Basics of Graphing: Study with Video Lessons, Practice Problems & Examples

The rectangular coordinate system, or Cartesian plane, allows for the representation of relationships between two variables, typically x and y. Points are described as ordered pairs (x, y), indicating their position on the grid. Understanding how to plot these points and identify intercepts—where graphs cross the x or y axes—is crucial. For example, the equation x + y = 5 has multiple solutions, represented as points on the graph. Mastering these concepts is essential for visualizing and solving multivariable equations effectively.

Introduction to Graphs & the Coordinate System

Hey, everyone. So up until now, all the graphing and plotting that we've seen in this course has 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. And, basically, what I am going to show you here is how we can take coordinates that are described with two numbers like 4 comma 3, and I am 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 am 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 a parenthesis and there's two numbers, an x and a y. So for example, there's going to be 4 comma 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 to 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 comma 3 means. It means you go 4 in the x and then 3 in the y, and that's why this location of a is equal to 4 comma 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 comma 0. It's where your graph starts. It's also basically where the x and y axes 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. 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 comma 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 am 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 am going to have to go up like this. So this is the point B, and this is negative 3 comma 2. Alright? Pretty straightforward. Let's keep doing a few more examples. So here we've got negative 2 comma negative 3. Remember, this is x comma y. So 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 comma negative 3. Alright. And now we have 5, negative 4. So 5, negative 4 is going to be positive 5. So here, I am 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 comma negative 4. Alright? We've got a few more. We've got the 0 comma 0, but we have already seen that before. 0 comma 0 is really just the origin. So that's just the location 0 comma 0. And then finally, we've got 0 comma 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 F, 0 comma negative 3. So this is a little bit 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 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 start at the top right hand corner, and then you're going to keep going in increasing numbers 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 to graphing in the coordinate system. Thanks for watching.

Example 1

Welcome back, everyone. So in this problem, we're going to graph these points that we 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 \( (1, -2) \). 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 \( -2 \). And so, my points 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 \( -4 \), 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 we're going to do is you're going to go to the left to \( -4 \), 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 this is 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 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 in Quadrant 2.

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

Equations with Two Variables

Hey, everyone. So up until now, when we've solved equations, they've always been of 1 variable, like x+2=5. But equations in this course won't always be so simple. Instead of just one variable, a lot of equations in this course will now start to involve 2 variables, and the most common ones you'll see are going to be x and y. So for example, instead of x+2=5, now we're going to have x+y=5. I'm going to show you the main difference between these two types of equations and how to solve them and, more importantly, how to visualize them. So let's go ahead and get started here.

So with equations with one variable, like, for example, x+2=5, you're really just trying to find the value of x that makes this equation true. So, for example, you would isolate and solve for x. You would subtract from both sides, and your answer ends up being 3. That's the number that makes this equation true. The way we visualize it is just by plotting it on a one-dimensional number line like this point over here.

Now let's take a look at x+y=5. So now we actually have 2 variables, x and y. Both are letters that can be replaced with numbers. So how do we solve this? Well, one thing you might notice is that if you try to isolate one of the variables, it's not really going to be much help because x=5-y doesn't give you any information about what x or y could be. Let's think about what this equation means. Can I think of 2 numbers that when I add them together, I get 5? And, actually, yes, I can. Because, for example, if x equals 1, what do I have to add to 1 to get to 5? Well, y could be 4. So that's a solution to this equation. Both of these numbers here, this combination makes this equation true. But is this the only combination of numbers that makes this true? Well, actually, no. Because what if x is equal to 2? If x is 2, then y could be 3, and that also satisfies these equations, x+y=5. In fact, you could have another one, x equals 5 already, and then y could just equal 0. So the whole point here is that with equations with 1 variable, the solution was always just one number. It was a single point on a one-dimensional plane versus when you have equations with 2 variables, as we can see, you end up with many solutions that satisfy this equation. And the way that we represent them is not on a number line, but actually as points as ordered pairs x, y on a two-dimensional plane. So, basically, what happens is I can take this x=1, y=4 and turn it into an ordered pair 1, 4. This becomes 2, 3, and this becomes 5, 0. And we can plot these on a 2-dimensional number or a 2-dimensional plane. So, for example, 1, 4, this point satisfies this equation. 2, 3, that also satisfies this equation. And 5, 0 also satisfies this equation. So all these things actually satisfy. In fact, there's actually an infinite number of solutions because if you sort of keep this pattern going on here, any points that is basically on this little line over here will actually satisfy between these two types of equations.

