Sistema de coordenadas cartesianas

(Video) Introduction to Graphing

by mathman1024

Here's a video by mathman1024 explaining the basics of graphing.

$\large{\text{Summary}}$

Graphing is done on the rectangular coordinate system which is made up of two axes. The horizontal axis is called the $x$-axis and the vertical axis is called the $y$-axis. The $x$-axis is a horizontal line. The $y$-axis is a vertical line. That means we are pairing up two real numbers. We can move left-right or up-down.

The coordinate plane is broken down into quadrants that go in a counterclockwise pattern.

The point of intersection of the two axes is called the origin. It has an $x$ value of $0$ and a $y$ value of $0$.

Points on the graph are given as ordered pairs which are of the form $(x,y)$. The $x$-coordinate tells you where on the $x$-axis the point is and the $y$-coordinate tells you where on the $y$-axis the point is. Each tells you half of the information needed in order to plot the point. $$(2,3)\neq(3,2)$$ If we graph these two points we see they're not the same. To graph a point, start from the origin. Then move to the left or right by the amount in the first part. Then move up and down by the amount indicated from the second part.

While we're just plotting points for now, we can also graph functions. These coordinate systems are also useful in statistics. If we plot a bunch of points we get something called a scatter plot.

(Video) Graphing on the Coordinate Plane

by Math Antics

Here's a video by mathantics that explains graphing.

$\large{\text{Summary}}$

Graphing is essentially taking mathematical relationships and turning them into pictures. Graphing is a way to make a visual representation of an equation or data set so you can understand it better.

The first major term for graphing is the coordinate plane.

The coordinate plane is the platform on which our graphing will take place. A coordinate place is similar to a number line...

...but it's got two dimensions, the $x$-axis and the $y$-axis!

You might remember that a number line starts at $0$ and increases (moves towards positive numbers) as you move to the right and decreases (moves towards negative numbers) as you move to the left; usually with marks showing you where each integer is along the way.

If you take the number line, copy it, and rotate it $90^{\circ}$, you get the coordinate plane. It's now a two dimensional representation of the number line.

With the number line, plotting points was easy. You only needed to know one number and you can place a point. With the coordinate plane, since it's two dimensional, you need two numbers in order to plot each point. These two numbers are called coordinates of a point. Coordinates use a special format to help you recognize them: $$(\#,\#)$$ The two numbers are put in-between parentheses, separated by a comma.

To understand how coordinates work we have to remember that our coordinate plane is made up of two perpendicular number lines. Each of these number lines are called an axis. One of the axes is horizontal, like the horizon.

The other axis is vertical.

Because of this, they're often called the horizontal axis and vertical axis. But even more often, they are referred to by variable names. The horizontal axis is called the $x$-axis and the vertical axis is called the $y$-axis. The reason we use variable names is because it's more flexible and because each of the coordinate numbers is a variable, $(x,y)$, that relates to a specific position along one of the two axes of the coordinate plane.

The first coordinate listed, $({\color{#29A603}{x}},y)$ will be called $x$ and the second coordinate listed, $(x,{\color{#8925AE}{y}})$, will be called $y$. The numbers will always be listed in that same order. $x$ first and $y$ second. Coordinates are often called ordered pairs because they're a pair of numbers always listed in the same order. So if you have the coordinates $(3,5)$ this means that $x=3$ and $y=5$.

Now we can begin plotting our coordinates. The first number in the coordinate tells us where on the $x$-axis the point is located and the second number tells us where the $y$-axis is located. The two numbers work together to give us one point. Each number only gives us half the information on where the point is.

Let's try it out by plotting the point $(3,2)$. First, we locate the $x$-value, $3$, on the $x$-axis. Instead of putting a point there, we either draw or imagine a dashed line perpendicular to the $x$-axis that goes through the $3$.

We do this because the first number only tells us where along the $x$-axis the point is but it can be anywhere along the $y$-axis. We won't know until we look at the second number. So we draw the line temporarily to represent every possible point that can have an $x$-value of $3$.

Now we locate the $y$ value, $2$, along the $y$-axis. Again, we don't put a point but instead draw a perpendicular dashed line through it.

