An Introduction to Graphs: The Algebra Lecture Series Chapter 2 Section 1

If you're familiar with a number line, then you know that each point on that number line corresponds to a number.  If we take two number lines which are perpindicular to one another (they cross each other to make a right angle) and have them cross at the others zero-point, then we have a method of analyzing two-dimensional data-sets and equations, data and equations with two variables which relate to one another.  This two-number-line situation is typically called either the x-y coordinate system (because the horizontal number line is labeled as the x-number line and the horizontal is labeled as the y-number-line) or the Cartessian Coordinate System (named after French mathematician René Descartes, supposedly the first person to come up with the system).  The two number lines, since they're in the context of a coordinate system, are each now called an axis (plural: axes).
The x-axis and y-axis of a Cartesian coordinate system.
Any point on this coordinate system is given by a pair of numbers separated by a comma and enclosed inside a pair of parentheses.  If we are looking at an equation with two variables, then the solution is a pair of numbers, ordered from the independent variable to the dependent variable.  For instance, we can have the point (2, 1), which represents two units to the right on the horizontal axis and one unit up on the vertical axis.  The order of these two numbers are important, because the first number will always represent the horizontal axis, regardless of what that number may be, and the second number always represents the horizontal axis, regardless of what that number may be.  For instance, the point (2, 1) and the point (1, 2) are two different points.  After all, going two miles east and one mile north will get you to a different location than one mile east and two miles north.  These are called ordered pairs since the order of the pair does matter.
I want to be at my friends place, not his momma's place!
The point where the two axes meet is called the "origin", because it is the point where we choose both of the number lines (axes) to be zero, and we label that point with either the point (0, 0) or the letter O.  [As a note: the concept of choosing where to place the point (0, 0) will be important in most Physics applications and in some Chemistry applications.]  For some reason (don't ask me why, I have no idea), most books and teachers emphasize the concept of labeling the four quadrants of the coordinate system with Roman Numerals I, II, III, and IV.  The upper right is the first quadrant is the first quadrant, and the labeling goes counterclockwise.  The axes and the origin are not in any quadrent, they merely make up the borders.
I've never used this.
When we are checking whether or not an ordered pair is a solution of an equation, then plug in the numbers of the ordered pair into the equation.  If the result makes sense (such as 1=1), then we say that point is a solution.  If the result makes no sense (such as 1=-5,000,000,000,005), then we say that the point is not a solution to the equation.  Also, regardless of how large the solutions are, there can still be solutions which are larger.  This means that any equation have infinitely many solutions.

A basic method to plot equations onto an x-y coordinate system, break it down into 4 steps:

1)  Set up a table of x-values and corresponding y-values.  Do not put any numbers yet!  That's for a future step!

y=2x+3
X
Y
Blank Cell
Blank Cell
Blank Cell
Blank Cell
Blank Cell
Blank Cell
Blank Cell
Blank Cell
Blank Cell
Blank Cell

2)  Chose a few x values.  Make sure the x-values are within the domain of the function!

Y=2x+3
X
Y
-2
Blank Cell
-1
Blank Cell
0
Blank Cell
1
Blank Cell
2
Blank Cell

3)  Plug into the equation the chosen x-values.

Y=2x+3
X
Y
-2
-1
-1
1
0
3
1
5
2
7

4)  Plot the points and draw a line through the points which goes to infinity.

y=2x+3

This procedure works for all equations, though we should be a little careful with the points we choose for the x-values.  Non-linear equations such as $y=/sqrt{x}$ and $y=frac{x^2}{x-2}$ have domains which are not $(- \infty , \infty )$, so you can not choose any values for x; only values within the domain of the function.

That's the end of this section.  If you have any questions, please leave them is the comments.  Like and share this post if you found it helpful.  And until next time, stay curious.

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