(15 points) Identify the type of conic section with the equation4x = 4y2 8y + 4and find the coordinates of the vertex/vertices and focus/foci. Then sketch it....

Identify the type of conic section whose equation is given and find the vertices and foci.

$ 4x^2 = y + 4 $

Hey, it's clears the one you married here. So we have four. X square is equal to ay square plus four. This gives us for X square minus Y square is equal to four. We divide by four to get X square over one. One is why square over four is equal to one. This gives us X square over one square Wyness, Why square over two square is equal to one. A square is equal to one. The square is equal to floor. So C square is equal to five. The folk I get plus or minus square root of five common zero and the vergis ese we get plus or minus one common zero.

Okay for this problem it covers again. The idea is that a parabola hyperbole or any lips and our key idea that we need is that Parabolas have just one variable squared hyperbole. La's both variables r squared. But you're subtracting and for ellipses you have both variable squared but you're adding Okay. So what we can tell right away from this problem four X squared Equals y Plus four. Again, I only have an X squared and then just a y know why squared. So that tells me right away. This is a parabola. Okay, awesome. One part now we have to get our information about problems so we have to remember that the standard Parabola looks like this X squared equals four. Py Okay. That has a vertical axis goes up and down. And if there's no shifting like there isn't in this one the vertex is at 00. Yeah. And the focus is that zero comma piece. Okay, so what we have to do now to identify my vertex and focuses, I need to get this equation over here for X squared equals y plus four. Into this form X squared equals four. Py Okay so that's gonna take some factoring. And so what I need to do is I need to take a look at this guy right here. Okay. I need to somehow factor this stuff, right? And so what I'm going to do is I'm going to factor out oh sorry that's not what I'm going to do, silly tammy. It's easier than that. I don't like this for here. So I'm just going to divide by that four and get X squared equals Y plus four divided by four. And again I need to get a number times The wise. So I'm just gonna instead of dividing by four, I'm going to think that I'm multiplying by 1/4. Okay perfect. Now this is telling me I have a shift in my wise, So my wives are shifting from Y. two y plus four. Okay, so again we've got a shift. Okay This guy here this 1/4. That is what for P. Is So four p. is the same as 1/4. Okay, so let's work that out here. four p. is 1 4th, Divide both sides by four. Can we get P. Is a 4th divided by four, which is a 16th. Okay. Already, super. And then finally There's no x shift here because it's just X. It's not X -2 or anything like that. So there is no shift in my exes. Alrighty now let's head over here. Let's work on our vertex and our focus. Okay, so for my vertex. All right, there's no shift in my exes. So my ex goes to X. So that means zero goes to zero because again, my vertex should be zero. But now my why goes to Y plus four because I do have a shift in my wise. So instead of why being zero, Y plus four is zero. So why is negative for And so that tells me my vertex is over zero down negative four. Perfect. All right. Let's work on the focus now. Again. No shift in the exes. So here with my ex zero. That just means X is going to be zero. But there is a shift in my wise. So, my y goes to y plus four. So, my y plus four is my p. All right. But we remember from over here We found Piers 1/16. So why plus four is 1/16? So, do that subtraction. Why is 1/16 -4? four is 64 16th. So, this is going to get me negative 63/16, awesome. And that is then going to let me conclude that my focus is over zero for the X And down negative 63/16 4 wise. Great. Kind of like these problems have a good rest of your assignment.

In this problem, we wish to identify the type of chronic section given by equation for x squared plus four, X plus Y squared equals zero. Hey, that we wish to find the corresponding vortices or vertex and fosse or focus of that conduct this question challenges our understanding of comic sections broadly. Remember the comic sections are either parabolas high, parabolas ellipses as they listed here below. To determine the correct form of Arconic. We need to solve for the simpler equation given here by completing the square and X we obtained for expose half squared plus y squared equals one from this. We see we have the form of an ellipse. So to determine the ellipse, Foxy advertises, we need to utilize parameters of ellipses I've ever given here because we have be greater than a We have a form to where the verticals are. Hk plus or minus B. The folks at hk plus you might have C where C is given here. Here we see the 18 negative one half, K is zero, is one half and be as one. So we can solve for C s 3/2. And for this we obtain vortices that negative one half plus and minus one. First side of negative one half plus or minus 3/2

Okay in this problem this is your typical, is it a probable hyperbole or your lips problem? And the key idea to these problems is which form is it in? Is it an X squared equals y or a watch? X equals y squared. Sorry, X equals y squared. Um In other words is one of the variables and only one of the variable squared or if both variables r squared, is it x squared minus y squared or y squared minus X squared? Or is it x squared plus y squared scissors? Are we subtracting the two squares or are we adding it? Those are the differences between the hyperbole A and the ellipse. So the first thing we need to do to start these problems is to try to get it in one of those forms. And this problem for X squared equals y squared plus four. It's pretty easy to get this one in that form. We're just going to subtract the y squared from both sides. Okay, so we're thinking it's a hyperbole to Okay, but again remember we want to make sure it's equal to one not equal to four. So what I'm gonna do is I'm gonna divide everywhere by four. Okay. And so if I do my fraction magic here, going to get X squared here and I'm going to write that X squared over one squared again, I can do that divided by one. Not a big deal. We'll see why we want to do that in just a minute minus Y squared over four, which is also two squared equals one. Okay, so again, I've got this form here that looks like an X squared minus or y squared. And so that is going to tell me that I have a hyperbole to they're both squared and I'm subtracting the two squares. So that means hyperbole to Okay, our next step is going to be can we find the vertex is And the folks I Okay. Alrighty. So to do this we have to remember a couple of other things about our form. So let me write those up here in our key idea area. So we have X squared. Okay to I X squared. If we go back and look at our hyperbole form it's going to be X squared over a squared minus Y squared over B squared equals one. Okay. And if that is the case then I know that my vert is is our Plus or -10. Those are my vortices. Yeah. And my focus I are plus or minus C. Yeah comma zero. Where a squared plus b squared equals c squared. Okay, so first of all, double check down here. I do actually have that form in my problem. There's no shifting going on. So I don't have to worry about moving my Perabo left and right or up or down. Okay so my vertex is okay. A. Is the number that squared under the X. So a. Is that 1? And B. Is the number under the Y. That's squared. So B. Is too. Okay. Alright so that tells me I can just read off my versus is are plus or minus again. A. Is one zero. No shifting in this problem. Right. My full sigh. Well I need see where A squared plus B squared equals C squared. So one squared for a two Squared for B equals C squared five. It's C squared. So C. Is the square root of five I think. And so my um force I are plus or minus square to fi on the zero. Thanks. Awesome. Great one. Much easier because there's no shifting. OK super. Keep up your work on the assignment