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Connecting Graphs, Tables, and Equations - Concept
In Algebra we do a lot of work with lines and their graphs. Connecting graphs, tables, and equations of lines is an important practice so that we can to help understand lines and how to graph them. When looking at graphs and tables, there are important characteristics that we need to be able to identify including the y-intercept and the slope.
When you guys are studying linear equations, you're going to learn about lots of different ways to graph them. Most times students start by learning how to make a table of values then they plot points on the dot and connect, or plot the dots on the graph and connect them with a ruler. But that can take a really long time especially if you're not good with order of operations it's easy to make mistakes.
So another way that students like to learn about graphing lines is by looking at shortcuts, shortcuts and making connections between the table of values, the graph and how steep the line is and also the equation itself.
So what I'm going to do is I'm about to show you guys what it looks like to use one of these calculators. Your teacher might want to have, might have one of these available for you in class but maybe not. It's not important that you know how to use the calculator, what I really want you guys to focus on is looking at similarities and differences between the table of values, the equations and the graphs. So let's go ahead and look at the computer.
When I turn on the graphing calculator this is the home screen but I'm going to be graphing some lines. It's not important that you guys know exactly what I'm typing in here, what's more important is that you guys are looking for patterns. So first thing I want to talk about is what would happen if I were to compare the lines y=x and y=x+2.
First of all, if you look at the equations, you notice they're really really similar right? They both start with y=x and the only difference is this guy has the +2. Let's look at the graphs of those. First we'll see the graph of y=x and then we'll see the line come up y=x+2. Here we go. This is y=x here, this is y=x+2. Notice first of all that these lines are equally steep. They're like parallel lines if you know what that means. The reason why these are equally steep is because they both start with y equal x or y equals 1x. These guys both have the exact same slope. The only thing that's different was that +2 business and that shows up here with the y intercept. I take the line y=x and then I move it up 2. +2 up 2, that makes sense. Let's look at the table of values and see if that makes sense also.
When I'm looking at the table, this first column here represents my x numbers, this column is my y numbers for y=x and this is my y values for y=x+2. So you'll see each one of these numbers +2, gives me these numbers. +2 +2. That's because this is y=x and this equal y=x+2. So it makes sense, they're all plus two.
Look at these two columns. You'll notice the x's and y's are exactly the same and that's because this represents y=x. They're the same numbers. So you can see what happens in the table. I just take my y values from here and I add 2 to them in order to get the line y=x+2.
What I want to do next is add in a couple more equations just so I can show you guys some really cool patterns and so that you can start to understand what's going on. I'm adding in here two more lines, y=x+4 and y=x take away 3 and we'll look at the graph. So before I hit graph, let me tell you what you're going to see. First you'll see y=x then you'll see y=x+2, then x+4 and then x take away 3. So here we go. This, there's all four of them. This is y=x, x+2, x+4 and then x-3. Look at what's the same. All of these graphs have the same slope or the same steepness because the coefficient in front of x for all of them is 1. The thing that's different is that plusing number. +2, +4 meaning I'm moving the y intercept all the way up to 4 or take away 3. I just used the word y intercept and I hope that makes that makes sense to you. What a y intercept is, is the place where the graph crosses this y axis. In the equation y=mx+b the y intercept is the b value.
Okay. So what we've just looked at is what happens if I change the x, excuse me, change the y intercepts by changing what number I'm adding out here. Let's try something different. What I want to do next, is looking at what happens if I change the slope or what happens if I multiply x by some constant. So I'll start by doing y=x, y=2x and also y=4x. Let's look at these graphs and see what they look like. y=x, y=2x, y=4x. Notice how they get steeper. This line was y=x. There's y=2x and y=4x. What's happening is that my y values are increasing more rapidly for this y=4x business because my x number is being multiplied by 4. It makes sense also if we look at the table. Here I have y=x in this column. This is y=2x. I'm taking this x value and multiplying it by 2 to get these guys. 2 times 2 is 4, 3 times 2 is 6, blah blah blah. So you guys can see that my y values are changing more rapidly. The word change is super important when we're talking about how steep lines are because change has to do with slope. The definition of slope that you guys might already know is change in y divided by change in x.
I'm going to go back in here and do something a little bit differently. Now, instead of multiplying by positive numbers, I'm going to show you what happens if I multiply by negative numbers. I have y=x, y=-x. Let's do -2x and let's do -4x just so we can see a whole bunch of different lines. So I already I'm predicting that these lines here, -2 and -4 are going to be steeper because the absolute value of the number multiplied by x is larger. Let's check them out. y=x, y=-x, y=2x and excuse me, -2x and y=-4x. This is tricky. This is the only line that goes in this positive direction because this line is y equals positive x. This guy here is the y=-x. See how it's decreasing from left to right? That's why we call it a negative slope. A negative slope means the line decreases from left to right.
So the thing I really want you guys to remember about what we're looking at here, is how, if I change the number that x is being multiplied by, it affects what we call the slope. I could also do this with a fraction. I'm going to go in just really briefly and show you guys what happens if instead of multiplying by an integer, I multiply by a fraction. I'm actually going to use the decimal value of 0.5 which represents a half and you'll see that instead of making my line steeper it's going to be actually less steep. See, there's y=x. That's more steep than y equals one half x, because my change in y is half as steep as my change in x. That's all slope business.
Let's look at a table of values so we can see what that means. If I look at my table here, this is y=x, this is y equals one half of x. You'll see that my change in y, each time I'm only changing by 0.5 or half. That relates to the slope.
Okay. The last thing I want to do before I set you guys lose on graphing these equationS using patterns, is show you a line that uses all of this stuff put together. Let's look at the graph of y=3x take away 2.
Before I hit graph, I'm going to try to think about what this means. It's a positive slope, so I know my line is going to be increasing from left to right. It's going to be pretty steep because the absolute value of 3 is larger than 1. But then I have this -2 business, this changes my y intercept. It tells me the graph is going to cross the y axis at -2. So here we go. There it is. My y intercept is down here at 2 and my slope is pretty steep. I could count it by going up 3 over 1. Up 3 over 1. If I were to look at the table of values, I would see that my y intercepts, which happens when x is zero, my y intercept is -2 and each time my y value is plusing 3, because my change in y is a positive 3.
So you guys, these are just some shortcuts things to keep in mind when you're graphing lines. You don't have to memorize any of this stuff, it's really just patterns that will help you connect graphs with tables and their equations.