Linear Relationships and Coordinate Geometry

Number Plane Review

• A Cartesian plane is also called a number plane.

• It has 2 axes: the x-axis and the y-axis.

• There are four quadrants.

• A point on the Cartesian plane is called a coordinate.

• The x-coordinate is written first, then the y-coordinate.

• The point (0, 0) is the origin.

Midpoint

• M represents the midpoint.

• The midpoint of a line can be found by inspection.

• For harder questions, use the following steps:

  1. Add the x-coordinates together, then divide by 2.

  2. Add the y-coordinates together, then divide by 2.

  3. Write the answer as a coordinate (x, y).

Example:
To find the midpoint of intervals, average the x-coordinates and the y-coordinates of the endpoints.

Pythagoras’ Theorem Review

• To find the length of an interval, or the distance between two points, Pythagoras’ Theorem is used.

  1. Join the two points together with a line.

  2. Create a right-angled triangle, making the line the hypotenuse.

  3. Find the hypotenuse using Pythagoras’ Theorem.

Length of an Interval

To find the length of an interval, or the distance between two points, we use Pythagoras’ Theorem to help us.

1. Join the two points together with a line.

2. Create a right-angled triangle, making your line as the hypotenuse

3. Find the hypotenuse using Pythagoras’ Theorem

Gradient

• The gradient of a line refers to the slope of the line, indicating how steep it is.

• A lowercase m represents the gradient.

Formula for calculating the gradient:

1. Find two points on the line and use them to make a right-angled triangle.

2. Count how many boxes this triangle has going up - this is your RISE.

3. Count how many boxes this triangle has going across - this is your RUN.

4. Write your answer as a fraction and simplify using a calculator.

   • Note: Double-check whether the line has a positive or negative gradient.

Equation of a Line

• To find the equation of a line, determine its gradient and y-intercept.

• The y-intercept is the point where the line crosses the y-axis.

• The equation of a line comes in the form: y = mx + c

  • Where:

    • m is the gradient.

    • c is the y-intercept.

Table of Values Review

• Complete the tables of value for different equations.

Graphing Linear Relationships

1. Use table of values to find coordinates

2. Plot coordinates onto a number plane

3. Join the dots together to form a line

Determining Whether a Point Lies on a Line

To see whether a point lies on a line, we substitute the x-value and y-value into the equation of the line. If the left hand side (LHS) equals the right hand side (RHS), then the point lies on the line

Comparing Straight Lines and the Y-intercepts

Use DESMOS or Geogebra to graph these four lines all onto the same graph 𝑦 = 𝑥 − 6 𝑦 = 𝑥 − 2 𝑦 = 𝑥 + 2 𝑦 = 𝑥 + 4
a) Does the constant term change the slope of the line?
b) What happens to the line as the constant term goes from -6 to +4 (i.e. as it gets bigger)?
c) Where does go through the y-axis? At y = 𝑦 = 𝑥 − 6
d) Where does go through the y-axis? At y = 𝑦 = 𝑥 − 2
e) Where does go through the y-axis? At y = 𝑦 = 𝑥 + 2
f) Where does go through the y-axis? At y = 𝑦 = 𝑥 + 4
g) Complete this sentence: The line would go through the y-axis at 𝑦 = 𝑥 + 7
h) This graph shows and 𝑦 = 𝑥 + 1 𝑦 = 𝑥 + 2
i) Draw a line for 𝑦 = 𝑥 + 1 1 2
ii) Draw a line for 𝑦 = 𝑥 − 7

Horizontal and Vertical Lines

• Horizontal Lines

  • Equation is y = a number (constant)

  • Only have a y intercept

• Vertical Lines

  • Equation is x = a number (constant)

  • Only have an x intercept

Parallel Lines

• Parallel Lines: Lines that never intersect.

• Remember, the gradient is the number in front of the x.