Welcome to a lesson on the Cartesian plane.
In this lesson we'll be discussing
ordered pairs and plotting points.
We now begin looking at the
relationships between two variables.
Typically one variable is considered to be the input,
and the other is called the output.
The input is the value that is considered first,
and the output is the value corresponding to
or is matched with the input.
The input/output designation may
represent a cause/effect relationship,
but that is not always the case.
But because of this, we sometimes say that
the input controls the output
or that the output depends on the input.
Whenever we have values that
represent the input/output relationship,
we can organize them in what's called an ordered pair,
where for an ordered pair we have the input value comma,
the corresponding output value in a set of parentheses.
Or in short, it's the input, comma,
the output in parentheses.
However, I do want to mention a common variable for
the input is X and a common variable for the output is Y.
So sometimes we see an ordered pair written as X, comma, Y
but the variables don't have to be X and Y
so we're going to refer to an ordered pair
as the input, comma, the output.
Example one, we're asked to complete the table.
So here we have an input of four, an output of negative 3,
and asked to write the ordered pair.
So the input is first, comma, the output is second,
and we organize this in a set of parentheses.
Next, the input is five and the output is eight.
So we have five, comma, eight, in parentheses.
Next, we're given the ordered pair
of zero, comma, negative four
and asked to give the input and the output.
So the input is the first value or first coordinate.
And the output is the second value
or second coordinate of negative four.
Next we have the ordered pair, negative two, comma, six.
So the input is negative two and the output is six.
Now, let's talk about the rectangular coordinate system,
also known as the Cartesian coordinate system.
It's formed by the intersection of two number lines
where we have a horizontal number line called
the horizontal axis, or sometimes the X axis,
and a vertical number line called the vertical axis
or sometimes called the Y axis.
We always find the input values along the horizontal axis
and we always find the output values
along the vertical axis.
The intersection point of these two axes,
this point here is called the origin.
The origin has the ordered pair
or coordinates zero, comma, zero.
So at this point or at this location,
the input is zero and the output is zero.
We can use this coordinate system to represent
the relationship between two variables graphically.
For example, every ordered pair here would be
a unique point or location on this coordinate system.
Let's go ahead and graph these four ordered pairs
by plotting them as points on the coordinate system.
So the first ordered pair is four, comma, negative three.
So we have an input of four
and an output of negative three.
So because the input is four,
from the origin we would go right four units.
Remember, the input is always along the horizontal axis
and then because the output is negative three,
which we find along the vertical axis,
from here, we would go down three units.
This would be the location of the point with coordinates
or the ordered pair four, comma, negative three.
Notice that this location, we have an input of four
and an output of negative three.
Next, we have the ordered pair five, comma, eight.
So because the input is five,
from the origin we would go right five units
and then, because the output is positive eight,
we would go up eight units.
This would be the location of the point with coordinates
or ordered pair, five, comma, eight.
Notice that this location we have an input of five
and and output of eight.
Next we have the ordered pair zero, comma, negative four.
So now the input is zero and the output is negative four.
So because the input is zero,
from the origin we don't move left or right but
because the output is negative four,
we would go down four units.
The location of the point with coordinates
or ordered pair zero, comma, negative four would be here,
right on the vertical axis.
And then finally, we have the ordered pair
negative two, comma, six.
So the input is negative two and the output is six.
So because the input is negative two,
from the origin we would go left two units,
and because the output is positive six,
we would go up six units from here.
This would be the location of the point with coordinates
or ordered pair negative two, comma, six.
Notice that this location the input is negative two
and the output is positive six.
I hope you found this helpful.