Stage 5 Mathematics Comprehensive Study Guide: Linear and Non-Linear Relationships

Assessment Context and Academic Framework

• Institutional Information:

  • School: St Mary and St Mina’s Coptic Orthodox College.
  • Motto: "In Whom are hidden all the treasures of Wisdom and Knowledge" (Colossians 2:3).

• Task Details:

  • Notification Year: 2026 Stage 5: Year 10 Paths & Year 9 Accelerated Mathematics.
  • Assessment Task Number: Task no. 3.
  • Format: In-class written examination.
  • Weighting: $20\%$
  • Task Total: $20\%$
  • Administered Date: Week 4, Monday (10/08/2026).
  • Task Date: Week 6, Friday (28/08/2026).
  • Examination Duration: $50\,\text{minutes}$ plus $5\,\text{minutes}$ reading time.
  • Equipment Required: Board Approved calculator, Ruler, Pencils, Pens, Eraser.

• Syllabus Outcomes Assessed:

  • MA5-EQU-P-02: A student solves linear equations of more than 3 steps, monic and non-monic quadratic equations, and linear simultaneous equations (Path: Adv).
  • MA5-EQU-P-01: A student solves monic quadratic equations, linear inequalities and cubic equations of the form $ax^3 = k$ (Path: Adv).
  • MA5-LIN-P-01: A student describes and applies transformations, the midpoint, gradient/slope and distance formulas, and equations of lines to solve problems (Path: Adv).
  • MA5-NLI-P-01: A student interprets and compares non-linear relationships and their transformations, both algebraically and graphically (Path: Adv).

• Curriculum Scope:

  • Textbook Reference: Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition.
  • Covered Chapters: Chapter 5 (Simultaneous Equations), Chapter 6 (Quadratics), Chapter 7 (Linear Relationships), Chapter 8 (Non-Linear Relationships).

• Assessment Performance & Administrative Criteria:

  • Marking Expectations: Showing correct solutions, clear and logical working, acceptable proof formats with valid reasons, detailed explanations, precise mathematical terminology, neatly drawn diagrams and graphs using appropriate equipment (sufficiently large and appropriately labelled), correct units, and stated extents of rounding.
  • Illness/Misadventure Policy: An absence requires a medical certificate and a misadventure form submitted to the head of faculty on the first day of return. Unexcused late submissions incur a penalty of $20\%$ per day late, and the task must still be completed.
  • Malpractice Policy: Assessment malpractice or plagiarism results in a zero mark and compulsory re-submission of the task.

Simultaneous Linear Equations

• System of Equations Definitions:

  • A system of simultaneous linear equations consists of two or more algebraic linear equations containing common variables. Solving a system means identifying the coordinate pair $(x, y)$ that concurrently satisfies all equations in the system.

• Graphical Solutions and Number of Intersections:

  • Graphing two linear equations on a Cartesian plane yields the point of intersection as the solution set.
  • Intersecting Lines: Gradients differ ($m_1 \neq m_2$). Exactly one unique solution exists.
  • Parallel Lines: Gradients are identical ($m_1 = m_2$) and $y$-intercepts differ ($c_1 \neq c_2$). Lines never meet; zero solutions exist.
  • Coincident Lines: Gradients are identical ($m_1 = m_2$) and $y$-intercepts are identical ($c_1 = c_2$). One line lies directly on top of the other; infinitely many solutions exist.
  • Perpendicular Intersections: Lines intersect at right angles ($90^\circ$) when $m_1 \times m_2 = -1$ or $m_1 = -\frac{1}{m_2}$. Exactly one unique solution exists.

