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.

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 >.

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.

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.

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.

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$x$).

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.

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.

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.