Common Core Algebra II.Unit 2.Lesson 7.Key Features of Functions

So today, what we're going to be doing is reviewing a lot of key features that functions have that you've seen in common core algebra one and a little bit in common core geometry. So let's launch into them. All right. In this first problem, we've got a function Y equals F of X given to us. Nice and graft. All right? And what we're going to do is we're going to answer a variety of questions given this function. So at each point, what I'd like to do is kind of pause and give you a chance to see if you can remember some of this terminology on your own. And then obviously we'll go through it. So what I'd like you to do is I'd like you to pause the video right now. And maybe take up to even ten minutes to think about this problem. And come up with as many answers as you can. Okay? And of course, those that you can't remember how to do, that's fine, and then we'll talk about them, okay? Go ahead and pause the video now. 

All right, let's go through them. Letter ray asks us to state the Y intercept of the function. The Y intercept of the function is simply the point where the function crosses the Y axis. In this case, that's Y equals two. Letter B, state the X intercepts of this function. Well, the X intercepts are very similar. They are where we cross the X axis. One, two, three, four, 5, 6. So X equals negative 6 and X equals one, two, three, four. The problem says, what is the alternative name that we give to the X intercepts? We call them the functions zeros. By the way, the word zeros is spelled both of these two ways. I tend to spell it with the E, but they're both acceptable spellings of the term, kind of a little weird.

 All right, letter C over the interval X is greater than negative one, less than two is the function increasing or decreasing. How can you tell? Well, let's go to negative one on the graph. And then let's go to two on the graph. So we're looking at the stretch of the graph that goes from here to here. Let me actually color that in green. So that we have it in a nice different color. Now, did you notice what I just did when I colored it? I colored it from left to right. And in that, the function was increasing. Okay, I'm still in green. Now, the way that I can tell that is that as X increases, in other words, as we move from left to right, Y goes up, and that's the key, right? Because you might say, well, look, I could go down that as well. I could go in this direction. But when we talk about a function increasing or decreasing what we're always saying is as X gets larger. In other words, read the graph this direction always read the graph this direction. 

Then you can tell, oh, at the beginning, the functions increasing, then it's decreasing that it's increasing that it's decreasing that it's increasing. All right, I'm going to erase some of this just so that we can see it a little bit better. All right, next question. Give the interval over which F of X is greater than zero. What is a quick way of seeing this visually? Well, remember, F of X is just another way of saying why. So what we want to know is where Y is greater than zero. Well, that's all of this up here. Down here, Y is less than zero. Well, so Y is going to be greater than zero. Between these two X values. So when it says give the interval, we want to give the X value. So one, two, three, four, 5, 6, and one, two, three. Oh, those are our X intercepts now. This is the correct answer. Or an interval notation, I know this looks a lot like a coordinate point. 

This quick question, why do we leave off the equal signs here? Why don't we have this instead of this? Think about that for a minute. Well, we don't have to equal sign because F of negative 6 is equal to zero. And F of four is equal to zero. And we want all the X values where the Y values are greater than zero. Not greater than or equal to zero. So we can't include those two X values. Letter E, state all the X coordinates of the relative maximums and relative minimums label each. All right, again, let me get rid of some of the writing here, so we're not confused by it. All right, relative maximums and relative minimums. There's a lot of those. There's one there. One there. Whoops, one there. Lost my pen there for a second. And one there. So this is at one two three four. We just have to state the X values, which is nice. And that's a relative maximum. And we've got that one at X equals negative one there, and that's a relative minimum. And then we've got this one at X equals two, I think it's one, two. Yep, and that's a relative maximum. 