We do this because the second number only tells us where along the $y$-axis the point is but it can be anywhere along the $x$-axis. So we draw the line temporarily to represent every possible point that can have an $y$-value of $2$.

But look! The exact point where the two lines intersect represents the only point on the entire plane where both $x=3$ and $y=2$. That intersection is the location of our point!

Hmm that seemed to take a while though. How about a simpler, and more intuitive, way to plot these points? Luckily there is one. We start by putting our point on the origin, or $(0,0)$ and use the coordinate as a set of instructions on where to move the point.

So to plot the point $(3,2)$ we look at the first number to see how far we move our dot in the $x$ direction. Since $x$ is positive $3$ we move our point three units in the positive $x$ direction.

Then, since our $y$ is positive $2$, we move the point two units in the positive $y$ direction.

Try plotting these point's by yourself. Click continue to see the answers.

1. $(-4,3)$
2. $(-3,-3)$
3. $(4,-2.5)$

Quadrants

Have you noticed that each of these points is plotted in a different region of the coordinate plane?

These four regions are called quadrants and their boundaries are defined by the two axes of the coordinate plane. The quadrants are named 1 through 4 so they can be easily referred to.

1. Quadrant 1 is the upper right corner and contains all the points where both $x$ and $y$ are positive. $(x,y)=(+,+)$
2. Quadrant 2 is the upper left and contains all the points where $x$ is negative and $y$ is positive. $(x,y)=(-,+)$
3. Quadrant 3 is the bottom left and contains all the points where both $x$ and $y$ are negative. $(x,y)=(-,-)$
4. Quadrant 4 is the bottom right and contains all the points where $x$ is positive and $y$ is negative. $(x,y)=(+,-)$

Usually we use roman numerals to label the four quadrants.

The Cartesian coordinate system is a system that is used to create visual representations of equations. These representations are made on a grid called the Cartesian Plane.

The Cartesian Plane can be thought of as a map. A map allows us to find the location of something or someone by using street names and numbers. Just like a map, The Cartesian Plane allows us to find the location of a point.

Let's begin with the map below. We can describe where the man, Charlie, is standing by using street names. i.e. Charlie is on 3rd Avenue and College Street.

The Cartesian Plane is a grid made up of two number lines that run perpendicular to each other. The two lines intersect at $0$ which is the middle or the origin of the grid. Below is a Cartesian Plane.

The two number lines that make up the grid are called axes.

The number line that runs up and down is the vertical axis or the $y$-axis. The number line that runs left to right is the horizontal axis or the $x$-axis.

Similar to the map, the location of something can be found using this grid. In the map, Charlie's location was described using the intersection of the roads he was on, the location on a coordinate plane works the same way. In the map below, let the black dot represent Charlie.

In order to describe his location, we will use ordered pairs.

Ordered pairs are used to describe the location of a point by using two numbers in the following format; $(\#,\#)$.

The first number in this format is the number of units to the right or left of the $y$-axis. This number is called the $x$-value.

The second number is the number of units above or below the $x$-axis. This number is called the $y$-value.

Notice that Charlie is $3$ units to the right of the $y$-axis and two units above the $x$-axis so Charlie is located at the point $(3,2)$.

The plane below represents where Charlie has moved to.

Notice that the Cartesian Plane is split into four parts. These parts are called quadrants and each one has a name. The quadrants in the Cartesian Plane below are all labeled.

The location of a point can be determined without graphing once you are familiar with the four quadrants and the format of the ordered pairs. All points in each quadrant will have a certain format.

All points in Quadrant $\text{I}$ will be in the following format: $(x,y)$. All values must be positive in order for the point to be in Quadrant $\text{I}$.

Points in Quadrant $\text{II}$ will have the following format: $(-x,y)$. This means that the $x$-value will always be negative while the $y$-value is positive.

Points in Quadrant $\text{III}$ will be in the format: $(-x,-y)$ so both the $x$ and $y$-values must be negative.

Points in Quadrant $\text{IV}$ will be in the following format: $(x,-y)$ meaning the $x$-value will be positive while the $y$-value is negative.