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- Maths and statistics
- Functions and graphs
Absolute values turn negative values into positive ones, and leave positive ones unchanged. Absolute value functions are useful for expressing situations where only magnitude matters, without regard to direction, like measuring distance, temperature differences, profit or loss diffeerences, and margin of error. Absolute values The absolute value of any number...
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- Maths and statistics
- Algebra
- Algebra
Let's combine your knowledge of fractions and algebra, and learn how to add and subtract algebraic fractions. By learning to find common denominators and simplify expressions, you'll enhance your ability to work with complex algebraic fractions, a skill crucial for solving equations and tackling advanced mathematical challenges. Video tutorial –...
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- Maths and statistics
- Algebra
Algebra focuses on understanding and manipulating mathematical symbols and expressions. Here, you'll explore key techniques such as simplification, factorisation, and rearranging formulas, which are essential for solving equations and understanding mathematical relationships. Algebraic operations Algebraic substitution Expansion Factorisation Adding and subtracting algebraic fractions Multiplying and dividing algebraic fractions Transposing formulas...
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- Maths and statistics
- Algebra
- Algebra
Algebraic operations are critical to many fields of study. In science, they are used to calculate forces to accelerate a car or determine the rate of a chemical reaction. In engineering, they can be used to design safe and efficient systems. They also underpin algorithms and data structures, which are...
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- Maths and statistics
- Functions and graphs
Circular functions produce periodic wave-like graphs and are foundational in trigonometry. They have broad applications across physics, engineering, and signal processing, where modelling repetitive phenomena and wave patterns is essential. Use this resource to explore circular functions. Circular functions include sine, cosine, and tangent. They describe relationships involving angles and...
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- Maths and statistics
- Trigonometry
Trigonometric functions such as sin, cos and tan are usually defined as the ratios of sides in a right-angled triangle. These ratios can be extended to angles greater than \(90^{\circ}\), using angles in a unit circle. Circular functions have applications in many STEM disciplines, such modelling oscillating motion in engineering,...
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- Maths and statistics
- Hyperbolic functions
The differentiation and integration of hyperbolic functions allow us to understand how these functions change and accumulate values which teaches us the essential tools for analysing complex mathematical and real-world systems. Keep reading to learn about these functions and put your new knowledge to the test with some exercises. See...
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- Maths and statistics
- Algebra
- Algebra
Brackets often appear in algebraic expressions and equations, but what do you do with them? Whatever is around the brackets affects everything inside the brackets, and being able to remove them helps set the stage for solving equations. The process of doing this is called expansion. Brackets in algebra Brackets...
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- Maths and statistics
- Functions and graphs
Simple functions and relations can be transformed into more complicated functions. Seeing how changes in equations affect their visual representation will give you a deeper understanding of functions. This is useful in modelling real-world situations and solving complex maths problems. Use this resource to learn about transformations of graphs. Summary...
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- Maths and statistics
- Functions and graphs
Hybrid functions combine different types of functions. They are useful for modelling real-world scenarios that have different behaviours under different conditions, like tax brackets or shipping rates. Other examples include analysing frictional forces in physics, designing profiles and shapes for vehicles, and modelling free-fall acceleration. Use this resource to learn...
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- Maths and statistics
- Hyperbolic functions
The hyperbolic functions are analogous to the circular (trigonometric) functions and are widely used in engineering, science and mathematics. This module introduces hyperbolic functions, their graphs and similarities to the circular functions. Whereas circular functions are defined on a unit circle, the hyperbolic functions are defined on a hyperbola. Hyperbolic...
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Information about functions and graphs to improve your maths skills in these areas. HF1 Hyperbolic functions This module introduces hyperbolic functions, their graphs and similarities to the circular functions. HF2 Derivatives and integrals of hyperbolic functions The hyperbolic functions are widely used in engineering, science and mathematics. HF3 Inverse hyperbolic...
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- Maths and statistics
- Functions and graphs
Interval notation helps to clearly express a continuous range of numbers in a simple, concise way. It is handy in many ways, such as to define acceptable weight limits for a machine, indicating a valid range of measurements in scientific experiments, and specifying temperature ranges in which a machine is...
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- Maths and statistics
- Functions and graphs
One-to-one functions can be inverted so that the inputs and outputs are reversed. They are crucial where we need effective reversal of processes and calculations, like undoing equations in physics, computer graphics transformations, or decrypting data. Use this resource to learn about inverse functions and how to graph them. If...
