Linear Relations: Cartesian Plane & Graphs

• Linear relations are explored

Algebraic Expressions

• Learning intentions include:
  • Identify and describe the key features of the Cartesian plane
  • Know that a linear relation forms a straight line when graphed representing the relationship between x- and y-coordinates
  • Use a linear rule to create a table of values and a straight-line graph
  • Identify x- and y-intercepts on a graph or table
  • Decide if a point lies on a line using the line's rule

Cartesian Plane

• The Cartesian plane has two axes (x-axis and y-axis) that divide the number plane into four quadrants
• A point is specifically positioned using a coordinate pair (x, y)
  • x describes the horizontal position
  • y describes the vertical position
• A linear relation is a set of ordered pairs (x, y) that, when graphed, form a straight line
• The y-intercept is where the graph intersects the y-axis
• The x-intercept is where the graph intersects the x-axis
• Examples of linear relationships include equations of the form:
  • y = mx + c
  • y = 2x + 2

Graphing Straight Lines Using Intercepts

• Two points needed to sketch a straight line graph
• Learning intentions include:
  • Knowing that two points are required to sketch a straight line graph
  • Knowing how to find the x- and y-intercepts from the rule for a linear graph
  • Can sketch a linear graph by finding intercepts and joining in a line

Sketch Rule

• To sketch a straight-line graph, two points are required

Intercepts

• The axis intercepts are commonly used to sketch a straight-line graph
• The y-intercept is the y-value where the line crosses the y-axis (where x = 0)
• The x-intercept is the x-value where the line crosses the x-axis (where y = 0)

Lines with One Intercept

• Learning intentions:
  • Knowing that vertical, horizontal lines, and lines through the origin only have one intercept
  • Understanding the equation form of vertical and horizontal lines, and lines that pass through the origin
  • Being able to sketch these three types of lines that have one intercept
• Three types of lines have only one intercept: vertical lines, horizontal lines, and lines through the origin

Vertical Lines

• Parallel to the y-axis
• The x-intercept is b
• Equation is x = b

Horizontal Lines

• Parallel to the x-axis
• y-intercept is c
• Equation is y = c

Lines Through the Origin

• The y-intercept is 0
• x-intercept is 0
• To find a second point, substitute any other value of x
• Example equations include: y = 2x, y = x, y = -x, y = -2x

Gradient

• The gradient is the ratio of the vertical change (rise) to the horizontal change (run) between two points
• Learning intentions include:
  • Knowing that the gradient is the ratio of vertical change to horizontal change
  • Knowing the gradient can be positive, negative, zero, or undefined.
• The gradient of a line measures its slope, represented by the letter m
• Determined by rise or fall (rise) between two points within a horizontal distance (run)
• Gradient can be positive (uphill), negative (downhill), zero (horizontal), or undefined (vertical)
• The gradient is calculated by:
  • rise/run
  • (y₂ - y₁)/ (x₂ - x₁)

Gradient-Intercept Form

• Learning intentions:
  • Being familiar with the gradient-intercept form of a straight line graph
  • Being able to identify the gradient and y-intercept from the rule of a linear graph
  • Using the y-intercept and gradient to sketch a straight line graph

Gradient-Intercept Form Equation

• The equation is y = mx + c where:
  • m represents the gradient
  • c represents the y-intercept

Sketching a Linear Graph

• Involves locating the y-intercept and finding a second point using the gradient
• Steps:
  • Start by plotting y-intercept
  • Use the gradient (rise over run) to find a second point
  • Connect the two points with a straight line
  • Ensure the equation is in gradient-intercept form first

Finding the Equation of a Line

• Learning intentions include:
  • Use the y-intercept and gradient to form a straight line equation.
  • Find the equation of a line given two points.

Steps for Success

• To find the equation of a line in gradient-intercept form (y = mx + c):
  • Find the gradient (m), either given or found using m = rise/run
  • Find the constant (c), observing the y-intercept or substituting a known point into the equation

Midpoint and Length of a Line Segment

• Learning intentions include:
  • Knowing a line segment has a midpoint and a length
  • Finding the midpoint of a line segment
  • Finding the length of a line segment

Midpoint

• The midpoint (M) of a line segment is the halfway point between two end points
• The x-coordinate of the midpoint is the average of the x-coordinates of the two endpoints
• The y-coordinate of the midpoint is the average of the y-coordinates of the two endpoints
• General formula: M = ((x₁ + x₂)/2 , (y₁ + y₂)/2)

Length of a Line Segment

• The length of a line segment (d) is found using Pythagoras' theorem
• Provides the distance between any two points
• Formula: d = √((x₂ - x₁)² + (y₂ - y₁)² )

Linear Modelling

• Learning intentions:
  • Identify whether a situation can be modeled by a linear rule
  • Form a linear rule to determine the variables
  • To sketch a graph by predicting the value from one variable based on another
• Many situations can be modeled using a linear rule or graph

Key Elements of Linear Modelling

• Finding the rule linking two variables
• Sketching a graph
• Use the graph to predict value of one variable based on another
• Finding the rate of change (which is the gradient)