Step by step
The question
Question: find the vertices, the asymptotes and the foci of x² over 9 − (y − 2)² over 16 = 1.
1 Find the center
The center comes from the brackets.
The sign turns: x alone gives 0, y − 2 gives 2.
So the center is (0, 2).
2 Find a and b
a² is the bottom under the plus part, the one with no minus.
It is under x.
So the curves open left and right.
a² = 9, so a = 3.
It is not the bigger bottom here: for a hyperbola, the plus part decides.
The other bottom is b² = 16, so b = 4.
3 Vertices
The vertices are where the curves start: a = 3 from the center, left and right.
Draw the grid.
Put the center on the grid.
And the two vertices.
4 Draw the box
A helper box: 3 each way across, and 4 each way up and down.
5 Asymptotes
The asymptotes go through the corners of the box.
The curves get closer and closer to them, but never touch.
Their slope is up over across of the box: 4/3.
One goes up, one goes down.
So the asymptotes are y − 2 = (4/3)x and y − 2 = −(4/3)x: the point-slope form, through the center.
Now the two curves: each starts at a vertex and bends out along the asymptotes.
6 Find the foci
The foci sit inside the curves, c from the center: c² = a² + b².
Careful!
For a hyperbola it is plus: the foci are further out than the vertices.
Minus is the ellipse rule.
Now put the numbers in: a² = 9 and b² = 16.
9 plus 16 is 25.
So c = 5, because 5 times 5 is 25.
Go c from the center, the way the curves open: (−5, 2) and (5, 2).
Put the foci on the grid, inside the curves.
7 Check
Let's check with a vertex, (3, 2).
Put it in the standard form.
We get 1, the right side.
So the vertex is on the hyperbola.
Right!
Let's look back.
First: find the center.
Then: find a and b.
Then: vertices.
Then: draw the box.
Then: asymptotes.
Then: find the foci.
Last: check.