 Welcome. Today we are going to be talking about parabolas. So recall from the previous video that the vertex form of a parabola looks like f of x or y equals a times x minus h squared plus k. Alright hopefully you had took some time to play with the website that says mathopenref.com quadvertexexplorer.html or anything else that you should have noticed on how the values of a, h, and k affect your parabola. Okay so a is going to, if it's positive, it changes how the parabola opens. So this parabola is going to open up. If a is negative then that's going to make the parabola open down. If you also do different values you know like sometimes it'll make the parabola skinnier or wider or certain things like that too but these are definitely the two most important pieces there. H, so h moves the parabola left and right. So if h is positive, now you also have to make note here there is a negative sign built into this formula. So when I say if h is positive that means you're going to have a subtraction in your formula. So if h is positive that's going to move your parabola to the right. It's also going to tell you then where the value of your vertex is. If h is negative it's going to move your parabola to the left. In h and k, so h is the x value of your vertex. Oopsie, let me erase all that stuff. I got a little crazy. H is the x value of the vertex. Vertex, x, k is the y value of your vertex. Vertex, y. Now k because it's on the outside of these parentheses here, k moves like you would think it would. So if k is positive it's going to move it up and if k is negative it's going to move it down. Okay, fantastic. So we're going to start with this next function that I made up here and I'm going to start without the calculator. If you want to go ahead and put in your calculator that's fine. I'm going to bring that up shortly. But it's good for you to build a practice with some of these. So that way if you know you want to move the parabola to the right you know your h is going to have to be positive. Or if you want to move it down you know your k is going to have to be negative. So being able to go through these different things in many different ways is a good idea. Okay, so which direction is the parabola open? Well that remember then is our a value. So looking over here our a is negative 2 because that's a negative value I know my parabola is going to open down. So let me write here a is negative. Okay, find the vertex. So for my vertex then, let me get rid of that, I need to be looking for my h and my k. So again comparing to the minus sign that was built into the formula because this is a plus sign that means my h is going to have to be negative. So my vertex is going to be negative 1 comma negative 3. Okay, so here I'll rewrite these parentheses as this is really the same as x minus a negative 1 squared. If that isn't crazy enough for you to look at. But you always want to remember then the signs get changed inside those parentheses. Outside they stay the same. So that was a minus 3. Okay, is it a min or is it a max? So thinking about this the parabola is going to open down. So that means my parabola is going to look something like that. So what does this function have? It's going to have a maximum. Okay, let me go ahead and bring the calculator up then. Let's go to the y equals. Clear out that equation from our factoring before. So I'm going to type in a negative 2 parentheses x plus 1 parentheses squared minus 3. Okay, I'm going to look at the graph of this one. So let's go ahead and graph it. And there's our beautiful parabola. It opens down as we suspected. So my max is going to be at negative 1 negative 3. Now, how would I know that by looking at this? Well, you can kind of eyeball the fact that this looks like I went left one and down three. But there is a built in feature on your calculator. So if you go to second calc right here, option number four, we would want to pick that one because we have a maximum. If our parabola opens up, we'd want to pick option number three, which is a minimum. So I'm going to pick option four. Now it asked me for a couple of things. It wants the left bound. So that means I need to get on the left side of that vertex. So I'm going to need to toggle to the right until my cursor gets over on the left side. Okay, that looks pretty good. Hit enter. Now it wants the right bound. So I'm going to toggle over on the right side. That looks pretty good. Hit enter. Now it wants a guess. So we can kind of get pretty close. That's good enough. Hit enter. And sure enough, it gives me exactly negative one, negative three. Now, if your calculator gave you like negative one point zero, zero, zero, zero, zero, one, know that that's negative one. Remember, this is just estimating things. Okay. All right. Next question says find the x intercepts. So where does this graph cross my x axis at? Looking at the graph, it doesn't. So there are none. So none. Where is my y intercept at? Let's bring that graph back up. If I look at that, it's kind of hard to tell here. So I'm going to take a look at my table. So second table, my x intercept or my y intercept remember means I want my x to be zero. So let me go ahead and toggle up here. My x is zero at negative five. So that's going to be the point zero comma negative five. Okay. And again, you know, if you want to go ahead and just plug in x equals zero, you can do that and then solve. So that would