And yes, they do tend to alternate. One, two, three, four, 5. And I bet you can figure out why that's the case. And that's a relative minimum. There they are. We also call these things turning points relative maximums and relative minimums. We also call them turning points. Or vertexes. But that's pretty much only for parabolas. Okay. Take a moment, pause the video if you need to, copy down anything. Anything you have to. Excuse me. All right, let's clear this up. So we keep going with this. Man, this graphical terminology never ends. All right, let's take a look at FG and H take a moment if you haven't worked on these problems yet. To take a shot at them. All right, let's go with them. Now, in the last problem we talked about relative maximums and minimums, letter F asks about what are the absolute maximum and minimum values of the function, where do they occur? Now, values of the function, those are always outputs. 

Always outputs, values of the function are outputs. Well, the absolute minimum output is right here. And that's one, two, three, four. So absolute min equals negative four, and notice how it says, where do they occur? At X equals one, two, three, four, 5, 6, 7, 8, at X equals negative 8. The absolute maximum little abbreviation there a bit better is way up here. One, two, three, four, 5, 6, 7, 8. At X equals two. All right, absolute maximums and minimums. The highest Y value we reach in the lowest Y value we reach. Domain and range, okay, the domain. The domain or the X values, right? The X valves. Well, what's the smallest X value? The smallest X value is negative 8. The largest X value one, two, three, four, 5, 6. It says interval notation, and we use square brackets because the negative 8 and the 6 are on there. The range, those are the Y values. Well, the smallest Y value as we found was negative four, and the largest Y value was paused at 8. And actually just comes from these two things. 

And we have, again, square brackets because we do hit negative four, and we do hit positive 8. All right, the last problem is sort of just out of nowhere. It seems. It says if a second function G of X is defined by the formula G of X equals one half of F of X plus two, what's the Y intercept of G? Well, the Y intercept of any function ever, ever, ever, ever, can always be found by putting in an X value of zero. And the thing that's really wonderful about function notation is that it is so very clear on what we're supposed to do if zero goes in for X zero goes in for X so then what are we supposed to do here? Well, zero plus two is obviously just two. Now the question is what's F of two? Well, F of two is one, two, three, four, 5, 6, 7, 8. F of two equals 8. So we're going to have one half of 8. Which means the Y intercept of that new function is four. 

And that's it. All right, lots in that problem. Lots, lots, lots. So pause the video now before we clear it out. Okay, here we go. All right, now we got to produce the graph ourselves. It says consider the function G of X equals two times the absolute value of X minus one. -8 defined over the domain negative four is less than or equal to X is less than or equal to 7. So they're telling me this is the domain. I've got nothing more than this. And letter a says sketch a graph of the function to the right. Now, any time I'm given the option of sketching a function and they don't say do it by hand, I'm going to use the graphing calculator to help generate a table of values. So let's review how to do that right now. Let's open up the TI 84 plus. All right, there's my calculator. Okay, so what I'm going to do is I'm going to put this function into Y one, then help it generate a table of values from negative four to 7. So let's do it. Let's hit equals. 

All right, if you have any equations in Y one, Y two, et cetera, clear them on out. Take a moment. All right, let's put that function in. All right, so I'm going to put in two. Now remember to get to the absolute value is a little bit tricky, but I'm going to hit the math button. Go over to the number menu, go down to absolute value or ABS, and hit enter, right? Now again, it's beautiful because it brings up those absolute value bars. Some don't. Some just bring up ABS with parentheses. But I'm going to type in X minus one. Now I have to use the right arrow key to get out of the absolute value bars, and then I'm going to put in -8. Take a look, make sure it looks good. No negative symbols there. They're all subtraction symbols. Okay, now what we're going to do is we're going to go into our table setup. So let's go into the second window. And I'm going to want my table to start at negative four. Yeah, that's just the deal. So I'm going to have my table start at negative four, make it go by once. And let's pop into the table. 

All right. Now, I could certainly write the table down. But what I'm going to do is I'm going to just start to plot the points. And that's okay as well. Generally speaking, so for instance, where the table says negative four two, I'm going to, and then where it says negative three zero, I'm going to plot that point. And then where it says negative two negative two, I'm going to plot that point. Negative one, negative four. I'm going to plot that point. Zero negative 6. One negative 8 to negative 6. Three negative four. Four negative two. 5 syrup. 6 to N 7. Four. All right, I'm going to use my line tool now. Because it's just going to make my life a little bit easier. But I'm not going to put arrows on. Okay, that's actually a little bit important. 