• Algebraic Solution Methods:

  • Substitution Method:
  • Best suited when one variable is explicitly expressed as the subject in one of the equations (e.g., $y = 2x - 1$).
  • Process: Replace that variable in the secondary equation with its algebraic equivalent expression to form a single-variable equation. Solve for the remaining variable, then back-substitute to find the second variable.
  • Equating Method: A variant of substitution used when both equations express the same variable as the subject (e.g., $y = 5x - 8$ and $y = -3x + 16$). Equate the right-hand sides: $5x - 8 = -3x + 16$.
  • Elimination Method:
  • Best suited when equations are written in standard form ($ax + by = c$).
  • Process: Add or subtract equations to eliminate one variable.
  • If coefficients of a variable are identical in magnitude and sign, subtract one equation from the other.
  • If coefficients are identical in magnitude but opposite in sign, add the equations.
  • If coefficients differ, multiply one or both equations by integer constants to make the magnitude of one variable's coefficient identical in both equations before adding or subtracting.
  • General Algebraic Solution Formula:
  • For the system $ax + by = e$ and $cx + dy = f$:       $y = \frac{ce - af}{bc - ad}$$x = \frac{de - bf}{ad - bc}$
  • A unique solution exists if and only if $ad - bc \neq 0$ (or $bc - ad \neq 0$).

• Applications and Worded Problems:

  • Step 1: Define unknown quantities explicitly using pronumerals (e.g., let $x = \text{cost of a short}$, $y = \text{cost of a T-shirt}$).
  • Step 2: Formulate two distinct linear equations based on given problem constraints.
  • Step 3: Solve the simultaneous equations using graphical, substitution, or elimination methods.
  • Step 4: Write the final answer in full sentences including context-specific units (e.g., $\$50$, $15\,\text{nuts}$, $10\,\text{bolts}$).
  • Step 5: Verify the values by substituting back into the original problem statements.

Quadratic Equations

• General Structure:

  • A quadratic equation is an equation of the second degree, expressed in standard form as:     $ax^2 + bx + c = 0$     where $a \neq 0$, and $a, b, c \in \mathbb{R}$.

• Algebraic Solutions:

  • Null Factor Law:
  • If $a \times b = 0$, then $a = 0$, $b = 0$, or both $a = 0$ and $b = 0$.
  • Requires factorising the quadratic expression first (using common factors, difference of two squares $a^2 - b^2 = (a-b)(a+b)$, monic trinomial factor pairs, or non-monic grouping).
  • Completing the Square:
  • Used when a quadratic expression cannot be easily factorised using rational numbers.
  • Requires the coefficient of $x^2$ to be $1$ ($a = 1$). If $a \neq 1$, divide every term by $a$.
  • Formula:       $x^2 \pm px = \left(x \pm \frac{p}{2}\right)^2 - \left(\frac{p}{2}\right)^2$
  • Procedure: Add and subtract $\left(\frac{b}{2}\right)^2$, convert the quadratic trinomial into a perfect square, write as a difference of two squares, and apply the Null Factor Law.
  • Equations Reducible to Quadratic Form:
  • Equations containing higher powers such as $ax^4 + bx^2 + c = 0$ or $a(f(x))^2 + b(f(x)) + c = 0$.
  • Perform a substitution: Let $u = x^2$, transforming the expression into $au^2 + bu + c = 0$.
  • Solve for $u$, reject non-viable solutions (e.g., $u < 0$ when $u = x^2$ since $x^2 \ge 0$ for real $x$), then solve $x = \pm \sqrt{u}$.
  • The Quadratic Formula:
  • Derived via completing the square on $ax^2 + bx + c = 0$:       $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Alternative form of the quadratic formula:       $x = \frac{2c}{-b \pm \sqrt{b^2 - 4ac}}$

• Graphical Solutions:

  • The solutions (or roots) to $ax^2 + bx + c = 0$ correspond to the $x$-intercepts of the parabola $y = ax^2 + bx + c$.
  • The number of real solutions is determined by the number of times the curve intersects or touches the $x$-axis.

