 34 
EXERCISE: The Product Rule

07:08  
Lecture 35 
Introducing The Quotient Rule

02:43  
Lecture 36 
Examples of using the Quotient Rule

06:11  
Lecture 37 
Examples of using the Quotient Rule with the Chain Rule

03:51  
Lecture 38 
EXERCISE: The Quotient Rule

07:54  
Lecture 39 
Introducing the "Identifying which method to use" Quiz

00:56  
Quiz 6 
Identifying which method to use: Chain, Product or Quotient Rule?

5 questions  
Section 4: Using Differentiation  
Lecture 40 
Introducing Stationary Points

03:42  
Lecture 41 
An Example of finding Stationary Points for a Polynomial

04:06  
Lecture 42 
EXERCISE 1: Stationary Points

07:06  
Lecture 43 
Examples of finding Stationary Points using the Chain Rule

10:44  
Lecture 44 
EXERCISE 2: Stationary Points

09:16  
Lecture 45 
An Example of finding Stationary Points using the Product Rule

03:53  
Lecture 46 
EXERCISE 3: Stationary Points

10:47  
Lecture 47 
An Example of finding Stationary Points using the Quotient Rule

03:19  
Lecture 48 
EXERCISE 4: Stationary Points

07:12  
Lecture 49 
Introducing the Second Derivative

06:15  
Lecture 50 
Examples of finding the Second Derivative

08:24  
Lecture 51 
EXERCISE: Finding the Second Derivative

07:10  
Lecture 52 
Local Minimums and Local Maximums

00:27  
Lecture 53 
Example of determining the Type of Stationary Point

08:26  
Lecture 54 
EXERCISE 1: Finding and Determining Types of Stationary Points

08:12  
Lecture 55 
EXERCISE 2: Finding and Determining Types of Stationary Points

05:59  
Lecture 56 
EXERCISE 3: Finding and Determining Types of Stationary Points

08:44  
Lecture 57 
EXERCISE 4: Finding and Determining Types of Stationary Points

05:46  
Section 5: Sketching Linear Graphs  
Lecture 58 
Introducing Sketching Linear Graphs

00:39  
Lecture 59 
Some important straight lines we need to know

02:18  
Lecture 60 
Transformations and the line y = x

03:18  
Lecture 61 
Finding where a Linear Graph crosses the coordinate axes

03:09  
Lecture 62 
Examples of Sketching Linear Graphs

04:53  
Lecture 63 
EXERCISE: Sketching Linear Graphs

06:44  
Lecture 64 
Introducing the "Linear Graphs" Quiz

00:20  
Quiz 7 
Linear Graphs

8 questions  
Section 6: Sketching Quadratic Graphs  
Lecture 65 
Introducing Sketching Quadratic Graphs

02:21  
Lecture 66 
Methods for Factorising Quadratics

04:29  
Lecture 67 
Using the Quadratic Formula

05:36  
Lecture 68 
Using the Discriminant

02:43  
Lecture 69 
Transformations of y = x^2

05:24  
Lecture 70 
Completing the Square

04:56  
Lecture 71 
An Alternative Method for Finding the Vertex of a Parabola

02:53  
Lecture 72 
Examples of Sketching Quadratic Graphs

13:20  
Lecture 73 
EXERCISE: Sketching Quadratic Graphs

11:46  
Lecture 74 
Introducing the "Quadratic Graphs" Quiz

00:20  
Quiz 8 
Quadratic Graphs

8 questions  
Section 7: Sketching Cubics and Higher Polynomials  
Lecture 75 
Introducing Sketching Cubics and Higher Polynomials

03:04  
Lecture 76 
Transformations of y = x^3

04:26  
Lecture 77 
Shapes of Cubics and Higher Polynomials

05:37  
Lecture 78 
Introducing the Remainder Theorem and the Factor Theorem

02:35  
Lecture 79 
Using the Remainder Theorem and the Factor Theorem

08:10  
Lecture 80 
Polynomial Division Method 1

06:50  
Lecture 81 
Polynomial Division Method 2

04:06  
Lecture 82 
Examples of Sketching Cubic Graphs

11:29  
Lecture 83 
EXERCISE: Sketching Cubic Graphs

14:15  
Lecture 84 
An Example of Sketching a Higher Polynomial

09:29  
Lecture 85 
EXERCISE: Sketching Higher Polynomials Part 1

07:30 
I am an ALevel Maths teacher from the UK, having achieved a Masters in Mathematics (MMATH) at the University of Southampton. I have taught ALevel Maths and Further Maths, as well as GCSE Maths, for six years. For the last two years, all of my classes have achieved 100% success, and last year my ALevel Maths and Further Maths classes achieved 70% and 100% high grades (As and Bs) respectively. I am a big fan of using teaching videos as part of instruction and have a growing following on YouTube as TLMaths.