Graphs of Functions

Domain and range: The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values).
Function notation: Functions can be denoted with a letter of the alphabet followed by parentheses, e.g., f(x).
Increasing and decreasing functions: If the y-value increases as the x-value increases, the function is increasing. If the y-value decreases as the x-value increases, the function is decreasing.
Maximum and minimum points: A maximum point on a function is a point where the y-value is greater than the y-values of the points immediately to its left and right. A minimum point is where the y-value is smaller.
Odd and even functions: An even function is symmetrical about the y-axis, while an odd function is symmetrical about the origin.
Composite functions: Composite functions are functions composed of two or more simpler functions.

Modulus

Definition: The modulus of a real number is its absolute value. It is always positive or zero.
Properties: The modulus of a product is the product of the moduli, and the modulus of a quotient is the quotient of the moduli.
Modulus equations: These equations involve the absolute value of a variable expression. Solution may involve squaring both sides or splitting the equation into two separate cases.

Degree of cubics: A cubic equation is one where the highest power of the variable is 3.
Shape of cubic curves: All cubic curves have a single point of inflection. They can have either one or three roots.
Roots of cubics: The roots of a cubic lie either on the x-axis (real roots) or off the x-axis (complex roots).
Factorising cubics: Cubics can be factored into linear and quadratic terms by using synthetic division and the factor theorem.

Interpreting inequalities: Inequalities describe the relative size of different values. This can be visualised on a number line.
Solving inequalities: Solving inequalities often involves reasoning similar to solving equations, but the direction of the inequality sign must be preserved.
Quadratic inequalities: Quadratic inequalities can be solved using factorisation or the quadratic formula, but remember to consider when the quadratic expression is positive or negative.
Graphing inequalities: Inequality in two variables usually bound a region in the coordinate plane, which can be shaded or hatched on a graph.

Methods of solving: Simultaneous equations can be solved by substitution, elimination, or graphical methods.
Nonlinear simultaneous equations: These involve at least one non-linear equation and can typically be solved using substitution.
Applications: Simultaneous equations are used in many real-world applications including finance, physics and geometry.