The Ellipse - Problem 2
Graphing a ellipse that has been transformed, so graphing ellipse that has been moved. So basically how we find this center of an ellipse that's been moved is the exact same way as we find the center of a circle, so we basically look at each term and see what you have to plug into it to make it 0.
So looking here x plus 2 that tells us that our center is shifted back 2 units, y minus 3 that tells us that our y it's value of it has been shifted up 3, so what that tells me is my center is back to up 3 and our center is going to be right around in here.
Now looking at what we know is that the term underneath the x coordinate relates to the x radius and the term underneath the y relates to the y radius. In this case the x term is smaller, so I know this is going to be the minor axis, the y term is bigger, I know that's going to be the major axis.
So what this tells me 4 is 2², so I know that the half of the minor axis, the x radius is going to be 2, so what I can do is go 2 units in either direction from the center and that will tell me where my widest point is. So I start at 2, I go over to 0 and I go back over 2, -4 and that's going to be where my widest portion is.
The 25 tells me that my y radius, half my major axis is going to be 5, 5² is 25 so then we go from 3 up until 8 three, four, five, six, seven, eight and we also go from 3 down 5 so we go down to -2 we then have our general points and we try our best to connect them to make an ellipse. I'm a horrible drawer, it's going to be funny but let's see what we can do. It's probably going to be more pointed than it should be, yeah it's pretty awful but hopefully you get the idea of what is going on.
So the last thing we're asked in this problem is the foci and the main thing for the foci is to remember the relationship between the major axis, the minor axis and the distance between the distance between the center and the focus. And that is going to be a² minus b² is going to equal to c² where a is the major radius, b is the minor radius and c is the distance from the center to the focus. Major, minor radius is horrible terminology but hopefully you know what I mean.
So a² is the major radius squared, that's going to relate to our larger term, so in this case it's just going to be 25. B² refers to our minor radius, our smaller radius squared which is just going to be 4 and this is going to equal c², leaving us with 21 is equal to c², c is equal to square root of 21 plus or minus.
So what that tells me is that our foci is always on our major axis and our major axis for this problem is vertical. So I know that my x coordinate or my foci is always going to be 2. So I have two foci, they both have an x coordinate of 2 and my y coordinate is going to be up 3, 21 from the center and also down root 21 from the center. So for the up I just take 3 plus root 21 and for the down I subtracted end up with 3 minus root 21.
So what we end up with is probably a point somewhere in here and probably a point somewhere in here. Those points are completely off, but you hopefully see that they will be on this major axis up a bit from the center and down a bit from the center.
So graphing a ellipse that's been transformed basically find the center as you would any circle, and then just take into consideration what you know about the major, minor axis and lastly to find your foci, just use your relationship a² minus b² equals c² and move a set distance down the major axis from the center to find where those foci lay.