• The Discriminant:

  • The discriminant is the expression under the radical in the quadratic formula:     $\Delta = b^2 - 4ac$
  • $\Delta > 0$ and a perfect square: Two distinct rational solutions; the quadratic is factorisable over rational numbers.
  • $\Delta > 0$ and not a perfect square: Two distinct irrational solutions (surds).
  • $\Delta = 0$: One distinct rational solution (a repeated root); the parabola's vertex touches the $x$-axis.
  • $\Delta < 0$: No real solutions; the parabola does not touch or cross the $x$-axis.
  • Line and Parabola Intersection: Equate $y = ax^2 + bx + c$ and $y = mx + k$ to form $ax^2 + (b-m)x + (c-k) = 0$. Calculate $\Delta$ for this new equation to determine if the line is a secant (2 points, $\Delta > 0$), a tangent (1 point, $\Delta = 0$), or non-intersecting (0 points, $\Delta < 0$).

Linear Relationships and Coordinate Geometry

• Gradient Formula:

  • The gradient ($m$) defines the steepness and direction of a line through points $(x_1, y_1)$ and $(x_2, y_2)$:     $m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$

• Forms of Linear Equations:

  • Gradient-Intercept Form: $y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept $(0, c)$.
  • Point-Gradient Form: $y - y_1 = m(x - x_1)$, where $m$ is the gradient and $(x_1, y_1)$ is a known point.
  • Two-Point Form:     $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$
  • General Form: $ax + by + c = 0$, where $a, b, c \in \mathbb{Z}$ and $a \ge 0$

• Horizontal and Vertical Lines:

  • Horizontal Lines: Equation $y = c$. Gradient $m = 0$. Parallel to the $x$-axis.
  • Vertical Lines: Equation $x = a$. Gradient $m$ is undefined. Parallel to the $y$-axis.

• Parallel and Perpendicular Line Conditions:

  • Parallel Lines: $m_1 = m_2$ with $c_1 \neq c_2$.
  • Perpendicular Lines: $m_1 \times m_2 = -1$ or $m_2 = -\frac{1}{m_1}$.

• Distance Formula:

  • Derived from Pythagoras' theorem, the straight-line distance $d$ between $(x_1, y_1)$ and $(x_2, y_2)$ is:     $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula:

  • The midpoint $M(x, y)$ of a line segment connecting $(x_1, y_1)$ and $(x_2, y_2)$ is:     $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
  • Finding an unknown end point $(x_2, y_2)$ given midpoint $M(x, y)$ and end point $(x_1, y_1)$: $x_2 = 2x - x_1$ and $y_2 = 2y - y_1$

• Collinear Points:

  • Three points $A$, $B$, and $C$ are collinear (lie on the exact same straight line) if:     $m_{AB} = m_{BC}$

• Perpendicular Bisector:

  • A line that divides a line segment into two equal halves at a $90^\circ$ angle.
  • Steps to determine its equation:
  1. Calculate the midpoint $M\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ of the segment.
  2. Calculate the original line segment gradient $m_1 = \frac{y_2-y_1}{x_2-x_1}$.
  3. Determine the perpendicular gradient $m_2 = -\frac{1}{m_1}$.
  4. Substitute $m_2$ and midpoint $M$ into $y - y_M = m_2(x - x_M)$.

• Linear Inequalities and Half-Planes:

  • A linear inequality divides the Cartesian plane into half-planes.
  • Boundary Line Types:
  • Dashed line (broken): Used for strict inequalities ($<$ or $>$); points on the boundary are excluded.
  • Solid line (unbroken): Used for inclusive inequalities ($\le$ or $\ge$); points on the boundary are included.
  • Determining Required Regions:
  • Test a point not lying on the boundary line (typically $(0,0)$). Substitute into the inequality.
  • If the resulting statement is true, shade or mark the region containing that test point. If false, select the region on the opposite side of the boundary line.

