In the last unit, we focused on equations and inequalities. We focused on solving them and figuring out how many solutions there were. In this section, we're going to look at graphing. So, let's ease into graphing and think about a city. So, what are some of the benefits of organizing a city in this way? Check any of these reasons that you think apply.
I would say that all of these are useful properties of a gridded city, and many of these same properties wind up being useful in mathematics.
Let's say this street was named 2nd Avenue, and this one could be named 4th Avenue, and this one 6th Ave. And what if this was named 2nd Street, and this one named 4th Street? Now, I wonder, can you tell me which intersection on this map would be the intersection of 3rd Avenue and 5th Street? Keep in mind, I haven't told you these streets, but I bet you can make a good guess anyways. Choose the, the best answer.
The answer is, this intersection. I solved this by guessing that there was a pattern to these street names. Which would mean that this is 3rd Avenue. And this is 5th Street. I know this is 3rd Avenue, because, well, because it's between 2nd and 4th Avenue. And this must be 5th Street, because it comes after intersection. Now, this is pretty powerful. Using only two numbers, and in this case, the street and the avenue, it's possible to figure out exactly what intersection I was talking about. Pretty cool.
Here's a different city map. And in this city, the naming starts here. We have avenues run north south. 1st Avenue, 2nd Avenue, and so on. Maybe this is the intersection of the 0th Street, and 0th Avenue. Now, 0th Street and 0th Avenue are pretty silly names for streets, but that's okay. And maybe this intersection is important. It's where the first building may have been built, so residents called it the origin. I'll just label it 0. Now, I want to think about this intersection at 5th Street and 3rd Avenue in relationship to the origin. How would you complete the following statement? How many blocks is this intersection to the right of the origin? And above the origin. Fill in the numbers here.
Well, this intersection was 1, 2, 3 blocks to the right of the origin and 1, 2, similar to east and up is similar to north.
It looks like two numbers convey a lot of information. In algebra we use a similar system to identify locations. This is system is called the coordinate plane or the Cartesian plane. The system uses two lines known as the x axis and the y axis. The x axis is the horizontal line and the y axis is the vertical line that runs through plane. By drawing this graph we can create an infinite world of numbers and intersections. Where the x axis and y axis intersect is called the origin, the center. Just like we labelled streets and avenues on our maps, we can label our axes with numbers, 0, 1, 2, and so on. We can do this in the horizontal direction and in the vertical direction. To indicate a point on this Cartesian system, how many numbers would we need to specify? This can be tricky, so think back to our map. How many numbers would we need to specify a point like this? I'm not asking you for the location, I'm asking you for the number of points. Put your answer in this box.
I only need 2 numbers. For example to talk about this point I would indicate how far it is along the x axis and then how far along it is on the y axis. I would move 2 units to the right and then 4 units up. So this point would be 2, 4. I went 2 in the horizontal direction and then 4 in the vertical direction.
To say this mathematically, I would have to have an ordered pair, or an x and a y. The x tells me the horizontal distance, in this case 2. And the y tells me the vertical distance, in this case 4. We call this an ordered pair or we could think about calling these the coordinates of the point. So, where would you find the point for 3? Choose the best answer.
The point four three would be four to the right of the origin and three above. The x coordinate is four so we travel one, two, three four and then three units vertically up, one, two, three. So this must be the point.
So far we've only seen positive values for x and y. We've only dealt with points in this region of the graph. I have a challenge for you. Don't worry if you don't get it right just yet. Where would you find the point negative 3, 2? Choose the best answer here.
Well, this ordered pair, or point would be here. If you got this right, well done. I haven't even told you about negative coordinates yet. To think about a negative coordinate, instead of moving right, we want to move left. We move left because it's the x coordinate. We move in the horizontal direction, this way. So from the origin, we move three units left and then two units up. I know it can't be either of these points because well the x coordinate would be positive and ours is negative, so these are out. The same would be true for this coordinate, the x value would be positive 3. If you chose this point, you were a close second. This point is really 3 to the left or negative 3. And then down 2, so negative 2 in the y direction. But we don't have a negative 2, we have a positive 2 for our y coordinate. Once we move three units to the left of the origin, we must move two units up. So one, two.
