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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.
Cubics and Higher Polynomials:
Exponential and Logarithmic 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 A-Level Mathematics, or for those wanting to test and improve their mathematical ability before tackling a Mathematics-related undergraduate degree course at university.
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|Section 1: Introductions|
Who is this course for?Preview
What will be covered on this course and what do I need to know?Preview
How will my progress be assessed?Preview
Sketching vs Plotting - what is the difference?Preview
Introducing the "Algebra Skills Practice" QuizPreview
|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|
Introducing Function Notation
EXERCISE: Using Function Notation
Introducing the "Using Function Notation for Substitution" Quiz
Using Function Notation for Substitution
Introducing Stretches in the x and y-direction
Introducing Reflections in the x and y-axes
EXERCISE: Describing Transformations
Introducing the "Transformations" Quiz
|Section 3: Introducing Differentiation|
Differentiation: before we begin...
Differentiating Linear and Constant Terms
EXERCISE: Differentiating Polynomials
Introducing the "Differentiating Polynomials" Quiz
Differentiating sin(x) and cos(x)
EXERCISE: Differentiating sin(x) and cos(x)
Differentiating exp(x) and ln(x)
EXERCISE: Differentiating exp(x) and ln(x)
Introducing the "Differentiation so far" Quiz
Differentiation so far
Introducing the Chain Rule
Basic Examples of using The Chain Rule
More Examples of using the Chain Rule
EXERCISE: The Chain Rule
Introducing The Product Rule
Examples of using the Product Rule
Examples of using the Product Rule with the Chain Rule
EXERCISE: The Product Rule
Introducing The Quotient Rule
Examples of using the Quotient Rule
Examples of using the Quotient Rule with the Chain Rule
EXERCISE: The Quotient Rule
Introducing the "Identifying which method to use" Quiz
Identifying which method to use: Chain, Product or Quotient Rule?
|Section 4: Using Differentiation|
Introducing Stationary Points
An Example of finding Stationary Points for a Polynomial
EXERCISE 1: Stationary Points
Examples of finding Stationary Points using the Chain Rule
EXERCISE 2: Stationary Points
An Example of finding Stationary Points using the Product Rule
EXERCISE 3: Stationary Points
An Example of finding Stationary Points using the Quotient Rule
EXERCISE 4: Stationary Points
Introducing the Second Derivative
Examples of finding the Second Derivative
EXERCISE: Finding the Second Derivative
Local Minimums and Local Maximums
Example of determining the Type of Stationary Point
EXERCISE 1: Finding and Determining Types of Stationary Points
EXERCISE 2: Finding and Determining Types of Stationary Points
EXERCISE 3: Finding and Determining Types of Stationary Points
EXERCISE 4: Finding and Determining Types of Stationary Points
|Section 5: Sketching Linear Graphs|
Introducing Sketching Linear Graphs
Some important straight lines we need to know
Transformations and the line y = x
Finding where a Linear Graph crosses the coordinate axes
Examples of Sketching Linear Graphs
EXERCISE: Sketching Linear Graphs
Introducing the "Linear Graphs" Quiz
|Section 6: Sketching Quadratic Graphs|
Introducing Sketching Quadratic Graphs
Methods for Factorising Quadratics
Using the Quadratic Formula
Using the Discriminant
Transformations of y = x^2
Completing the Square
An Alternative Method for Finding the Vertex of a Parabola
Examples of Sketching Quadratic Graphs
EXERCISE: Sketching Quadratic Graphs
Introducing the "Quadratic Graphs" Quiz
|Section 7: Sketching Cubics and Higher Polynomials|
Introducing Sketching Cubics and Higher Polynomials
Transformations of y = x^3
Shapes of Cubics and Higher Polynomials
Introducing the Remainder Theorem and the Factor Theorem
Using the Remainder Theorem and the Factor Theorem
Polynomial Division Method 1
Polynomial Division Method 2
Examples of Sketching Cubic Graphs
EXERCISE: Sketching Cubic Graphs
An Example of Sketching a Higher Polynomial
EXERCISE: Sketching Higher Polynomials Part 1
I am an A-Level Maths teacher from the UK, having achieved a Masters in Mathematics (MMATH) at the University of Southampton. I have taught A-Level 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 A-Level 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.