Graphs and Coordinates: Study with Video Lessons, Practice Problems & Examples

The rectangular coordinate system, or Cartesian plane, consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Points are represented as ordered pairs (x, y), indicating their position on the grid. The origin (0, 0) separates positive and negative values, creating four quadrants. Understanding how to plot points like (4, 3) or (-3, 2) is essential for graphing relationships between variables. This foundational knowledge is crucial for exploring multivariable polynomials and their properties in future lessons.

Graphs & the Rectangular Coordinate System

Hey, everyone. So up until now, all the graphing and plotting that we've seen in this course have only involved one variable like x, and we've always plotted it on the horizontal number line. But a lot of graphing for the rest of this course is going to involve the relationship between 2 variables. So we're going to have to plot points and also equations. And in order to do that with multiple variables, we'll need to be familiar with the rectangular coordinate system. So that's what we're going to talk about in this video. Basically, what I'm going to show you here is how we can take coordinates that are described with two numbers like 4, 3, and I'm going to show you that these are really just locations on this two-dimensional grid, and it has to do with their x and y values. So I'm going to show you how to plot these kinds of points. So let's get started here. The rectangular coordinate system, sometimes called the Cartesian plane, is really just where you have a horizontal number line and a vertical number line that are sort of together and crossing. These are 2 perpendicular number lines that come together to form a 2-dimensional grid instead of just a one-dimensional line. So now we're going to describe locations not just as an x-coordinate but also as a y-coordinate as well. So let's get into the specifics. This horizontal axis that we've been familiar with so far is called the x-axis, and so you're going to see a little x written out here along this number line. And then over on the vertical axis, that's going to be the y-axis. So, basically, what we can do is now instead of describing just one point on this number line with one number, we can actually describe it using two numbers and one for the x and one for the y. And the way that we ascribe points or sometimes these are called ordered pairs, is basically just a position, and it's always in the form where it has parentheses and there are two numbers, an x and a y. So for example, there's going to be 4, 3. That's an x coordinate and a y coordinate. That's called an ordered pair. Basically, what you're going to do here is you're going to start from the sort of center of this diagram, and you're going to go along the x-axis until you hit 4. So this is going to go 4 in the x, and then you're going to go 3 in the y from there. So that's what the coordinate 4, 3 means. It means you go 4 in the x and then 3 in the y, and that's why this location is equal to 4, 3. Alright? So that's what a point or an ordered pair is. So for this example, we're just going to be plotting out a bunch of ordered pairs on this graph. So let's keep going. Now notice how in b, here, I've got a negative number inside for the x-coordinate. So what does that mean? Well, in order for us to understand that, we'll talk about the origin. The origin really is just the center of this diagram, which we've already sort of labeled over here, and it's just the point 0, 0. It's where your graph starts. It's also basically where the x and y axis intersect. And notice what happens is it also separates positive, sorry, positive and negative values. So for example, what you'll see is that the x values are positive, the y values are positive to the right and above the origin. So x values are positive, and y values are positive to the right and above the origin. And then they're negative when they're to the left or below the origin, as we can see over here. Alright? So how do we graph the coordinates negative 3, 2? Well, now what this is saying is that on the y-axis, we're going to go to negative 3. So instead of going to the right like I did for a, I'm going to have to go to the left, negative 3, and then I have to get to 2 on the y-axis. So do I have to go up or down? Well, I have to get to positive 2, so I'm going to have to go up like this. So this is the point b, and this is negative 3, 2. Alright? Pretty straightforward. Let's keep doing a few more examples. So here we've got negative 2, negative 3. Remember, this is x, y. Here I have to go to negative 2 by going to the left, and then I have to go down to get to negative 3. That's over here. So this is the coordinate c, and this is negative 2, negative 3. Alright. And now we have 5, negative 4. So 5, negative 4 is going to be positive 5. So here, I'm actually going to go to the right. I'm going to have to go to the right 5, and then I have to get to negative 4, so I have to go down from here. So this is 1, 2, 3, and 4. This is negative 4. So this over here is the point d, which is 5, negative 4. Alright? We've got a few more. We've got the 0, 0, but we actually have already seen that before. 0, 0 is really just the origin. So that's just the location 0, 0. And then finally, we've got 0, negative 3. Again, what this means is that you're going to go 0 on the x-axis, so you're not really going to go left or right. Then you're going to have to go down just from the origin until you hit to negative 3. So this is the coordinate f0, negative 3. Alright? So this is a little bit sort of cluttered here. We've got a lot of points, but, hopefully, this makes sense in how to sort of graph them. The last thing I want to talk about here is that a lot of these points have sort of fallen into 2, 4 different corners of this diagram, and these are called quadrants. Basically, what happens is that the x and y axes divide the graph into 4 regions or 4 corners, and these just get names. They're called quadrants. And, basically, they all have numbers, and quadrant 1 is going to stop at the is going to start at the right top right-hand corner, and then you're going to keep going in increasing number as you go counterclockwise around. So this is quadrant 2, this is quadrant 3, and this is quadrant 4. Sometimes they get Roman numerals, but you don't really need to know that. Alright, folks. So that's just an introduction into graphing in the coordinate system. Thanks for watching.