So, a negative x coordinate is to the left of the origin. And a negative y coordinate is below the origin. Can you match the following points to the appropriate places on the graph? Write the letter in the corresponding box. For example, if you think this point is letter c, you would write c in the box. Good luck.
We know an ordered pair is always x, and then y. So for this first ordered pair, I move one unit to the right from the origin, and then two units down. One unit right, two units down. That's C. For the second one, I move one unit left of the origin, and then two units down. That must be B. For the third choice, I have to move two units right, and then one unit down. Two units to the right of the origin. And then one unit down, so answer choice D. This, of course, leaves us with the last one, which is A. We move two units left, for negative 2, and then one unit up. If you got all these right, great work on this quiz. And if you didn't get all of them right, think about which ones gave you trouble. Keep in mind which direction you should move first, right or left. And then, up or down.
Points are good and we use them a lot when we look at real world data, but what's even better than a point? A line. This is an example of a line on a coordinate plane. Now this line might not seem that interesting, but its got a whole personality that we're going to talk about. Notice that this line has a ton of points on it. I have points here, here, here and even anywhere along this line all of these are points, but there are two points on this line that I find most interesting. I want you to check the two most mathmatically interesting points on this line and I should say this is a debatable one. So if you get a different answer from me it just means we are thinking differently which is okay.
Well, I think these two are the most interesting. Notice that they both appear on an axis, this one appears on the x-axis, and this one appears on the y-axis.
This is the point where our line intersects the x axis. They meet right here and this is the point where our line intersects the y axis. They meet here. Now thinking back to before, what are the coordinates of this x intercept and this y intercept? Write the numbers in here.
Well, this x intercept is three units to the left of 0, so that's negative 3 for the x coordinate. And I don't move vertically up or down, so the y coordinate must be 0. For the y intercept, we don't move left and we don't move right, so the x coordinate is 0. Then, I move two units up, so positive 2 in the y direction.
Now can you see a pattern here? Something's going on. I want you to really think about what the x-intercept and the y-intercept mean, and then complete these sentences. For the x intercept the x or the y coordinate is always equal to what? And for the y intercept the x or the y coordinate is always equal to what? So choose out of the x or y coordinate for each sentence and then choose the number that it's equal to.
Well, check it out. When I'm at the x intercept, I'm right on the x axis. So this means that the y coordinate is zero. Likewise, for the y intercept, I'm right on the y axis. Which mean the x coordinate is zero. These statements will always be true for x intercepts and y intercepts.
Okay. So, we know we can draw lines on this Cartesian grid. The line is just a visual for a linear equation. Let's explore the relationship between linear equations and lines. For now, I'll just tell you that this line is described by the equation negative 2 x plus 3 y equals 6. Now, we know that this x-intercept is negative 3. But how do we find it? Let's see if you can figure it out. Would you set y equal to 0 and solve for x, set x equal to 0 and solve for y, or solve the equation for y in terms of x? Think about what this point means and think about how you could actually find the coordinate to the point. Good luck.
We know this point is negative 3,0 so the y coordinate is 0. We want to set y equal to 0 and solve for x, so these other choices won't work. So, I'm going to let the y-value equal 0, and then I'm going to solve for x. 3 times 0 is 0, and then I divide both sides by negative 2. So, we get x is equal to negative 3. That's our x-intercept. This makes a lot of sense because the x-intercept happens when the line crosses the x-axis. So, we can't be vertically up or down on the y-axis. The y must be zero. This x-axis is just an entire line, with all the y-coordinates being equal to zero. So, plugging in zero for y into our equation gives us the x-intercept.
How would we find the y-intercept? What variable should go here, what number should go here, and what variable should go here? Fill in the blanks to make the statement true. Again, think about what this y-intercept means. What's going on here?
We know the y intercept is 0, 2. This is the ordered pair. So, we must set x equal to 0, since the x-coordinate is 0 here, and then solve for the y-coordinate.
