Question

How do I graph `16x^2+y^2+32x-18y=119` algebraically?

Answer 1

Get the equation into a familiar form, and then figure out what each number in that equation means.

Explanation:

This looks like the equation of a circle. The best way to get these into a graphable form is to play around with the equation and complete squares. Let's first regroup these...
`(16x^2 + 32x)+(y^2-18y)=119`

Now take out the factor of 16 in the x "group".
`16(x^2 + 2x) + (y^2-18y)=119`

Next, complete the squares
`16(x^2+2x+1)+(y^2-18y+81)=119+16+81`
`16(x+1)^2+(y-9)^2=216`

Hmm... this would be the equation of a circle, except there's a factor of 16 in front of the x group. That means it must be an ellipse.
An ellipse with center (h, k) and a horizontal axis "a" and vertical axis "b" (regardless of which one is the major axis) is as follows:

`(x-h)^2/a+(y-k)^2/b = 1`

So, let's get this formula into that form.
`(x+1)^2/13.5 + (y-9)^2/216 = 1` (Divide by 216) That's it!

So, this ellipse is going to be centered at (-1, 9). Also, the horizontal axis will have a length of `sqrt13.5` or about `3.67`, and the vertical axis (also the major axis of this ellipse) will have a length of `sqrt216` (or `6sqrt6`), or about `14.7`.

If you were to graph this by hand, you would draw a dot at (-1, 9), draw a horizontal line extending about 3.67 units on either side of the dot, and a vertical line extending about 4.7 units on either side of the dot. Then, draw an oval connecting the tips of the four lines.

If this doesn't make sense, here's a graph of the ellipse.
graph{16x^2 + y^2+32x-18y =119 [-34.86, 32.84, -8, 25.84]}