Non-Linear Relationships

• Parabolas ($y = ax^2 + bx + c$):

  • Graphs of quadratic relationships are symmetrical curves named parabolas.
  • Concavity and Turning Points:
  • Concave up ($\cup$): Occurs when $a > 0$. Possesses a minimum turning point (vertex).
  • Concave down ($\cap$): Occurs when $a < 0$. Possesses a maximum turning point (vertex).
  • Transformations of $y = x^2$:
  • Dilation: $y = kx^2$. Stretches graph vertically by factor $k$. Narrower if $|k| > 1$; wider if $0 < |k| < 1$
  • Reflection: $y = -x^2$. Inverts the parabola across the $x$-axis.
  • Vertical Translation: $y = x^2 + c$. Shifts graph $c$ units up if $c > 0$, or $c$ units down if $c < 0$.
  • Horizontal Translation: $y = (x - b)^2$. Shifts graph $b$ units right if $b > 0$, or $b$ units left if $b < 0$.
  • Turning Point / Vertex Form:     $y = k(x - b)^2 + c$
  • Turning point coordinate: $(b, c)$.
  • Axis of symmetry equation: $x = b$
  • General Form:     $y = ax^2 + bx + c$
  • $y$-intercept: $(0, c)$.
  • Axis of symmetry equation: $x = -\frac{b}{2a}$
  • Vertex $x$-coordinate: $x_{TP} = -\frac{b}{2a}$. Substitute $x_{TP}$ back into the original equation to find the vertex $y$-coordinate.
  • Factored / Intercept Form:     $y = a(x - p)(x - q)$
  • $x$-intercepts: $(p, 0)$ and $(q, 0)$.
  • Axis of symmetry lies midway between intercepts: $x = \frac{p + q}{2}$.

• Exponential Graphs:

  • General form: $y = k(a)^{\pm x} + c$, where $a > 0$ and $a \neq 1$
  • Key Features of $y = a^x$:
  • $y$-intercept: $(0, 1)$.
  • Horizontal asymptote: Line $y = 0$ ($x$-axis). Graph approaches but never touches $y = 0$
  • Range: $y > 0$
  • Transformations:
  • Vertical translation: $y = a^x + c$ shifts horizontal asymptote to $y = c$; $y$-intercept becomes $(0, 1 + c)$.
  • Reflection in $x$-axis: $y = -a^x$ flips curve downwards; $y$-intercept $(0, -1)$.
  • Reflection in $y$-axis: $y = a^{-x} = \left(\frac{1}{a}\right)^x$ flips curve horizontally.

• Hyperbolas:

  • General form: $y = \frac{k}{x - b} + c$ or $(x - b)(y - c) = k$
  • A hyperbola is a discontinuous graph consisting of two separate symmetrical branches.
  • Asymptotes:
  • Vertical asymptote: Line $x = b$ (value where the denominator becomes zero, making the expression undefined).
  • Horizontal asymptote: Line $y = c$ (value the function approaches as $x \to \pm \infty$, since $\frac{k}{x - b} \neq 0$).
  • Transformations:
  • Base graph $y = \frac{1}{x}$: Asymptotes at $x = 0$ and $y = 0$. Branches in quadrants 1 and 3.
  • Reflection: If $k < 0$, branches flip into quadrants 2 and 4.

• Circles:

  • Definition: Set of all points in a plane that are at a fixed distance (radius $r$) from a fixed point (centre).
  • Standard Equation (Centre at Origin $(0,0)$):     $x^2 + y^2 = r^2$
  • Standard Equation (Centre at $(a, b)$):     $(x - a)^2 + (y - b)^2 = r^2$
  • General Circle Equation:     $x^2 + y^2 + ax + by + c = 0$     Convert to centre-radius form by completing the square on $x$-terms and $y$-terms independently.

• Simultaneous Linear and Non-Linear Equations:

  • Systems combining a linear equation ($y = mx + k$) and a non-linear equation (parabola, hyperbola, or circle) can yield 0, 1, or 2 real intersection points.
  • Solution Method: Substitute the linear expression into the non-linear equation to obtain a quadratic in one variable, solve for $x$, and back-substitute to find corresponding $y$-coordinates.