work out to or simplify it. All right. Find the axis of symmetry. Go ahead and bring up my graph again. When you look at the graph then, you know that the axis of symmetry is going to cut this parabola right here in half. So that's got to go through my vertex and it's got to be an x value. So it's just going to be x equals, because it's a vertical line, the x value of my vertex, which is negative one. So again, this value here and this value here, those should always match up. All right. Fantastic. Now this one gives me the vertex. It wants me to find the vertex, but it gives it to me in standard form. You notice this says ax squared plus bx plus c. That one's a little bit harder. So let's go ahead and graph that on the calculator since I can't just look at it and pick it off. So I'm going to go to my y equals, clear out this parabola, type in x squared minus 8x plus 5. Okay. Go ahead and hit graph. This one has a minimum, which we can just barely miss here. So I'm going to go ahead and change my window. I'm going to go to my y min since I'm missing the bottom part of that graph. And I'm going to try negative 15. Let's see if that helps us see it any better. Sure enough, that looks pretty good. So I would eyeball this to be, let's see, one, two, three, four on the ax and maybe negative 11, negative 12, somewhere like that on the y. But I don't like to guess. So I'm going to have the calculator do it for me. But this is a minimum this time. So I'm going to go to second calc. We're going to pick option number three since we have a minimum with this graph. Again, it wants the left bound. I tend to get a little bit closer though. So I'm going to toggle a little bit closer to that minimum. That's pretty good. All right, hit enter. Now it wants the right bound. So I'm going to toggle over a little bit more. Keep going, keep going. All right, that's pretty good. I'm going to go ahead and hit enter. And then my guess, well, it looks like my y might have been a little bit, my mind, I'm pretty darn close. Okay, hit enter. Now it should pop up an answer. Okay, this is what I was talking about earlier. You notice it said 4.00000021. We're going to call that four. Okay, so my minimum's at four negative 11. Okay, and I'm going to go ahead and write the word min here just so we can get practice with that. And we said that was at four negative 11. So if you were doing this for me on the test, I would want you to kind of draw out this and put a minimum here so that way I know how you found it. So that way you can tell me you did it on your calculator. Okay, fabulous. Symmetry. So we've been talking a little bit about that, but I want to go into a little bit more detail here. It says all parabolas are symmetric around their axis of symmetry. So remember again, that's the x value of your vertex. You can use the symmetry to find many pairs of points on the graph. So I took a picture out of the book looking at this. So this said, what do you notice about where the axis of symmetry lies with respect to the points 012 or 011? So that's the point right here in 811. Okay, you can't really read what's in here, which I did that purposely. So if you were to count these spaces, this is 1, 2, 3, 4 to get to my axis of symmetry. And then if I go in the other direction, it's going to count 1, 2, 3, 4. So what do you notice that axis of symmetry is right in the middle? So it's in the middle, so to speak. It's right in half. It cuts the parabola in half. What do you notice about where the axis of symmetry lies with respect to the points? Okay, so let me clear these points out of here. If we do 3 negative 4 and 5 negative 4, so we have 3 negative 4s down here, 5 negative 4s down here. Again, how many spaces is it to go between my point and my axis of symmetry there? 1. My point axis of symmetry, 1. So again, the axis of symmetry is right in the middle. This time, let's say in the center. Center doesn't really work. I like middle better. It cuts it in half. What do you notice about the y values of these symmetrical points? Well, let's take a look at them. This one is 11. This one is 11. Those are symmetric. Let's look at the next set of points. This one's negative 4. This one's negative 4. Those are symmetrical points. So what do you notice? The y values are the same. That's how they are symmetric. Fantastic. Okay, find f of 1. We're just going to have to eyeball. So f of 1, remember the inside of the parentheses at your input, so that's my x value. If I go up here on my graph, I think I would say that my y value is 4. So f of 1 equals 4. Now, what is a symmetrical point to 1, you notice I left that one blank? 4. Because remember, we said that was our y value. So we got to go directly across this parabola. So if I go 1, 2, 3, we'll hit my axis of symmetry. So that means I need to go 3 in the other direction. 1, 2, 3. So that's going to give me an x value of 7. And a y value of, well, it's got to be 4. It's got to have the same y value. Okay? Now, if you're more visual, you can also think about this on the number line. So stick 4 in the middle. So I had to go 3 this direction, and that will take me to 1. Then I'd have to also go 3 in that direction, and that would take me to 7. So that's how I know then what my x values are. So remember, this one here is my axis of symmetry. So that's x equals 4. Okay. Well, that does it for parabolas today. Thank you very much, and good luck practicing.