Now how do we actually use this in problems? Well, let's take a look at our example here. In our example, we have this equation, x+y=5. This is the same exact equation we've been using before, and we want to first determine if these points satisfy the equation. We've got 3, 2, 4, 1, 0, 0, and -1, 3. So in order if you're ever asked to determine whether points x,y points satisfy an equation, what they're really just asking you to do is they're asking you to replace the x and y values to check if the equation is true. Remember, that's what satisfying an equation means. So what we're doing in part a is we're when we have a coordinate like 3, 2, this is really just giving us an x and a y value, and we just plug it into this equation, x+y=5, and we just figure out if that makes the equation true. So does it? Well, if I have if I basically just place x with 3 and y with 2, then 3 plus 2 does, in fact, give me 5. So this definitely does satisfy the equation. Let's get started with the next one, which is 4, 1. Again, does this satisfy x+y=5? Well, if I replace x with 4 and y with 1, this does indeed get me 5. So both of these points do satisfy the equation. What about 0, 0? I do 0, 0 into x+y=5, then I'm just replacing x with 0, y with 0, and 0 plus 0 does not give me 5. So this actually does not turn out to be a solution to this equation. And last but not least, what about -1, 3? So into x+y=5. Well, this is going to be -1 plus 3, and this also does not give me 5. It gives me 2. So it turns out that not all of these points are going to satisfy the equation. The first two work. The second two didn't work. And now we're going to go ahead and plot them. Right? So as we can see the equate, the point 3, 2 is going to be here. That's the point 3, 2, right? That's this point over here. We've got 4, 1. This is also going to be a solution to the equation. And then we've got 0, 0, which is over here, and then -1, 3. So if you've seen here, there's a pattern that happens. Basically, when points do satisfy an equation, when they do make the equation true, then they are on the graph of that equation versus when they do not satisfy an equation as we've seen with these points over here, they are not on the graph of that equation. So that's the basic difference between equations with 2 variables versus 1 variable. Alright? So, hopefully, that made sense. Thanks for watching.

Graphing Two Variable Equations by Plotting Points

Hey, everyone. So up until now, when we've seen equations with 2 variables, their graphs were already given to us. For example, we saw that x+y=5 was a line that looks like this. But sometimes their graphs aren't going to be given to us, and you're going to have to go graph them yourself. That's what I want to show you how to do in this video, and, basically, the way we're going to do this is by plotting points. To graph an equation, we're going to calculate and plot a bunch of ordered pairs that make the equation true, and the way we do that is by plugging a bunch of x-values into this equation, and we're going to get points out of it. So those points we can take and plot them on a graph and then connect them with a line. That's basically what I'm going to show you how to do in a step-by-step way. Let's get started.

Alright. So we're just going to actually dive right into our equation or our example here so I can show you how this works. So we're going to graph this equation -2x+y=-1, and we're going to do this by creating a bunch of ordered pairs using some x-values that are already chosen for us. So here's the first step. You're going to take an equation, and if it's not already isolated to one side, you're going to have to isolate y to the left side. So, basically, you always want y= and then the rest of the equation on the right side. And the reason to do this is because you're going to plug x-values into this equation, and it's just going to be way easier to see what y is if it's isolated. Alright? So how do we do this? How do I take this equation over here and isolate y? Well, basically, all I have to do is just take the 2x, the negative two x, and move it to the other side. So I'm going to add 2x over here and add 2x to both sides. And, basically, what I end up with when I bring it down here, this is actually just going to end up being 2x-1. So now that y is isolated, I can move on to the next step.

