 Welcome back to our lecture series math 1050 college algebra for students at Southern Utah University as usual I'm your professor today. Dr. Andrew Misseldine This video is going to be the first lecture in our first video in lecture 15 And this is also the last section in our chapter 2 about linear functions So we've learned about linear functions with an emphasis on modeling. We talked about Solving systems of linear equations where we might have multiple linear Equations but also variables at the same time how we could find unique solutions and you know when other things happen And we've recently learned about how we can use matrices to help us with that process So oftentimes it's not good enough just to have one linear function And I don't mean like a system of linear equations sometimes The function at hand is actually a piecewise linear function. That is it could be created by piecing together different Different linear functions like tax brackets kind of work this way that maybe lower income have one You know type of tax bracket then when you get higher There's like a jump and then you get something maybe like this and then there's like another jump and it gets even steeper, right? You can piece together and they can they don't necessarily have to even be disconnected. It could be something like this Right we use piece together linear functions. These are important now probably of all the all the piecewise linear functions in the world The most important one is going to be the absolute value function Which we don't often think about absolute value in that way, but it is a piecewise linear function So the absolute value function is the function that forgets the sign of the number So when you take the absolute value of a number if it's positive You'll just get back the number x unaffected if it's zero you'll also just get back zero Which is the original number but on the other hand if your value is negative We're going to switch it to be positive and really if you have a negative number if you times it by Negative one that becomes a double negative and since it's a positive So we actually can realize absolute value as this piecewise function And when you think of the graph of the absolute value you get a graph that looks like the following right here When x is greater than or equal to zero This function right here will look just like the line y equals x the identity line It's you know makes that 45 degree angle with the x-axis It goes through all these points one one one one two two That's the function we're talking about right here now on the other hand when you take a negative number Your function will look like this line right here, which this line is just y equals negative x It has a slope of negative one so it is decreasing will go through points like negative one one Negative two two you know because just taking the absolute value of the things now It turns out that because both lines intersect at the origin We see that this graph is continuous that there was no breaks at it does contain the origin right here And when you look when you take a step back and you look at the absolute value function It actually looks like a v-shape which is somewhat fortuitous given the course we were talking about the absolute value It's a very nice mnemonic device that v is stands for value here absolute value And that can be that can help us think about this graph here Now some things I want you to do I want you to be aware of this graph. It does have that v-shape It does come to a corner when you hit the origin The origin is a point on the graph and this point we often refer to as the vertex of The absolute value v the vertex right here This will be significant because as we transform the graph the vertex will move around But it'll be a critical element to helping us graph this thing Then some other things to keep track of this is the absolute value goes to the origin 0 0 the standard absolute value function also goes to the point 1 1 and negative 1 1 these two points will be important to help you gauge The function as we transform it which will do it in just a second and so we get this absolute value function I do want to mention that the reason why absolute value is so important is that when we talk We do like story problems absolute value often comes into play when we talk about distance of some kind So like for example, if we have some variable x and I take three steps away from it Then we're gonna take x minus three. This would give me the distance that we've now gone Absolute value often measures distance because if I take three steps forward or three steps back That doesn't make much difference on the distance distance is irrelevant I should say distance doesn't care about the direction that one takes So we often use absolute value when we talk about distance functions So let's consider graphing an absolute value function So let's consider the equation f of x equals two times the absolute value of x minus three plus four so this is a Transform absolute value function and the presence of the absolute value is gonna be very helpful for you Because everything inside of the absolute value will be the horizontal zone So any numbers you see inside of the absolute value symbol will affect Horizontal transforms and everything outside of the absolute value symbols will affect vertical transforms And so that there's a clear distinction about the horizontal versus the vertical going on here So let's analyze what's going on here. We have this multiplication by two multiplying by a number always affects of some type of scaling so we are going to stretch or compress press the graph It's a positive number so no reflection is going on there. It is outside of