The figure shows the parabola 𝑥 equals two 𝑦 squared minus 16𝑦 plus 22 with its vertex 𝑣 marked. What are the coordinates of 𝑣?
So notice when we usually see a parabola, it’s opening upwards, not sideways like this. The reason why is because this is 𝑥 equals two 𝑦 squared minus 16𝑦 plus 22. Usually, it’s 𝑦 equals instead of 𝑥 equals. This is what makes it turn sideways. So this equation isn’t a normal standard form, and if we wanna know the vertex of this we, need to put it in the vertex form, and we can do that by completing the square.
Our first step is to group the first two terms together. And now let’s take out a GCF, a greatest common factor. And when we take a two out from two 𝑦 squared, we get one 𝑦 squared, and when we take a two out of negative 16𝑦, we get negative eight 𝑦.
Now notice we left a little space. We’re gonna be adding something in there. We will fill it with 𝑏 divided by two squared. So in a normal polynomial, 𝑏 is the coefficient in front of 𝑥 or 𝑦, not the squared term and not the constant. So for this, 𝑏 would be negative eight, and negative eight divided by two is negative four. And when we square that, we get 16, so we will add a 16 in that spot.
Now you can’t just add things to an equation; we have to keep things balanced. So if we just added in a 16, then technically that 16 is a 16 times two, so really that represents we’ve added in a positive 32. So to keep it balanced, we also have to subtract 32 from that same side. So adding 32 and subtracting 32 at the same time means we’re really not doing anything; it’s just zero.
Now the whole point of doing this is this polynomial should be something squared, so what number multiplies to be 16 and adds to be negative eight? Well, that’s negative four and negative four, which is 𝑦 minus four squared, and then 22 minus 32 is negative 10.
So now we’re in vertex form and our vertex- so our vertex is negative 10, four. So we go left 10 and up four to get to our vertex. Now usually, that negative 10 in our equation represents going up and down, but since it’s 𝑥 equals, it’s left and right, and same thing with the four.
In our normal equation, it’s 𝑦 minus the number, so 𝑦 minus, we must have plugged in positive four. So it’s negative 10, four as our vertex, and that four moved it up and down. And again, usually when it’s 𝑦 equals, that’s left and right, but for this, it’s up and down.