Welcome to a lesson in working with scale
on the Cartesian plane or a coordinate plane.
We're asked to give the coordinates
of each point shown below.
So, each point on the coordinate plane
is identified by an ordered pair,
or coordinates.
Where for each ordered pair of the first value
is the input, which we find along the horizontal axis.
And the second value is the output,
which we find along the vertical axis.
Sometimes, you'll hear the ordered pair
referred to as x, y.
But, the variables are not always x and y.
So, we're going to generalize and say
each ordered pair is the input, the output.
Notice right now, the horizontal axis
is scaled by 25s,
and the vertical axis is scaled by 1000s.
To more accurately identify the coordinates
or ordered pair for each point,
let's modify the scaling on both
the horizontal and vertical axis.
Let's first focus on the interval
from zero to 25.
Notice how because we're on the coordinate plane,
there are one, two, three, four, five
smaller partitions from zero to 25.
And, because, 25 divided by five is equal to five.
If we cut our partitions interval from zero to 25,
here,
into five equal pieces.
We can now count by fives.
We'd have five,
10,
15,
20,
25,
and so on.
If we divide the interval from zero to negative 25,
into five smaller partitions, or pieces,
or sub-intervals.
We can count by fives in the negative direction.
We'd have negative five, negative 10,
negative 15, negative 20, negative 25, and so on.
Now, let's look at the vertical axis,
which is currently scaled by 1000s.
Notice, how from zero to a 1000,
because we're on the coordinate plane.
We can see there are one, two, three,
four, five, smaller partitions, or sub-intervals
from zero to 1000.
And, because 1000 divided by five is equal to 200.
Each sub-interval equals 200 units,
so we can count by 200s up to 1000.
We'd have 200,
400,
600,
800,
and a 1000.
And, of course, we can continue all the way to 2000.
In the negative direction,
even if we cut our partition
interval from zero to negative 1000,
into five sub-intervals or five partitions.
We can count by 200s in the negative direction.
So, negative 200,
negative 400,
negative 600,
negative 800,
negative 1000,
and so on.
Using this scale will make it much easier
to accurately identify the ordered pair for each point.
So, looking at point A,
again, the ordered pair is the input, the output.
The input is along the horizontal axis,
so the input is negative 30.
And the output is along the vertical axis,
so the output would be 600.
So, the ordered pair is negative 30,
comma 600.
For point B.
Notice how the input would be negative 15,
and the output would be negative 1600.
So the ordered pair is negative 15,
comma
negative 1600.
Notice point C is right on the vertical axis,
so if we were to go straight to the horizontal axis
notice how we'd be at zero.
So, zero is the input, and the output
we can see is going to be 800.
And, therefore, the ordered pair
for point C is zero,
comma 800.
Any point on the vertical axis,
would have an input value
of zero.
Next, point D
has an input of 10,
and an output of 1400.
So the ordered pair is 10,
comma
1400.
Point E,
has an input of 35,
and an output of negative 400.
So the ordered pair is 35,
comma
negative 400.
It is important to remember
each ordered pair does go in a set of parenthesis.
Now, for example two.
We have to plot the given points on the graph below.
So, we're going to have to scale the horizontal
and vertical axis
to accommodate all of the input values,
and all the output values.
And again, remember,
all of the inputs here
are along the horizontal axis.
And, all the outputs here,
must be along the vertical axis.
Let's work on scaling the horizontal axis,
so that all the input values
will be somewhere on the horizontal number line.
Let's begin by identifying which of these values
is furthest from zero.
We'll notice negative 950
would be the value that's furthest from zero.
So, we want to reflect a nice value,
or a value that we can easily divide
into smaller partitions, or pieces
to the left of negative 9500.
So, let's use negative 1000.
So, on the horizontal axis, on the far left,
we'll mark this value as negative 1000.
And, because the axis must be scaled
the same on the right of zero.
We'll go to the far right over here and label this 1000.
Now, because of the grid on the coordinate plane,
notice from zero to 1000, it's already divided
into smaller partitions, or pieces.
We have one, two, three, four, five,
six, seven, eight, nine, 10, smaller pieces,
partitions, or sub-intervals
from zero to 1000.
Which means each of these smaller partitions
would be 100 units.
Because, 1000 divided by 10 is equal to 100.
But, let's just say instead of dividing this
into 10 smaller sub-intervals,
we only wanted five.
Because, 1000 divided by five is equal to 200.
If we divide the interval from zero to 1000
into five equal sub-intervals, or five equal pieces.
Let's say here.
We can now scale this by 200s all the way to 1000.
We'd have 200,
400,
600,
800, and a 1000.
In the negative direction,
if we divide the interval from zero to negative 1000
into five sub-intervals or five pieces.
We'd have negative 200, negative 400, negative 600,
and negative 800, and negative 1000.
Notice now, all the input values will be somewhere
on the horizontal axis.
Now, let's focus on the outputs and scale the vertical axis.
We'll begin by identifying which output value
in furthest from zero.
Notice 1.8 is the value furthest from zero.
Which means the largest value on the vertical axis
must be more than 1.8,
otherwise this point here would be right on the edge
of the coordinate plane.
We also want to reflect a nice value
that would be easy to divide
into smaller sub-intervals.
So, let's use two as the maximum value
on the vertical axis.
So, if we label this value here as two.
The axis must be scaled the same in the negative direction.
So let's label this value down here negative two.
Now, because of the grid, notice from zero to two
is naturally already divided into one, two, three,
four, five, six, seven, eight, nine, 10, sub-intervals.
And, because two divided by 10 is equal to 2/10s, or 1/5th.
Each of these smaller sub-intervals
would be equal to 1/5th or 2/10s.
So, for the vertical axis, let's go ahead and use
10 sub-intervals from zero to two.
And now we can count by 2/10s.
So, we'd have 0.2,
0.4,
0.6,
0.8,
one,
1.2,
1.4,
1.6,
1.8,
and two.
And, we'll use the same scale in the negative direction.
We'd have negative 0.2,
negative 0.4,
negative 0.6,
negative 0.8,
negative one,
and so on.
Now we can plot all of the points on the Cartesian plane,
or coordinate plane.
Starting at the origin,
the input tells us whether we go left or right.
And, the output tells us whether we go up or down.
So, for point A, because the input is negative 800.
From the origin we go left to negative 800.
And, because the output is 1.8.
We go up to 1.8, which would be here.
This would be point A.
Point B is 550,
comma 0.2.
So, because the input is 550,
from the origin we go right to 550.
So, this is 400, this is 600, this would be 500.
550 would be here.
And then, because the output is 0.2.
We go up to 0.2 which would be here.
So, this is point B.
Point C we have 180, zero.
So, from the origin we go right to 80.
Which would be just to the left of 200.
And, because the output, or the second
value of the ordered pair is zero.
We don't move up or down,
the point C is right here on the horizontal axis.
Point D, we have the ordered pair of zero, negative 1.5.
So, because the input is zero,
we don't move left or right, we stay at the origin.
And, the output is negative 1.5,
so we go down to negative 1.5.
Here's negative 1.2,
negative 1.4,
and here's negative 1.6, negative 1.5 would be between
negative 1.4 and negative 1.6.
So, this would be point D.
Point E is 425,
comma
negative 0.4.
So, from the origin we move right 425.
Which would be to the right of 400,
approximately here.
And then, because the output is negative 0.4,
we go down to negative 0.4.
Which would be here.
So, this is point E.
And then, finally, point F.
We have negative 950, one.
So, from the origin we move left to negative 950.
Which would be approximately here.
And then, up to positive one, which is here.
This would be point F.
I hope you found this helpful.