What is the vertex form of the function 𝑓 of 𝑥 equals negative 𝑥 squared plus six 𝑥 plus five?
In order to put this function into the vertex form, we need to complete the square. And our first step is to group the first two terms together. And then, we need to make the leading coefficient a positive one and since there’s a negative, we should take that out. And now that we’ve done that, once we’ve taken out the negative from negative 𝑥 squared, that turns into positive 𝑥 squared. And when we take out a negative from six 𝑥, that makes it negative six 𝑥.
So now, we need to work with a formula. So 𝑎 is our leading coefficient — it’s one — 𝑏 is negative six, and 𝑐 is five. And we need to take 𝑏 divided by two and square it. So we plug in negative six for 𝑏 and negative six divided by two is negative three and negative three squared is nine. And what we will do with the nine is to add to the inside of the parenthesis. But we have to keep the equation balanced. So since there is a negative upfront, technically this is a negative nine. So if we technically added in a negative nine to the equation, we need to add in a positive nine to counter it so that way it stays balanced, which we’ve done here.
So we call it completing the square because we should create something that is squared. And inside the parenthesis is actually going to be something squared. Two numbers that multiply to be nine and add to be negative six are negative three and negative three. So we can replace what’s in the parenthesis with 𝑥 minus three times 𝑥 minus three. And we can rewrite that as 𝑥 minus three squared. And then, five plus nine on the outside is 14.
And now, we’re in vertex form. So 𝑓 of 𝑥 equals negative 𝑥 minus three squared plus 14 will be our final answer. And the reason they call this the vertex form is because these numbers will represent where the vertex would be. And it’s three for 𝑥 because it’s 𝑥 minus and then whatever we plug in. So 𝑥 would be three and 𝑦 would be 14. So the vertex of this function if it were graphed will be at three, 14.