In this video, in the series of quadratic functions, I want to focus on graphing these quadratic functions in more detail. Show you all the little parts and take you through this process. Let me start out with an example here, one that we are partially familiar with.
We've got x minus 1 squared minus 2. That's the given problem and we want to graph this in all of its detail. Now the first thing I notice that's understood in the problem, because it fits this formula here, a times x minus h squared plus k. Remember again that that's our standard form. That's the easy form to work from. and I have an understood value of a being 1 since there's no number out there and since that a since a is positive since a is positive we know that the graph opens up we know the vertex we've been practicing that now we know that the vertex is located at h k h takes on a value of 1 in this particular problem because it's x minus the h value k takes on a value of negative 2 so so what I'd like to do is begin graphing this let's get some graph paper going here and I'm looking at 1, negative 2 I'll say each one of those marks is one unit so you can easily read my graph.
And let's mark that vertex. 1, negative 2. 1, negative 2 should... set right there. I know that the graph is opening up but I don't want to graph that yet because I don't know how wide it's opening.
So what I'm going to do is I'm going to pause there and I'm going to mark this axis of symmetry that goes right through that vertex. Remember this is called, I'm going to write it again, this is the axis of symmetry. And this point here is the vertex.
OK, now to determine how wide it opens we just need to choose another value that's either side of this line. And you can choose anything either side of that line. I think I will choose or maybe something like the value 2 because that's just one unit over so I'm going to make myself a little chart here that says well what if X is 2 If x is 2, go back to my problem, it says 2 minus 1 squared, let me write that out, 2 minus 1 squared minus 2. 2 minus 1 is 1, and 1 squared is 1, and 1 minus 2 is negative 1. So I have the point 2, negative 1. Now, here's what I want you to notice.
The axis of symmetry says, well, if that's 1 unit over and has that y value, then I can mirror across there, and I can get that one. Well, I've got now an idea as to how wide this thing opens. So I will roughly sketch it. There's a pretty rough sketch. And this is a good time to go ahead and talk about a topic that we studied the first part of this unit of domain and range.
Domain and range. The domain of a quadratic function is always all real numbers. The domain... would be all real numbers abbreviate there. The range let me explain that domain before I go further make sure you've got that the reason it's all real is do you notice how the parabola is continuing to open out?
It will continue to open out open out and it'll fill up that whole number line eventually there's never a gap or anything it just fills up every x value throughout there. The range which measures the y value The lowest y-value is here indicated at this point, the vertex, and it opens up all the way to infinity. What's the lowest y-value, you ask? Well, that y-value comes from that vertex y-value.
So the lowest you get is negative 2. The highest goes to infinity. So you could say... say the range is all Y's that are greater than or equal to negative 2 greater than or equal to negative 2 another way to write that is you could say from negative 2 all the way up to infinity this one probably will make more sense but they both mean the same So we've taken a function, we've identified whether it opened up or down by the a value, we found its vertex, we plotted that, we drew an axis of symmetry through that vertex point. and then we answered how wide it opens by plotting at least one other point and then we declared its domain and range the last thing I would do to finish this off is just say what is this axis of symmetry what's the equation of it because you know it's a line and it's vertical because it goes up and down so what's the equation of a vertical line you say well it's the x value it goes through x equals 1 so we have a vertex clearly marked at 1 negative 2 we've got an axis of symmetry at x equals 1 and we have the domain and the range clearly marked I'll give you one more example of this in the next video