Step by step
The question
Question: name the conic and write in standard form y² − 4x + 4y + 8 = 0.
1 Name the conic
Look at the squares.
Only y² is here.
So it is a parabola.
2 Keep y on the left
Keep the y terms on the left.
Move − 4x and the plain number to the right: their signs change.
3 Complete the square for y
Half of 4 is 2.
2 times 2 is 4.
Add 4 to both sides.
Careful!
What we add on the left, we add on the right too.
Then the two sides stay equal.
4 Write as a square
The bracket is a full square: (y + 2)².
On the right, −8 plus 4 is −4.
5 Take out the number in front
Take 4 out of the right side: 4(x − 1).
That is the standard form of a parabola.
6 Check
Let's check with the vertex, (1, −2).
Put it in the first equation.
We get 0, like the first equation says.
So the standard form is right!
Let's look back.
First: name the conic.
Then: keep y on the left.
Then: complete the square for y.
Then: write as a square.
Then: take out the number in front.
Last: check.