the absolute value It's actually it's right in front of the absolute value. So this is going to have an effect of a vertical vertical scale and remember that vertical scaling Multiplying by two is going to cause a vertical stretch and so we're going to record that we have a vertical stretch by a factor of two Looking at this plus four addition or subtraction. This is has to do with shifting translating moving the graph around in the plane We have a plus four and it's outside of the it's outside of the absolute value So this is going to be a horizontal excuse me a vertical shift and because it's a plus four It means it's going to shift things up by by a factor of four And so now we enter the horizontal zone inside of the absolute value We see there's only this negative three going on right here And so because we are subtracting three This will be a horizontal shift, but in the horizontal zone things work a little bit different than we would expect Subtracting three actually moves in the positive horizontal direction, which we call right and so there'll be a horizontal shift right by three And so with these transformations in mind Let me shift over to what this graph is going to look like the standard absolute value, right? Look something like this if we were to graph it it goes through zero zero We get the point one one and then we all set the point Negative one one like I mentioned earlier. This is the standard graph and we want to figure out what's happening here So one thing to remember when it when it comes to graphing an absolute value I like to pay attention to the vertex the vertex of the original absolute value is always the origin and as we perform Transformations to the absolute value the absolute value is vertex is only affected by shifts the vertex Let's make a comment here. The vertex is only only transformed by the shifts Because reflecting doesn't change the vertex because it's already on the x-axis shifting doesn't affect The the slight I should shifting does affect it scaling doesn't because if you stretch it vertically Uh-uh doesn't matter the white coordinates already zero if you stretch it horizontally Uh-uh doesn't matter the x coordinates already zero so the origin is unaffected by reflections and stretches and compresses The origin will only be moved by shifting which is where the vertex is located And so let's then pay attention to what happened there It's vertically shifted up by four and right by three so we can count that off one two three four And then one two three So when we move the vertex the vertex is only affected by the shifting And so the shifting because we went to the right by three will be the horizontal shift will be the x-coordinate of the new vertex and The shift up will then give you the y-coordinate of the new vertex three comma four And so that tells us where the vertex is going to be Just based upon the shifts now remember this is other points one comma one This one comma one will be affected by the shifting right, but it's also affected by the stretching notice that The standard absolute value there is a change of one when you go up so there's a rise of one and there is a Run of one to get to that new point when we started looking at stretching and compressing Basically changing the slope of this line if you stretch the graph vertically that means that from the vertex From the vertex you are going to go up to so your rise got doubled There was no horizontal stretching so the run is still one So basically this line will have a slope of two now instead of the usual slope of one And so if we go up two over one If we go up two over one that's going to change the x-coordinate by one So it's now a four it's going to change the y-coordinate by By two which gives us a six So using the vertical stretch we can find this point right here in two points determine a line We could draw that line using that Well, okay. What about the other side of the line? Basically, you just take the opposite slope. So instead of positive two you'll take negative two So you go up to over one And that gives us of course the point Two comma six the y-coordinates can be the same But another way I like to think of it is absolute value always has a axis of symmetry If you reflect across this line the you always get the exact same graph So taking the same vertical line if I went one unit this way I'm going to go one unit this way right here And that's why I got two comma six instead of four comma six If you can find these three points Then you can graph the absolute value because you just connect the dots and you form the v So finding these three points is sort of critical when it comes to absolute value graphs Let's look at another example This time let's consider the absolute value function the absolute value of two times x minus three And then you're going to subtract from that too Remember everything inside of the absolute value symbols is the horizontal zone So when you see multiplication by two in the horizontal zone That means we are going to do some type of horizontal scaling Multiplication in the horizontal zone actually causes a compression. So we're going to squish our function horizontally speaking This negative three right here This negatives and pluses Represent shifting in the horizontal zone. This would be a horizontal shift And so we're going to shift things to the right by three And then this negative two on the outside that's not in the horizontal zone So it'll be a vertical shift and it will be a shift down by a factor of two In order to get the horizontal shift correctly if there is a horizontal compressor scale make sure you shift are you factored away If you had if you saw something instead like the following If