What I want to do is I want to calculate y values for a bunch of x values that I choose, and usually 3 to 5 is enough. So, I've got my table here that we've got a bunch of x-values that are already chosen for me, -2 and so on. And what I'm going to do here is I'm just going to replace the x inside of this equation with whatever that x-value is, and then I'm basically just going to get a y-value out of it. So, for example, I'm just going to take this -2, and I'm going to plug it in. Right? So this is going to be 2×-2-1. So what does that give me? What y-value does that give me? Well, 2×-2=-4, and -4-1=-5. So this is going to be -5 over here. What that means here is that I plugged in -2, and I got out of it -5. So that's an ordered pair. The ordered pair that satisfies this equation is (-2,-5). You're just going to repeat this a bunch of times over here. So that's what we're going to do.

So this y=2×-1-1. So this is going to be -2,-1, which is going to be -3. So this is going to be (-2,-3). 0 is always a pretty easy point to do because usually what happens is you just replace x with 0 and one term just kind of goes away. So this whole term just cancels out, and then you get the -1, which is just -1. So the ordered pair is (0,-1). And then over here, we're going to get 2×1-1, so that's 2-1, and that's just going to give us 1. So this is going to be (1,1). And then finally, we've got 2×2,4-1=3. So the ordered pair is (2,3). Alright? So you just have to basically just plug and chug a bunch and fill out your table. Tables are going to be really good at organizing your information.

The third thing you're going to do is you're going to plot these xy points that you've just gotten in this table over here, and you're going to plot them on your graph. So let's go ahead and do that. So I'm going to take the ordered pair -2,-5, and that's going to be right over here. I'm going to take the points I'm sorry. This is going to be well, whoops. That's going to be -1. I'm sorry. Hopefully, you caught that. So -1,-3 is going to be this point over here, then we've got 0,-1. So that's over here. And we've got 1,1, which is over here. And then finally, we got 2,3 over here. So I just end up with a bunch of points, I'm going to plot them all, and the last thing I do is connect them with a line or sometimes a smooth curve. In this case, what happens is I can actually plot a line that goes through all of these 5 points, and so this is the graph of the line. So this is the graph of the equation, y=2x-1 or -2x+y=-1. That's how to graph these types of equations. The very last thing I want to sort of mention here is that most of the time these x coordinates won't be given to you, and if you're not given x values to evaluate, then you can always just feel free to choose your own. And usually, 3 to 5 is enough. Alright? So thanks for watching. That's it for this one.

Graph the equation $y-x^2+3=0$ by choosing points that satisfy the equation.

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Graph the equation $y=\sqrt{x}+1$ by choosing points that satisfy the equation. (Hint: Choose positive numbers only)

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Graphing Intercepts

Everyone, throughout some of your graphing problems, you may be asked to identify these things called intercepts, and that's what I want to show you how to do in this video. Basically, what we're going to see here is that intercepts are really just special places or special points where the graph crosses either one of the x or y axis. So I'm going to show you how to identify those types of points and also some special things about them, and we'll do an example. Let's get started.

Alright. In this diagram here, I've got 2 lines. I'm going to call this one a and this one b. Both of these graphs actually cross the x and y axis. If you look at line a, line a crosses the x axis over here. Line b crosses the x axis over here. These things are called the x intercepts because that's where they cross the x axis. So the x intercept is where the graph crosses the x axis. Now, similarly, these graphs also cross the y axis over here and over here. So these points over here are called the y intercepts. That is basically the y value where the graph crosses the y axis.

Alright. So pretty straightforward here. If you take a look at line a, line a crosses over here. So this is an x intercept, where x=-2. And over here, this is where x=-4. So it's basically just the x value where that graph crosses the x axis. Alright. And over here, we've got y=4, and over here, we've got y=-3. Alright. So that's what the y intercepts and x intercepts are.

Now, what's sort of sort of confusing about this is what happens to the other value. So, for example, in the x intercepts, what you're going to see here is that the y value at any point along this line is always equal to 0 because you're basically right on the line, you have no height above it. So the x intercept is where it crosses the x axis, but it's also where the y value is equal to 0. Now similarly, what happens for the y intercept is that the x value is going to be 0. So notice how here, anytime you're on the y axis, you're not to the right or to the left of the origin. So the y intercept is where it crosses the y axis, but the x value is always 0. That's kind of like the opposite. Alright. That's really all there is to know about x and y intercepts.

