7.5 Conic Sections

[STARTS AT 4:15]

Instructor: So we are going to take this equation here. X squared minus 4x minus 8y plus 12 equals 0. We want to put it into standard form and then graph the parabola. 

Now, one thing you should notice pretty quickly is because I have an x squared term, that's going to be an opening up or down versus a left or right. If it was y squared, that would tell me it's opening left or right. So to put this in standard form, first, x squared minus 4x plus I'm going to complete the square here. 

Because I need an x minus h squared, all right? Minus 8y plus 12 equals 0. To complete the square there, that would be a plus 4, so add 4 to both sides. So this will be x minus 2 squared minus 8y plus 12 equals 4. 

Now we want to get y alone. So we'll subtract 12, will be a negative 8. 12, so that will be a negative 8. And I'm going to go ahead and subtract this term over as well. 

Let's make that minus 8. That's negative x minus 2 squared equals negative 8y. Dividing by negative 8, we get y equals negative 1/8 x minus 2 squared plus 1. So my vertex is at the point 2.1. 

Now, my p-value, I want to go ahead and note that 1/4p is equal to negative 1 over 8, which makes our p value equal to 2, which means my distance to the focus, my distance to the focus is 2. And since it is opening up or down, this gets actually opening down, we'll have a point going to in either direction, or actually p is negative 2, which is why we have it opening down there. All right, so it's going to be facing down. We have our vertex, negative 1/8. 

And I don't really want to guess at this. But I can go ahead and tell you my directrix is this line right here. My directrix is y equals negative 1 based off our p-value And where my focus is, all right. Now let's see, negative-- oh, that's actually a positive 1/8, that's why this is not looking right, positive 1/8. 

OK, 2,1 my p-value is positive, OK? p-value is positive 2. So that all lines up, OK. My graph matches. 

All right, now, we can get some other points. These parabolas are always symmetric about their vertex, which we've already labeled. So let's see, 0 4 03/2. So we'd have a point like right in here. 

So it should s of this form right here. It will look something like that. The 1/8 value actually makes it wider in this case. It always does. If it's bigger than 1, it will make it narrower or shrink, horizontal shrink. 

[ENDS AT 8:10]