Let's put our knowledge to use. What are the x and y-intercepts for these lines? Some of these intercepts might be fractions and that's okay. Whenever you write an x-intercept or a y-intercept be sure you write it as a point, remember there are two coordinates here. For these questions, I just want you to enter the numbers here and the same in these two. I've already included the parenthesis and commas for you.
To find the first x intercept, I take my equation and I set y equal to 0. I know any number times 0 is 0 so I have negative 2x equals 8. Dividing both sides by negative 2, I get both sides equal to negative 4. So, our x intercept occurs at x equals negative 4 and the point is negative 4, comma 0. For the Y intercept, I set x equal to 0. Negative 2 times 0 is 0 and I'm left with 4Y is equal to 8. We divide both sides by 4 to get y is equal to 2. So, our y intercept occurs at y equals 2, but the point is 0, 2. For the second equation, we can find the x intercept in the same way. We set y equal to 0. So this term drops out. And I get 3x is equal to 6, then we divide both sides by 3, and x equals 2, so our x intercept occurs at x equal 2 and we have the point 2 comma 0. This y intercept is a little bit harder, we do the same process for y intercepts we set x equals to divide by 9. So I have y is equal to 6 9ths. If we remember back from fractions, we know that a 3 goes into both the numerator and the denominator, so I can simplify this fraction. 6 divided by 3 makes 2, and 9 divided by 3 makes last equation, we can find the x intercept again by setting y equal to 0, then solve for x and divide both sides by 4. 0 divided by 4 makes both sides 0. So, our x intercept occurs at point 0,0. To find the Y intercept, you let x equal 0, so I get y is equal to 0. Whoa. This is the same point and their both the origin so something is going on with this line. We'll learn about this equation in the upcoming lessons but if you're interested try graphing this and see if you can figure out what's going on.
That was all great work. You now know how to plot points in a Cartesian coordiante system. You can look at lines and identify x and y intercepts. And you can calculate intercepts from an equation which is really just a description of a line. And in the next section we will investigate lines and equations in more detail. We'll look at slope and figure out what that word actually means. Try some practice on this, and then stay with me to see what's next.
Now that you've learned about the Cartesian coordinate system, let's practice. For this problem I want you to match the points to the ordered pairs. You can put the letter that corresponds to each of the points in these boxes right here. Good luck.
Let's see which of these ordered pairs matches with these points. For negative 4 direction and that leads us to point f. So f is the answer to the first problem. For 3, -2, we know we're going to go 3 in the X direction and -2 in the Y direction. So we start at the origin and we count 3, 1, 2, 3 and -2 in the y direction means we move down, 1, 2, so we get to point A. For negative 3, 4, negative 3 in the x-direction, 4 in the y-direction. So we start at the origin. Negative 3 for x, 1, 2, 3, positive 4 for y, 1, 2, 3, 4, leads us to point c. 4, leads us to point b. And 0, 2 means 0 in the x-direction so that means we don't go left or right at all. And we simply go up to 1, 2 leads us to point D. And for the last one, negative 2, negative 3, means we go negative 2 in the x direction, negative 3 in the y direction, which leads us to point E. If you got those right, you did a wonderful job. Let's do something a little different.
For this problem we're going to practice finding the x and y intercepts. We're given an equation, 2x minus y equals 6. And we want to find the x-intercept and the y-intercept. You can put your answers in these boxes as ordered pairs.
Let's see how you did. The x intercept for this equation is 3, 0 and the y intercept is 0, negative 6. Let's see how we got those answers. Remember when we are finding an x intercept we set the y value to 0. When we set y equal to 0 in this equation we get. 2x minus 0 equals 6. Which is the same as 2x equals 6. And if we divide both sides by 2 we get x equals 3. So the x intercept is going to be 3 comma zero. To find the y intercept we want to set x equal to 0 in the equation. When we put x equals 0 in this equation, we get. 2 times 0 minus y equals 6. 2 times 0 is 0. So we have negative y equals 6. And, we can divide both sides by negative 1, which gives us y equals negative 6. So, the y-intercept is 0. -6. If you got the answer right, great work. If you didn't get it right, don't worry, just go back and look at the video, and try again.