So let's go ahead and do a quick example. Here, we're going to take this graph, and we're going to write the x and y intercepts of the graph that's shown below here. Remember, the x intercepts were crosses the x axis. If you look at this line or this sort of curve-shaped over here, this curve crosses the x axis twice, once over here and once over here. So what happens here is that the x intercepts are just the values. It's just the number of the x coordinate. So in other words, our x intercepts are negative 3 and positive 5. You don't have to write the ordered pair. You just have to write the numbers. Then what happens for the y intercept? Well, this graph also crosses the y axis over here. That's going to be a y intercept. So your y intercept over here is going to be y=-4. So that's really all there is to it. Right?

So let's take a look at another example here. Now the question is written a little bit differently because instead of having to write the x intercepts and y intercepts, we're just asked to find the intercepts of the graph below. Okay? So let's take a look at that. In this graph here, we've got this sort of like circle-looking thing, and notice how this graph actually crosses the x axis right over here. So basically, what happens here is when they're asking for the intercepts, they're actually not asking for the x value. They're asking for the ordered pair. So that's sort of the real confusing thing with these types of problems. But, basically, if they ever ask for an x or a y intercept whenever they actually reference the letter by name, then all you have to do is just write the x or y value. But if they just ask you to find the intercepts like they're asking you for in the second example, then you're going to write the ordered pair. So, really, all we have to do for the second example is just write the ordered pairs for these two numbers. And in this case, it's just going to be 2 comma 0, and then also 4 comma 0. So these are the intercepts of the graph, but these are the x and y intercepts of this graph. Alright. So that's really all there is to intercepts. Thanks for watching.

Here’s what students ask on this topic:

What is the rectangular coordinate system and how do you plot points on it?

The rectangular coordinate system, also known as the Cartesian plane, consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin (0,0). Points on this plane are described as ordered pairs (x, y). To plot a point, start at the origin. Move x units along the x-axis (right for positive, left for negative), then move y units along the y-axis (up for positive, down for negative). For example, to plot (4, 3), move 4 units right and 3 units up.

How do you graph an equation with two variables?

To graph an equation with two variables, such as y = 2x - 1, follow these steps: 1) Isolate y if necessary. 2) Choose several x-values and calculate the corresponding y-values to create ordered pairs. 3) Plot these points on the Cartesian plane. 4) Connect the points with a line or curve. For example, for y = 2x - 1, if x = -2, y = -5; if x = 0, y = -1; if x = 2, y = 3. Plot (-2, -5), (0, -1), and (2, 3), then draw a line through them.

What are intercepts and how do you find them on a graph?

Intercepts are points where a graph crosses the x-axis or y-axis. The x-intercept is where the graph crosses the x-axis (y = 0), and the y-intercept is where it crosses the y-axis (x = 0). To find intercepts, set y to 0 in the equation to find the x-intercept, and set x to 0 to find the y-intercept. For example, in the equation y = 2x - 1, the y-intercept is (0, -1) and the x-intercept is found by setting y to 0: 0 = 2x - 1, so x = 0.5, giving the x-intercept (0.5, 0).

How do you determine if a point satisfies an equation with two variables?

To determine if a point (x, y) satisfies an equation with two variables, substitute the x and y values into the equation and check if the equation holds true. For example, for the equation x + y = 5, to check if the point (3, 2) satisfies it, substitute x = 3 and y = 2: 3 + 2 = 5, which is true. Therefore, the point (3, 2) satisfies the equation.

What are quadrants in the Cartesian plane?

The Cartesian plane is divided into four regions called quadrants by the x and y axes. Quadrant I is the top-right region where both x and y are positive. Quadrant II is the top-left region where x is negative and y is positive. Quadrant III is the bottom-left region where both x and y are negative. Quadrant IV is the bottom-right region where x is positive and y is negative. These quadrants help in identifying the signs of coordinates in different regions.