 Welcome to a video today on graphs of quadratic functions. So quadratic functions, they come in two different forms. One is called the standard form, so that form is going to look like f of x, or y if you want to write it that way, equals ax squared plus bx plus c. And the squared form here, you notice that's going to be different than lines. Lines would just be the bx plus c part, but now the quadratic has a new term to it. The vertex form, which looks a little bit different, but also a little bit the same, is a parentheses x minus h squared plus k. And I want to make a note here that for both of these forms, a can never equal zero. And the a in the standard form and the a in the vertex form is actually the same thing. And you notice, like I said, the difference between quadratics and lines is the fact that we have a square in them. Not really to memorize these forms, but just make sure that you can recognize them. Alright, the shape of quadratic functions. So if you remember this from previous algebra courses, they are parabolas. So either your parabola, which is kind of a u shape, is going to open up or your parabola can open down. So let me write out that word parabola for you. Okay, some of the properties of these quadratic functions. So your quadratics always have a vertex. So the vertex is either the lowest point if your parabola opens up, or it's the highest point if your parabola opens down. So this is either the lowest or the highest point on your parabola. Those are also called minimums or maximums. And these are points. I want to make sure to emphasize that. So these are minns or maxes. Alright, which will come to those definitions So the axis of symmetry, what that is is basically then, it's an axis, so that just means it's going to be a line. It's going to run parallel to the axis. So this is a vertical line in this case that is going to split this parabola in half. And don't mind my messy drawings here. Your parabola should be symmetric. So in this case it's a vertical line that splits your parabola in half. And if you notice, how does the axis of symmetry and the vertex, what do they have in common? Well, hopefully you notice in both of my drawings here, no matter which way it opened that axis of symmetry goes through the vertex. So these two things are very much related. So once you find one, you usually know what the other one is. Okay, increasing and decreasing. So if you are following your parabola, so pretend like you're walking around it, your parabola is going to have a part of it where it's going to be either increasing or decreasing. So you always reach your graph left to right. Make a note of these. So in this case, if I'm starting at the left of this graph, it is going to be decreasing. So it's going down to my vertex. But then after my vertex, it is increasing. Okay, so the parabola is going to be going one way or the other. If I look at my other parabola though, this parabola is increasing into my vertex. It's going up and it's decreasing after my vertex. Okay, and you're always going to want to write these increasing and decreasing as intervals. So increasing means it's going up and decreasing means it's going down. And these are intervals of x values. So that's always the hard part about reading these graphs sometimes as you're looking at what's happening to the y values, but you're actually giving your answer x values. Okay, so the minimum and maximum. So the minimum is going to happen on a parabola that opens up and it's the lowest point on that parabola. So it's the minimum and it is your vertex. So it's the vertex of a parabola that opens up. For your maximum, it's the opposite. So maximums occur on a parabola that opens down. So I'll write this one a little bit nicer. So this is the vertex on a parabola that opens down and it's your highest point. So in this case, this one has a maximum and this other parabola has a minimum. You won't recognize these parabolas by the time I'm done here. Alright, the way it opens, that's either going to be up or down. So you just have to look at it as your graph a smile that means it opens up or is your graph a frown. That means it opens down. The x intercept, so that's no different than what they were before where the graph crosses the x-axis. But you'll notice with parabolas now, let me write this down, you could have two of them as the two parabolas that I've drawn. This parabola could be bumped up so you would have none of them. Or the vertex could be right on the x-axis. So that means you would only have one of them. So you could either have zero, one, or two. So every problem is going to be a little bit different. I wouldn't memorize when you have one and when you have two or anything. Just graph them on your calculator and look at the graph, but you could have that many of them. The y-intercept again is just like it was before. So where the graph crosses the y-axis and you better only have one of these. I guess that would kind of be possible to have zero, but those parabolas you're not going to see very often. But if you have two of these, that means it's not a function, so you better not have two of those. Domain is going to be your x values and typically for your domain it's going to be all real numbers because your parabola will go forever in both directions. As far as the range goes, that's your y values. And this one is not all real numbers. So it's either going to be the numbers that are above your vertex or the numbers that are below your vertex. So this has to do with the vertex. So either above or below. Let's do some examples so hopefully all the stuff will get clarified a little bit more. Okay, fantastic. So let's go ahead and go to an example. So here's the graph of a parabola. You notice that I did not give you the function though. It's just the graph. So this says for what x values is this curve increasing. Okay, so this arrow means it's going to keep going forever. So if I were to start at the left of this graph, it's going to be increasing until it hits that vertex which looks like an x value of 2. The y value is 6 but again it asks for the x values. So I'm going to either write this as negative infinity to 2 if that makes sense to you. Or you can say x is less than negative or sorry positive 2. Either way you want to write that. Decreasing. So it's going to be decreasing after the vertex. So here it's going down. Here it was going up. So this one you want to write as 2 to infinity because this arrow tells me it's going to keep on going forever. Or you can say x is greater than 2. So again once you've identified where that vertex is at, you pretty much know where this parabola is going to be broken. You just have to know which one is going to be bigger or less than. Okay, then the vertex. So my vertex we've already identified is at 2, 6 and we know that that is a maximum because it's at the top. The x intercepts so I'm going to eyeball these. They're not exactly very nice numbers on this one. So I'm going to eyeball that to be negative 0.5 comma 0 and my other one's at, let's call that 4.5 comma 0. And those are just eyeballs. They're not going to be perfect. The y intercept though on the other hand so that's where it crosses over y axis. That looks like it's exactly at 0, 2. Okay, f of 3. So that means my x is 3. So I need to look at my graph. Go over here to 3. So that's about halfway between 2 and 4 and go up to my graph. And it looks like my point is about right there. So I would eyeball my y value to be 5. So then again that's the y value there. Okay, what x values will make f of x equal to negative 10? So this time they're giving us the output. So that's going to be my y value. So I need to go down to negative 10 and notice that it's going to hit my graph here and it's also going to hit my graph over here. So my x values are negative 2 and x is also going to equal to a positive 6. Alright, what is the domain? So again we talked about these arrows. That means it's going to go forever in both directions. So negative infinity to infinity. What is the range? So this one, my vertex is at the top so it's going to be everything below what my y value is here which is 6. So I'm going to say negative infinity to 6 or you can also say y is less than 1 to 6 which technically means that I should be putting a bracket on the 6 since that 6 value is included. Okay, fantastic. So now here's a couple of data sets and the question asks can the following data sets be modeled with a quadratic function? So what you want to do either in your calculator or just graph it by hand. We can do that really quick. See what kind of a shape does this have. So here we're looking at shape. Does our shape look like a parabola either opening up or opening down or is the shape different? Okay, so when I plot 0, 5 that goes here 1, 3 we'll say goes about right here 2, 3 goes about right here 3, 5 goes about right here and 4, 9 goes up here so certainly if I were to connect these dots I would say this has a u shape so the answer would be yes. Okay, I'll show you in the next section then or in another video how we can actually come up with a quadratic function that would do that. Alright, let's take a look at this one. So 0, 7 we'll say is roughly here. 1, 4 is about right here. 2, 1 is looking here 3 negative 2, this isn't looking good here and 4 negative 5 is somewhere down here so I would say the answer to this one is no. This one looks linear. Also you will notice here our y-values repeat itself so that is a good indication that this is a parabola. None of these y-values repeat themselves so that doesn't really help. Okay, more on parabolas to come next time. Thank you very much.