instead it was like two x minus six minus two You would be best off to factor this thing first because this might be misleading on how much of a shift you are We're not going to shift six. We're actually going to shift three So you should always factor the horizontal zone The horizontal zone should be factored so that you can get the translations correctly And so then what we see here if we were to graph this thing I'm actually using the shifts first to find the new vertex. So going down by two And then over by three one two three This is my new vertex. And so this is going to have the coordinates three comma negative two All right, and then Using the horizontal compressing we've now changed the run Right, uh, we've changed the run so that we're going to go Well, well, we can we can change it like we did before but let me actually I pause for a second because I'm hesitating to say it but actually want to kind of mention something That's really interesting here. So um, what what we're going to do if we if we proceed with the way we've done it Right, we would go up by one and then go over by a half because So one up and then point five to the right Because that's what that's what a horizontal compression by factor of due the horizontal the run got cut in half You can do that But a nice little trick about absolute values is the following notice how I said it was factored earlier, right? If you take the absolute value of two times x minus three Absolutely value minus two one nice thing about absolute value is that if you have the absolute value of a product This is actually equal to the absolute value of the factors individually So you can basically take out the factor of two So you get two times x minus three inside the absolute value minus two So instead of thinking this as a horizontal compress, we could actually think of this as a vertical a vertical stretch And so absolute value has this ability to transition from horizontal compressing to vertical stretching and I actually like this a lot because Again things in the horizontal zone kind of seem to act backwards to what we expect So if you can take something out of the horizontal zone and make it into a vertical transformation You're probably less likely to make a mistake. Um, and this is what I often like to refer to as the play-doh effect The play-doh effect how a horizontal compression turned into a vertical stretch and the idea is the following If you have like a ball of clay dough or gack or clay or whatever, you know Magic sand or whatever whatever like Messy malleable toy you had as a child, right? What happens when you start to horizontally compress it if you like take that ball of play-doh and squeeze it in your hands What's gonna happen is that it's gonna get like all squished, right? It's squishing it as you squish it horizontally what that causes it to do it causes it to ooze out vertically So compressing it horizontally actually causes a vertical stretch to happen to it And that's a the play-doh effect as you squish it horizontally it becomes a vertical stretch And similarly if I was to compress it Like if I press it Vertically that can cause it to expand horizontally this play-doh effect It doesn't affect every single function, but it turns out absolute value is made out of play-doh Did you know that because compressing it horizontally is the same thing as stretching it vertically And so with that in mind we could then think of this as a vertical stretch by a factor of two And so like the last example it has a rise of two Run of one and so however you want to do it you're going to find this second point For me the second point would be four comma zero And then by reflection on the other side you're going to get two comma zero That's if you think about it as a vertical stretch But if you want to think of it as a horizontal compression You'll instead get the point Right here three point five comma negative one and then this point over here Which would be two point five negative one and you can see that not just a business about horizontal zone versus not horizontal zone If I could avoid fractions I like to kind of do that And that's sort of like a nice thing to do and so one thing I like to mention about Absolute value functions is absolute value functions can always be written in the following form Y equals a times the absolute value of x minus h Plus k with a little basically using you know Factoring the horizontal zone and then factoring the absolute value if necessary Every absolute value function can be written in this standard form The standard form of the absolute value function For which then case we see that the horizontal shift is h the vertical shift is k And so then the vertex of your absolute value function is going to be the point h comma k And so you can find the vertex of the absolute value very very quickly and then you look at this number a right here a a is going to measure the slope The slope of the right hand side of the function Right, so we adjusted it so we got a slope of two right here And so that was the slope of the right hand side And which also incorporates the fact that if a here was negative If you had a negative slope for a then your right hand side would actually go down and the left hand side is going to go down as well Uh, so that so the v either points up or points down based upon this if a is positive the v will point up If if the a's negative will be a negative slope because you can get the slope right here So if you have the vertex and you have the slope of the right hand side You have the graph because the left hand side isn't constructed by reflection So we can very quickly graph any absolute value function if we put it into this standard form