Year 10 Advanced Mathematics: Linear Relationships Study Notes

Assessment Overview

• Subject: Year 10 Advanced Mathematics
• Task Title: Linear Relationships
• Task Type: In-class examination
• Date of Assessment: Wednesday 26th August (Week 6, Term 3)
• Time Allocation: Period 3 (Duration: 1 period)
• Equipment Requirements:
  • Students must bring their own pens, ruler, and calculator.
  • Borrowing of equipment is strictly prohibited.
  • Prohibited Items: Phones, smartwatches, and computers are not allowed during the assessment.
• Provided Materials:
  • There are no study notes allowed.
  • A formula sheet will be provided covering gradient, distance, and midpoint formulas only.
• Examination Structure:
  • Multiple choice questions
  • Short answer questions
  • Long answer questions
  • Questions requiring working out to be shown
  • Questions involving the creation or interpretation of graphs and drawings

Fundamental Line Calculations

Midpoint of a Straight-Line Interval

• The midpoint is the exact center of a line segment connecting two points.
• It can be determined via graphical methods (plotting and finding the center) or geometrical methods (using the formula).
• Midpoint Formula:$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Distance Between Two Points

• The distance represents the length of the interval between two coordinates on a Cartesian plane.
• Distance Formula:$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Gradient of an Interval

• The gradient ($m$) measures the steepness and direction of a line.
• Graphical Method: Construct a right-angled triangle between two points and determine the ratio of vertical change to horizontal change.
• Gradient Formulas:$m = \frac{\text{rise}}{\text{run}}$$m = \frac{y_2 - y_1}{x_2 - x_1}$

Properties and Classifications of Gradients

• Positive Gradient: The line slopes upwards from left to right ($m > 0$).
• Negative Gradient: The line slopes downwards from left to right ($m < 0$).
• Zero Gradient: The line is perfectly horizontal ($m = 0$).
• Undefined Gradient: The line is perfectly vertical ($m = \text{undefined}$; the "run" is zero).

Linear Equations and Intercepts

The Gradient-Intercept Form

• Linear relationships are typically expressed in the form:     $y = mx + c$     where:
  • $m$ is the gradient.
  • $c$ is the $y$-intercept.

Intercepts

• $y$-intercept: The point where the line crosses the $y$-axis. This occurs when $x = 0$. In the formula $y = mx + c$, the value of $c$ identifies this point.
• $x$-intercept: The point where the line crosses the $x$-axis. This occurs when $y = 0$.

Special Equations and Axes

• The x-axis: Has the equation $y = 0$.
• The y-axis: Has the equation $x = 0$.
• Horizontal Lines: Lines parallel to the $x$-axis have equations in the form $y = k$.
• Vertical Lines: Lines parallel to the $y$-axis have equations in the form $x = k$.

Graphing and Modeling

• Graphing via Gradient and Intercept: To graph a line, start by plotting the $y$-intercept ($c$) and then use the gradient ($m$) to find the next point following the "rise over run" principle.
• Inspection of Graphs: You must be able to look at a straight-line graph, identify the numerical value of the gradient and the vertical intercept, and use those values to construct the line's equation.

Coordinate Geometry and Line Relationships

Point Testing and Collinearity

• Determining if a Point Lies on a Line: A point $(x, y)$ lies on a line if its coordinates satisfy the equation of that line when substituted.
• Collinear Points: These are three or more points that lie on the same straight line. This can be verified by checking if the gradient between all pairs of points is identical.

Parallel Lines

• Parallel lines never intersect and have the exact same steepness.
• Rule: Parallel lines have the same gradient ($m_1 = m_2$).

Perpendicular Lines

• Perpendicular lines intersect at a right angle ($90^{\circ}$).
• Rule: Lines are perpendicular if the product of their gradients results in $-1$ (negative one).     $m_1 \times m_2 = -1$$m_2 = -\frac{1}{m_1}$

Formulating Linear Equations

• From Gradient and Intercept: Given the gradient ($m$) and the intercept ($c$), the equation is formed by direct substitution into $y = mx + c$.
• From Two Points: To find the equation given two points, first calculate the gradient ($m$) using the gradient formula, then substitute one point and the gradient into the point-gradient formula or solve for $c$ in $y = mx + c$.
• Parallel or Perpendicular to a Given Line: You must find the equation of a line relative to another line by identifying the required gradient based on the parallel or perpendicular rules and passing it through a specific coordinate.