Year 10 Mathematics Advanced Task 3 Study Guide Flashcards

Vocabulary flashcards covering key algebraic, trigonometric, and parabolic definitions, rules, and formulas from Chapters 5, 6, and 7.

Distributive Law

An algebraic expansion rule given by $a(b + c) = ab + ac$, where the outside term is multiplied by every term inside the bracket.

Difference of Two Squares

An algebraic identity stating $a^2 - b^2 = (a - b)(a + b)$.

Monic Quadratic

A quadratic expression or equation in which the coefficient of $x^2$ is equal to $1$.

Completing the Square

An algebraic technique that rewrites $x^2 + bx$ as $\left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2$.

Null Factor Law

The algebraic property stating that if $AB = 0$, then $A = 0$ or $B = 0$.

Quadratic Formula

The formula used to solve any standard quadratic equation $ax^2 + bx + c = 0$, given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Discriminant

The value given by $\Delta = b^2 - 4ac$ which determines the number of real solutions or x-intercepts of a quadratic function.

SOHCAHTOA

A mnemonic for right-triangle trigonometric ratios: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.

Angle of Elevation

An angle measured upward from a horizontal reference line.

Angle of Depression

An angle measured downward from a horizontal reference line.

True Bearing

A direction angle measured clockwise from North, written as a three-digit number (e.g. $045^\circ\text{T}$).

Sine Rule

A trigonometric relationship for non-right triangles stating $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$.

Cosine Rule

A trigonometric relationship for non-right triangles given by $c^2 = a^2 + b^2 - 2ab \cos(C)$ or $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$.

Trigonometric Area Formula

The formula used to calculate the area of a triangle given two sides and the included angle: $\text{Area} = \frac{1}{2}ab \sin(C)$.

Unit Circle Coordinates

The coordinates of a point on the unit circle represented as $(x, y) = (\cos(\theta), \sin(\theta))$.

Reference Angle

The acute angle formed between the terminal arm of an angle and the x-axis on the unit circle.

Turning-Point Form

The parabolic equation $y = a(x - h)^2 + k$, where $(h, k)$ represents the turning point and $x = h$ is the axis of symmetry.

Axis of Symmetry Formula

The vertical line equation $x = -\frac{b}{2a}$ for a parabola given in standard form $y = ax^2 + bx + c$.

Average Rate of Change

The gradient of the secant line passing through two points on a graph, calculated as $\frac{y_2 - y_1}{x_2 - x_1}$.

Instantaneous Rate of Change

The gradient of the tangent line to a curve at a single given point.

Direct Variation

A relationship where $y$ is proportional to $x$, represented by $y = kx$ where $k = \frac{y}{x}$ is constant.

Inverse Variation

A relationship where $y$ is inversely proportional to $x$, represented by $y = \frac{k}{x}$ or $xy = k$.