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Factor Theorem
Factor theorem is the process of validating whether a factor is a factor of an equation. A factor is a number where the a % b = 0 . For example, $ x + 3 $ is a factor of $f x = x^2...
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Turning Point
To get the local turning point of a quadratic, you can either: 1. Complete the Square Completing The Square 2. Use differentiation anx Equation Remember that the $gradient = 0$ and...
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Comparing Coefficients
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GCSE Recap
Vector vs. Scaler A vector has both a magnitude and direction , whereas a scaler has only a magnitude .
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Vector to Unit Vector
To convert a vector to a unit vector, you need to: 1. Calculate the magnitude using the Pythagoras theorem 2. Divide both axes with the magnitude Demonstration TODO: Write iframe c...
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Binomial Expansion
TODO Write and explain techniques. Choose Example $$ \begin{array}{0} 1+ax ^10 = 1^10 + 1^9 {10\choose {1}} ax ^1 + 1^8 + {10\choose {2}} ax ^2 \dots \end{array} $$ Year 13 Techniq...
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Definite Integration and Limits
Limits can be represented as either $ x ^a {b}$ or $\int^a {b}x$. The limits represent both the upper and lower bounds of a definite integral the interval over which to calculate t...
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e - Exponent Constant
WARNING This document was partially AI generated, however adapted. $e$ is used as a base because it has the unique property where the gradient of $y = e^x$ is equal to $e^x$ itself...
mathsyear-1pure-mathschapter-14-exponentials-and-logarithms
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Introduction to Integration
Integration is the reverse of Differentiation. It uses the formula $y = \frac{k}{n+1}x^{n+1} + c$ as long as $n \not= 1$. It can be summarised in three steps: 1. Add one to the pow...
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Logarithms
A logarithm is the exponent to which a base must be raised to produce a given number. Equation $$ \begin{array}{0} a^x = b \\ x = \log {a}b \\ \end{array} $$ Usage Rules You cannot...
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Stationary Points
A stationary point on a curve is where the gradient is $0$, such as $ 0, 0 $ on $x^2 + 5x$ The Local Maximum and Minimum Building upon thi, there is local maximum and minimum point...
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(an)x^n Equation
$anx^n$ or $ax^n$ is an equation that can be used for differentiation. Example $$ \begin{array}{0} \end{array} $$ TODO TODO.
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Pascal's Triangle
WARNING This does not have any practical example. It is just a demonstration. See Binomial Expansion for technique. $$ \begin{array}{0} x + 2y ^2 = 1x^3 2y ^0 + 3x^2 2y + 3x 2y ^2...
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Negative and Fractional Indices
$$ \begin{array}{1} a^\frac{1}{m} = \sqrt m { a } \\ a^\frac{n}{m} = \sqrt m { a^n } \\ a^ m = \frac{1}{a^m} \\ a^0 = 1 \end{array} $$
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Calculating the Midpoint of a Straight Line
Formula This formula calculates the midpoint of both the $x$ and $y$ axes individually. $$ \begin{array}{1} \left \frac{x {1} + x {2}}{2}, \frac{y {1} + y {2}}{2} \right \end{array...
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Completing The Square
NOTE Revise. $$ a x + h ^2 + k $$ To get the co ordinates of the maximum/minimum turning point. $ x + h = 0$ or $ h$ provides the $x$ co ordinate. $k$ provides the $y$ co ordinate....
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Factor Theorem
Polynomial: $x^a$ Exponential: $a^x$ Factor theorem is when $A$ is a factor of $f x $ if $f a = 0$. Textbook Examples Question 1: Prove $ x + 4 $ is a factor. $$ \begin{array}{0} x...
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Polynomial Long Division
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Calculating the Gradient
$$ m = \frac{\Delta y}{\Delta x} $$
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Finding the Length
To find the length of a line, you can use Pythagoras' Theorem. $$ A^2 = B^2 + C^2 $$ as: $$ d = \sqrt{ x^2 x^1 ^2 + y^2 y^2 ^2 } $$
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Parallel Lines
Parallel lines are two lines which have the same gradient but never touch. Example $$ \begin{array}{1} y {1} = 5x + 2 \\ y {2} = 5x + 4 \end{array} $$ These linear lines are parall...
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Perpendicular Lines
Perpendicular lines are two lines which meet at 90°. If a line has the gradient of $m$, a line perpendicular will have the gradient of $ \frac{1}{m}$. If two lines are perpendicula...
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Sine, Cosine and Tangent Graphs
Sine and Cosine $\sin x $ and $\cos x $ are periodic functions with a period of 360° or $2\pi$ radians . Both sine and cosine graphs oscillate between 1 and 1. The sine graph start...
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