This section delves deeper into quadratic equations, introducing the discriminant and its role in determining the nature of roots. It also covers completing the square and solving hidden quadratics. The page concludes with an introduction to simultaneous equations.

Vocabulary: The discriminant is the expression b² - 4ac in a quadratic equation ax² + bx + c = 0.

Example: For the equation 3x² + 1 - 8 × 3^x + 27 = 0, we can rewrite it as a quadratic in terms of 3^x: 3$3^x$² - 8$3^x$ + 27 = 0.

Highlight: The discriminant helps determine the nature of roots: b² - 4ac > 0 indicates two real roots, b² - 4ac = 0 suggests one real root, and b² - 4ac < 0 means no real roots.

The page also introduces the quadratic formula and explains how to use it to solve quadratic equations. This is a crucial tool for A Level Maths students, often appearing in exam questions.

This page covers inequalities, including quadratic inequalities, and introduces polynomials and algebraic division. It explains how to solve and represent inequalities graphically, which is a common topic in A Level Maths questions and answers.

Definition: A polynomial is an expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents.

Example: To solve the quadratic inequality 2$x² - x - 2$ > 0, first solve the equation 2$x² - x - 2$ = 0, then plot the results on a number line to determine the solution.

Highlight: When representing inequalities on a graph, use solid lines for ≤ or ≥, and dotted lines for < or >.

The page also covers algebraic division, an essential technique for factoring higher-degree polynomials and solving more complex equations.

This section focuses on the Factor Theorem and its application in factorising polynomials. It provides a step-by-step guide to using the Factor Theorem in combination with algebraic division to fully factorise polynomial expressions.

Definition: The Factor Theorem states that $x - p$ is a factor of f$x$ if and only if f$p$ = 0.

Example: To factorise x³ + 4x² - 11x - 10, first find a factor using the Factor Theorem, then use algebraic division to find the remaining quadratic factor.

Highlight: The Factor Theorem is a powerful tool for factorising higher-degree polynomials, which is crucial for solving complex equations in A Level Maths.

The page also covers algebraic fractions, explaining how to simplify them by factorising both the numerator and denominator and cancelling common factors.

This page introduces improper algebraic fractions and their division process. It also covers graphing functions, including finding turning points, y-intercepts, and x-intercepts (roots).

Vocabulary: An improper algebraic fraction is one where the degree of the numerator is greater than or equal to the degree of the denominator.

Example: To divide the improper fraction $x² + 3x² - 4x + 2$ ÷ $x - 1$, perform polynomial long division to get a quotient and remainder.

Highlight: When graphing functions, remember that y-intercepts are found by setting x = 0, while x-intercepts $roots$ are found by setting y = 0.

The page provides a comprehensive guide to analysing function graphs, which is essential for understanding the behavior of various functions in A Level Maths.

This section covers reciprocal graphs and solving equations graphically. It explains the characteristics of reciprocal graphs and provides examples of how to solve equations by finding the intersection points of graphs.

Definition: A reciprocal graph is the graph of a function in the form y = k/x, where k is a constant.

Example: To solve the equation 2x³ + 5x² + 2x = 0 graphically, plot y = 2x³ + 5x² + 2x and y = 0, and find their points of intersection.

Highlight: Reciprocal graphs always have asymptotes at x = 0 and y = 0, and their shape depends on whether k is positive or negative.

The page also covers solving more complex equations graphically by finding the intersection points of two or more graphs, a technique often used in A Level Maths exam questions.

This page introduces proportional relationships and functions, including direct proportion, inverse proportion, and composite functions. It also covers function notation, domain and range, and mapping diagrams.

Vocabulary: The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

Example: For the function f$x$ = x², with domain -5 ≤ x ≤ 5, the range would be 0 ≤ y ≤ 25.

Highlight: Composite functions involve applying one function after another, often written as f(g(x)).

The page also introduces the concepts of one-to-one and many-to-one functions, which are important for understanding function behavior and invertibility.

This section focuses on modulus functions and their graphs. It explains how to sketch modulus function graphs and solve modulus equations. The page also covers function transformations, including translations and stretches.

Definition: The modulus function |x| returns the non-negative value of x without regard to its sign.

Example: To sketch y = |f$x$|, first sketch y = f$x$, then reflect any part of the graph below the x-axis in the x-axis.

Highlight: When solving modulus equations like |2x - 4| = |x - 1|, consider both positive and negative cases for each modulus expression.

The page provides detailed explanations of various function transformations, which are crucial for understanding how changes to function equations affect their graphs.

The final page of the document continues the discussion on function transformations, focusing on translations and stretches. It provides examples of how these transformations affect function graphs.

Vocabulary: A translation moves every point of a graph by the same distance in a given direction, while a stretch enlarges or shrinks the graph by a certain factor.

Example: The graph of y = f$x$ + 1 is a vertical translation of y = f(x) by 1 unit upwards.

Highlight: Horizontal translations work in the opposite direction to what you might expect: y = f$x + a$ shifts the graph a units to the left.

The page concludes with a summary of how different transformations affect function graphs, providing a comprehensive overview of this important topic in A Level Maths.

This page introduces fundamental algebraic concepts and techniques crucial for AQA A Level Maths students. It covers laws of indices, manipulation of surds, and quadratic graphs. The page also explains how to find roots, turning points, and y-intercepts of quadratic equations.

Definition: Laws of indices are rules for simplifying expressions involving powers, such as a^m × a^n = a^$m+n$.

Example: To find the roots of a quadratic equation y = ax² + bx + c, you can either factorise or use the quadratic formula.

Highlight: The shape of a quadratic graph depends on the sign of 'a' in the equation y = ax² + bx + c. Positive 'a' results in a U-shaped graph, while negative 'a' produces an inverted U-shape.

The page also introduces the concept of rationalising the denominator when dealing with surds, which is an important technique in simplifying algebraic expressions.