Graphs and Coordinates - Example

Welcome back, everyone. So, in this problem, we're going to graph these points that were given W, X, Y, and Z. We're given their ordered pairs, and then we're going to identify the quadrant of each one of these points. So let's get started. Now remember, whenever you're given coordinates or ordered pairs, the two numbers, the first one corresponds to the x and the second one corresponds to the y. So, the first thing you do is walk along the x-axis right or left, and then you go to the y-axis up or down. So let's get started. For this first point, we have 1, negative 2. So I have to walk along the x-axis to 1 over here, and then because I'm going into the negative, I have to go down to negative 2. And so my point W ends up being over here.

Let's go to the next one. X is \( (5, 2) \). That means 5 in the x, 2 in the y. So you walk along the x-axis to 5, that's over here, and now you go upwards because you're going positive and both of them are positive. So you go up to the right and up, and that's going to be 2. So this is going to be your X coordinate.

So now about Y. Y is \( (-3, -4) \). So now both these things are negative, so you're going to have to go into the left until you get to 3. So you're going left here, and then you're going to go down to negative 4, so it's down over here. So this is the point Y.

And then last but not least, we have the Z coordinate \( (-4, 3) \). So now what you're going to do is you're going to go to the left to negative 4, and you're going to have to go up now 3. So 1, 2, 3, and that's going to be your Z coordinates. Alright? So these are all your coordinates. It's always going to be really helpful to familiarize yourself with the coordinate system and how to graph some points.

So now we're just going to identify the quadrants of each one of these points over here. So what about W? So remember the quadrants go, you start at Q1, that's quadrant 1, and then you go counterclockwise, and then it goes in increasing order. So this is quadrant 2, quadrant 3, and quadrant 4. So that means that quadrant sorry. The W coordinate or the W point over here is actually in Q4. So this is going to be in quadrant 4. What about X? Well, X over here, we located X. X is this point, and this is clearly in quadrant 1. That's the top right corner. Now what about Y? Y is in the lower left, and that's quadrant 3. So that's quadrant 3. And then Z over here is going to be in these coordinates. So that is going to be the top left, which is quadrant 2.

So, hopefully, you got that right. Let me know if you have any questions. Thanks for watching.

Here’s what students ask on this topic:

What is the rectangular coordinate system?

The rectangular coordinate system, also known as the Cartesian plane, consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin, denoted as (0, 0). Points on this plane are represented as ordered pairs (x, y), where 'x' indicates the horizontal position and 'y' indicates the vertical position. This system allows for the plotting of points and the visualization of relationships between two variables, which is fundamental for graphing equations and understanding multivariable functions.

How do you plot points on the Cartesian plane?

To plot points on the Cartesian plane, you need to use ordered pairs (x, y). Start at the origin (0, 0). Move horizontally to the x-coordinate: right for positive values and left for negative values. Then, move vertically to the y-coordinate: up for positive values and down for negative values. For example, to plot the point (4, 3), move 4 units to the right and 3 units up. For (-3, 2), move 3 units left and 2 units up. This method helps in accurately locating points on the grid.

What are the quadrants in the Cartesian plane?

The Cartesian plane is divided into four quadrants by the x and y axes. Quadrant I is in the top right, where both x and y are positive. Quadrant II is in the top left, where x is negative and y is positive. Quadrant III is in the bottom left, where both x and y are negative. Quadrant IV is in the bottom right, where x is positive and y is negative. These quadrants help in identifying the sign of the coordinates of points.

What is the origin in the Cartesian plane?

The origin in the Cartesian plane is the point where the x-axis and y-axis intersect, denoted as (0, 0). It serves as the reference point for plotting all other points on the plane. The origin separates the plane into four quadrants and is the starting point for measuring distances along the x and y axes. Understanding the origin is crucial for accurately plotting points and interpreting their positions relative to this central point.

How do you interpret negative coordinates in the Cartesian plane?

Negative coordinates in the Cartesian plane indicate positions to the left of the origin for the x-coordinate and below the origin for the y-coordinate. For example, in the point (-3, 2), the x-coordinate is -3, meaning you move 3 units to the left from the origin. The y-coordinate is 2, meaning you move 2 units up. Similarly, for the point (5, -4), you move 5 units to the right and 4 units down. Understanding negative coordinates is essential for accurately plotting points in all four quadrants.