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- Maths and statistics
- Hyperbolic functions
Providing a function is one to one, it is possible to find an inverse function. This module discusses inverse hyperbolic functions, which are used in advanced calculus for integration and in the solution of differential equations. See Hyperbolic functions for a list of definitions. Inverse hyperbolic functions may also be...
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- Maths and statistics
- Functions and graphs
Linear functions are the most basic type of function. They are used extensively across STEM and other disciplines, especially to model the rate of physical, biological or economic change. Use this resource to learn about linear functions and graphs. Linear functions create a straight line when graphed. They have the...
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- Maths and statistics
- Algebra
- Algebra
Let's build on your understanding of basic fraction operations and algebraic expressions by looking at multiplying and dividing algebraic fractions. By mastering these skills, you'll be able to simplify and manipulate complex fractions, which is essential for solving more advanced equations and applying algebra to a variety of contexts. Video...
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- Maths and statistics
- Algebra
- Algebra
Partial fractions are used to simplify more complex algebraic fractions into simpler components. The ability to find partial fractions is used widely in calculus, like when solving problems involving Laplace transforms in engineering, analysing frequency response in electronics, and simplifying expressions in mathematical models. Use this resource to learn to...
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- Maths and statistics
- Trigonometry
Pythagoras’ theorem shows the relationship between the sides of a right-angled triangle. Knowing the length of two sides of a right-angled triangle, the length of the third side can be calculated. Understanding Pythagoras' theorem helps you solve problems in fields such as architecture, engineering, and computer graphs, where precise calculations...
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- Maths and statistics
- Functions and graphs
Quadratic functions have a distinct parabolic shape. Graphs of quadratic functions have applications in a wide range of fields, like modelling projectile motion, optimising business revenue, analysing population growth and designing bridges and arches. Use this resource to learn about parabolas and how to sketch them. A quadratic function has...
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- Maths and statistics
- Algebra
- Algebra
Let's take a look at algebraic fractions where the denominator is a quadratic expression. Such fractions are common in mathematics and engineering. Before you continue with the content on this page, it is important that you are confident about identifying common factors, factorising quadratics using common factors and the difference...
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- Maths and statistics
- Functions and graphs
Understanding relations and functions is crucial for exploring more advanced topics in math and their applications in various scientific and engineering areas. Use this resouce to learn about what functions and relations are. Relations A relation is a set of ordered pairs. For example, \((1,2),(2,6),(3,4),(x,y)\) are ordered pairs. If the...
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- Maths and statistics
- Trigonometry
Explore the relationships between the angles and side lengths of a right-angled triangle. You will learn how to use trigonometric ratios to find unknown angles and lengths, enhancing your ability to solve geometric problems. Mastering this concept is vital for fields such as engineering, physics, and architecture, where precise calculations...
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- Maths and statistics
- Trigonometry
A calculator may only give you one answer to an inverse trigonometry question between \(0\) and \(90^{\circ}\). When you use the unit circle, you can visualise the many different solutions for the angle. Understanding this concept will help you in many STEM fields, such as satellite navigation in physics, and...
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- Maths and statistics
- Trigonometry
Like the sine rule, the cosine rule comes in handy for triangles that are not right-angled. Use this resource to learn about the cosine rule, how it is applied to find unknown sides and angles, and its importance in solving complex geometric problems involving any type of triangle. The cosine...
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- Maths and statistics
- Trigonometry
The trigonometric ratios are handy for real-life situations where right-angled triangles are involved, but what about triangles that are not right-angled? We use the sine rule. Use this resource to learn what the sine rule is and how it is applied. The sine rule shows that the ratio of the...
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- Maths and statistics
- Trigonometry
Angles are commonly measured in degrees but sometimes, it is useful to define them in terms of the length around the unit circle. This resource introduces radians as a measure of angle. Being able to use radians is crucial for solving problems in engineering, such as analysing the motion of...
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- Maths and statistics
- Algebra
- Algebra
The transposition of formulas involves rearranging equations to solve for a specific pronumeral or variable. You need to be able to first isolate your desired variable in order to solve problems various fields. Mastering transposition will enhance your ability to manipulate and understand mathematical relationships effectively. Video tutorial – transposition...
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- Maths and statistics
- Trigonometry
Trigonometric identities are essential tools for simplifying expressions and solving equations in mathematics. They are used in a wide range of disciplines, including architecture and engineering, physics, computer graphics, astronomy and medical imaging. Use this resource to learn about some of the fundamental identities, double angle formulas, sums and difference,...
