Today we’re going to graph the absolute value of functions. And you’ve used absolute values of numbers before. You’ve taken absolute values of numbers and in the past you have learned the absolute value of a number is basically asking you how far from zero is this number. And we have dealt with number lines. And we’ve talked about it in that capacity but we haven’t really talked about linear functions. So we are going to take a look at y equals the absolute value of x plus two. And it’s still going to be taking the positive of that number. So it’s still talking about distance.

But absolute values lend itself very nicely to teach arts. So I’m going to draw a little t-chart here. And I’m going to put my absolute value of x plus two in the middle. So I’m going to plug in some values for x. and then input that into my little function and then out put my y so I will know what my ordered pairs look like. For absolute values it’s always a good idea to do some negatives and positives. But you can always be certain that absolute values have a very specific way of showing up on that graph. They’re not the traditional lines that you’ve been working with. So zero is always the easiest one that I plug in to begin with. I always use zero as oneofmypointsin.SowhenIputthatin,Igetayoftwo. SoI’mgoingtoplot that in, zero, two on my graph. And I always choose a few positive. I’m going to choose one and two since they are pretty simple. I’m going to put a one in and I get absolute value of three which is three. And I’m going to substitute my two in and two plus two is four and the absolute value of four is four. So I’m substituting one and I get, two I get four. So that’s what my points look like.

Traditionally what you have done, is that was enough to draw your line. And your line would look something like this. But these are absolute values and absolute values never look like straight lines. And if you were to graph this, this would be incorrect. This would not be a correct answer. If you were taking an exam for this, this would be an incorrect answer. Your teacher would mark it wrong. So what I am going to do is I’m going to try a couple of negatives, since I don’t know what the graph looks like. I’ll just throw a few in. It’s always a good idea to do positives, negatives and zero for absolute value. I’m doing four just so I get a good idea of what the graph looks like.

I’m going to substitute that negative one in. Negative one plus two is one the absolute value of that is one. The point would be negative one, one.

If I substitute negative two in, negative two plus two is zero, the absolute value of that is zero. I am still working on my straight line.

Negative three plus two, the absolute of negative one is a positive one. All of a sudden it is starting to look a little different. I get to negative three and I hop up to one. I no longer have a straight line.

If I substitute negative four in, I get a negative two the absolute value of negative two is a positive two. So I am going to go to negative four, two and I am going to plot that point. So once you have those points, you can tell that you now have a V for your graph and if were to continue with the same series of negative numbers, the y would go on and on and on and do the same as the positive numbers. So they do tend to look like an elbow or a V. It’s not always the case. You’ll see some graphs like that. Either way you want to pick several different numbers. Always choose positives. Always choose negatives. Let’s try a couple different graphs.

The next one we’re going to do – let’s use a variation of what we just did. We are going to use the absolute value of x plus two but notice we’re taking the absolute value of X, I’m not taking the absolute value of the quantity x plus two. So again, let’s look at the t-chart. And I’m going to substitute what Y equals in the middle here. If you did not know what Y was set equal to, you would have to solve your equation first for y. You would have to put it into the y equals mx plus b format. So far I’m giving you equations that are in that format. So I’m going to choose a couple variations. I’m going to do two positives, two negatives and zero. And I’m going to choose numbers that are not right next to each other. Negative two, negative one, zero, one, two on my last graph could have fooled me into thinking I had a straight line. So I am going to do something that has just a little bit of a variation. If I get a straight line again I’m going to have to go further out with my numbers. Maybe choose like negative seven or eight or even ten.

So if I substitute negative five in, I end up with negative five the absolute value of that is five plus two is 7, so my ordered pair is negative five, seven. Let’s see if I can graph this all the way up here.

If I substitute negative one in, the absolute value of negative one is a positive one plus two is three. So I have negative one, three. If I substitute a zero in, I end up with two, zero two. Still if I were to graph this at this point it looks like a straight line. Let me substitute my one in. The absolute value of one is two OR – the absolute value of one is one plus two is three. So my ordered pair is one, three.

And now I’m starting to veer over to a different looking line. The absolute value of five is five plus two is seven so this particular ordered pair is five, seven. So now I have my V or elbow, if you will, and my line because I cannot connect my dots here. My line is not looking not like a line but an absolute value.

Ok, let’s use another variation of x and 2. This time we’re going to graph y equals negative the absolute value of x plus two. Once again, do a t-chart. And this time I’m going to choose since I have no idea what this is going to look like, I am going to choose a wider range of numbers, negative eight, I’ll stick with negative five, negative one, zero, two and five.

So I have a nice group of numbers. When I start substituting in and you can do one point at a time or you can just fill it in the entire t-chart row or column or each row. And then find your y at the end. There are a couple of different ways to do it. Either way you just want to remember to choose a wide variety of choices for your x values.

So if I go back to my negative eight. Negative eight plus two is negative six. The absolute value of that is six but I have this negative out in front so it tells me my answer should be negative so my answer is negative six. When I substitute in negative five, the absolute value of negative five plus two is three but this negative again out front tells me the answer should be negative.

Substituting negative one in negative one plus two is one, the absolute value is one. But I have a negative one because of that negative out in front.

Zero plus two is the absolute value of that is two and then I have a negative two.

Plugging a two in I get four, the absolute value of that is four but I have a negative so it’s a negative four.

And then five plus two is seven, absolute value is seven, with the negative out in front, it’s a negative seven. So If I were to graph that series of points, to get my line, then I have oh, negative eight, six. Way out here off the chart almost or off the graph. Negative five, three. Negative one negative one. And then zero negative two. And then two negative four and then five negative seven. I’m not really sure where my point is going to be so I’m going to actually plot another point, negative two to see what that looks like and that’s going to give me a zero. Definitely on up higher here and to see if that is the point or negative three is the point, I’m going to graph a negative three very quickly. Negative three plus two is negative one, the absolute value of that is one, and when I pull the negative out there, negative three, negative one so it looks like the tip of my V is at negative two, zero. This is what that graph will look like.