No arrows. Let me write that down. No. Arrows. All right, now that I've got the thing there, I don't need my graphing calculator anymore. So I'm going to put that away. All right, graph and calculator is gone. And now I can just analyze my function just like I was doing before. Letter B, state the domain interval over which this function is increasing. All right. Well, there's lots of different things that we can say. All right? It's definitely increasing over this span, right? It's going up over the span. Most people will say that it's increasing from negative one less than X less than 7. That would be okay. You could even say negative one is less than or equal to X is less than or equal to 7. It would also be okay to include the equalities. You could do a little bit of interval notation negative one to 7, like this. 

You could also do negative one to 7 like that. Okay. Letter C, state the zeros of the function on this interval. All right, well, where are those zeros? Those zeros are my X intercepts. They're here and here. So that's an X equals negative three, and one, two, three, four, 5. And X equals 5. Letter D state the interval over which G of X is less than or equal to zero. Notice this time we put the equals. Therefore, I want everything down here, right? In fact, I want everything on this interval from negative three to 5. Now this time I include the negative three and the 5, those equal signs. Because at negative three and at 5, G of X is equal to zero. And I've included that equality. All right. Letter E says evaluate G of zero by using the algebraic definition of the function, what point does this correspond to on the graph? So real quick, let's get a little bit of a workout on here. I mean, I know we could just look at the graph, but let's just make sure we can do it by hand. Absolute value of negative one is one. And that's going to be two -8, which is negative 6. And of course, that corresponds to the Y intercept. 

Well, there it is. Let it rest. Are there any relative maximums or minimums on the graph if so, which and what are their coordinates? And you bet there is a relative minimum at one comma negative 8. Right there. Okay. That's it. But it was fun because we had the calculator out. We had to graph that. And then we were able to answer all this different type of terminology. This is a very, very terminology laden lesson, and it's really important that you come out with it with having reviewed that terminology. Okay. Pause the video now if you need to and I'm going to clear out all this mess. Okay, here we go. Exercise three. A continuous function F of X has a domain of negative 6 to 13 with selected values shown below. The function has exactly two zeros and has exactly two turning points. One at three comma negative four, and one at 9 comma three. 

Let's talk about this piece of terminology really quickly. I continuous function. All right? Not all functions are continuous, but what a continuous function means is that I can draw it without ever picking up my pencil. So that's a continuous function. I'll erase these eventually. This. Is not a continuous function. All right? Because I had to pick my pencil up, it's still a function, but it's not a continuous function. So continuous function is one where there's no breaks, no gaps, no holes, no jumps, no skips. No hops, apparently. There we go. Man, I tell you. I'm going to make a bigger figure eraser. Oops. Okay, but let's get into it. But it raises state the interval over which F of X is less than zero. All right. Oh. Um. Well, here's a place where it's less than zero. All right. So it's tempting to say from zero to 5. But it's actually also less than zero between these two things. How do we know that? Well, because we've got zeros here. 

So it doesn't go positive anywhere in here, because then we'd have to have another zero. So literally F of X will be less than zero. When X is greater than negative one, but less than 8 or an interval notation like that. Letter B state the interval over which X F of X is increasing. State the interval over which F of X is increasing. All right? Well, it's definitely decreasing in here. Then it's increasing in here, then it's decreasing again and here. So where is it increasing? Well, it's going to be increasing from this turning point. All the way up to this turning point, right? You can almost imagine it. That one turning point down at three comma negative four is a low point, and then it goes to the point 9 comma three. So we are increasing over the interval three to 9. Error of X goes from three to X goes to 9 or increasing. Going up. All right. So dealing with this terminology even with tables.