How do I graph the equation 16x^2-9y^2+32x+18y-137=0 on a TI-83?

Sep 6, 2014

First you would have to complete the square on both variables, x and y:
16x^2−9y^2+32x+18y−137=0

Group together the x terms and y terms, and move the constant to the other side.
$16 \left({x}^{2} + 2 x\right) - 9 \left({y}^{2} - 2 y\right) = 137$

Now, complete the squares. Half of the linear coefficient, square it, add inside the parentheses on the left and multiply by the factored out number before adding to the right.

$16 \left({x}^{2} + 2 x + 1\right) - 9 \left({y}^{2} - 2 y + 1\right) = 137 + 16 - 9$

Completed squares are now factorable:

$16 {\left(x + 1\right)}^{2} - 9 {\left(y - 1\right)}^{2} = 144$

I can already tell by the form of the equation, that the graph will be a hyperbola. Now, when you solve for y = , there will be two parts to graph.
$- 9 {\left(y - 1\right)}^{2} = 144 - 16 {\left(x + 1\right)}^{2}$

Divide both sides by 9, and take the square root:

$\sqrt{{\left(y - 1\right)}^{2}} = \sqrt{\frac{144 - 16 {\left(x + 1\right)}^{2}}{9}}$

So y - 1 = $\pm \sqrt{\frac{144 - 16 {\left(x + 1\right)}^{2}}{9}}$

And last, add 1: y = $1 \pm \sqrt{\frac{144 - 16 {\left(x + 1\right)}^{2}}{9}}$

You will have to type in two separate equations into your calculator! (TI-83 or 84 models)
y1 = the part with the "+" sign
y2 = the part with the "-" sign

In other online applications or in the TI-nspire, you do not have to solve for y = any more. After this step: $16 {\left(x + 1\right)}^{2} - 9 {\left(y - 1\right)}^{2} = 144$
just divide both sides by 144 and simplify:
$\frac{{\left(x + 1\right)}^{2}}{9} - \frac{{\left(y - 1\right)}^{2}}{16} = 1$
Here is what the TI-nspire looks like now: