Number Plane and Cartesian Geometry
Theory, Formulae and Practice
Welcome to Sydney Based Aussie Math Tutor NSW, your go-to destination for expert private math tutoring online. On this page you will learn the basic of graph in a clear easy way. The page is made by our experienced tutors in Sydney for the students in Sydney. Thus, master concepts such plotting the points, quadrants and plotting a line equation in this lesson.
CARTESIAN PLANE
A Number Plane is a flat-surfaced two-dimensional grid that extends infinitely and is formed by the intersection of two number lines. One number line runs horizontally and is called the X-axis. The other number line runs vertically and is called the Y-axis. These lines intersect at a point called the origin and is denoted as (0,0).
Quadrants: The four sections of the Cartesian plane divided by the x-axis and y-axis. The X-axis and the Y-axis divides the Number plane into four sections called as the Quadrants. The x and the y values of any point maybe positive or negative depending on which quadrant it lies.
The image below shows the number plane along with the Quadrants
The Co-ordinates of a point on a Number Plane
The Co-ordinates of any point is determined by its distance from the X-axis and the Y-axis. The x co-ordinate is the distance of the point away from the Y-axis and the y co-ordinate is the distance of the point away from the X-axis.
To find these distances, we draw a segment perpendicular to the X-axis and the Y-axis as they are the Shortest Distance to the Axes.
The image below shows how to find x and y co-ordinates of a give point.
Exercise: Practice Questions on Plotting the Points
Question : Plot the following points on a graph:
a) ( 0, 3 ) b) ( 1, 2 ) c) ( 2, 1 )
d) ( 3, -4 ) e) ( -4, -4 ) f) ( -5, 3 )
Answers
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Equation of a Line
The most common form of the equation of a line is called as the slope-intercept form: y = mx + c
Where:
(x,y) are the co-ordinates of a point on the graph
m is the slope of the line. The slope represents the steepness of the line.
c is the constant. The constant represents the distance away from the centre O (0,0).
The formula for the slope of a line is given by:
Distance and Midpoint
Relationship Between Slope and Parallel or Perpendicular Lines
Parallel Lines
Definition: Two lines are parallel if they have the same slope and never intersect.
Slope Relationship:
If two lines are parallel, their slopes are equal.
If line 1 has a slope m1 and line 2 has a slope m2, then for the lines to be parallel: m1=m2
Perpendicular Lines
Definition: Two lines are perpendicular if they intersect at a right angle (90 degrees).
Slope Relationship:
If two lines are perpendicular, the product of their slopes is −1
If line 1 has a slope m1 and line 2 has a slope m2, then for the lines to be perpendicular: m1×m2=−1
Drawing a Line from a Linear Equation
The most common way to draw a line from a linear equation is by converting the equation in the slope-intercept form: y = mx + c.
Substitute different values of x into the equation to obtain corresponding y values, creating at least two points. While plotting more points can increase the accuracy of the graph, only two points are necessary to draw a straight line representing the equation."
Two Points Suffice: While more points can improve accuracy, two points are sufficient to draw a straight line. All points on the line will satisfy the original equation.
Example on how to Draw a Line from a Linear Equation
Question : Draw the line for the equation x + y =3
Step 1: Convert the equation in the slope-intercept form: y = mx + c. Hence, the equation become, y= - x + 3.
Step 2: Substitute different values of x into the equation to obtain corresponding y values. Hence,
When x = -1 , y = 4. Therefor the point on the line is ( -1, 4 ).
When x = 0 , y = 3. Therefor the point on the line is ( 0, 3 ).
When x = 1 , y = 2. Therefor the point on the line is ( 1, 2 ).
When x = 2 , y = 1. Therefor the point on the line is ( 2, 1 ).
Step 3: Plot the points on the graph:
From the above graph, it can be seen that all the points lie on in a single line. Hence, The points are collinear points.
Step 4: Draw a line connecting all the points. Finish naming the graph, lines and the points.
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