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## Forms and features of quadratic functions

# Interpret quadratic models: Vertex form

CCSS.Math: , , ,

## Video transcript

- [Instructor] We're told that
Taylor opened a restaurant. The net value of the restaurant
in thousands of dollars, t months after its opening is modeled by v of t is equal to
two t squared minus 20t. Taylor wants to know what the restaurant's
lowest net value will be, underline that, and when
it will reach that value. So let's break it down step by step. The function which describes how the value of the restaurant, the net value of the restaurant, changes over time is right over here. If I were to graph it, I
can see that the coefficient on the quadratic term is positive, so it's going to be some form
of upward-opening parabola. I don't know exactly what it looks like, we can think about that in a second. And so it's going to have
some point, right over here, which really is the
vertex of this parabola, where it's going to hit
its lowest in that value, and that's going to happen at some time t, if you can imagine that this
right over here is the t-axis. So my first question
is, is there some form, is there some way that I can re-write this function algebraically
so it becomes very easy to pick out this low point, which is essentially the
vertex of this parabola? Pause this video and think about that. All right, so you can imagine the form that I'm talking about is vertex form, where you can clearly spot the vertex. And the way we can do that is actually by completing the square. So the first thing I will do is, actually let me factor out a two here, because two is a common
factor of both of these terms. So v of t would be equal to
two times t squared minus 10t. And I'm going to leave some space, because completing the square,
which gets us to vertex form, is all about adding and subtracting the same value on one side. So we're not actually changing
the value of that side, but writing it in a way so we have a perfect square expression, and then we're probably going to add or subtract some value out here. Now how do we make this a
perfect square expression? If any of this business about completing the square
looks unfamiliar to you, I encourage you to look
up completing the square on Khan Academy and review that. But the way that we complete the square is we look at this first degree
coefficient right over here, it's negative 10, and we say all right, well let's take half
of that and square it. So half of negative 10 is negative five, and if we were to square it, that's 25. So if we add 25 right over here, then this is going to become
a perfect square expression. And you can see that
it would be equivalent to this entire thing,
if we add 25 like that, is going to be equivalent
to t minus five squared, just this part right over here. That's why we took half
of this and we squared it. But as I alluded to a few seconds
ago, or a few minutes ago, you can't just willy
nilly add 25 to one side of an equation like this, that will make this
equality no longer true. And in fact we didn't just add 25. Remember we have this two out
here, we added two times 25. You can verify that if
you redistribute the two, you'd get two t squared minus 20t plus 50, plus two times 25. So in order to make the equality, or in order to allow it
to continue to be true, we have to subtract 50. So just to be clear, this isn't some kind of
strange thing I'm doing, all I did was add 50 and subtract 50. You're saying wait, you added 25, not 50. No look, when I added 25
here, it's in a parenthesis, and then the whole expression
is multiplied by two, so I really did add 50 here,
so then I subtract 50 here to get to what I originally had. And when you view it that way, now v of t is going to be equal
to two times this business, which we already established
is t minus five squared, and then we have the minus 50. Now why is this form useful? This is vertex form, it's very
easy to pick out the vertex. It's very easy to pick
out when the low point is. The low point here happens
when this part is minimized. And this part is
minimized, think about it, you have two times something squared. So if you have something squared, it's going to hit its lowest point when this something is zero, otherwise it's going
to be a positive value. And so this part right over here is going to be equal to zero
when t is equal to five. So the lowest value is
when t is equal to five. Let me do that in a different color, don't wanna reuse the colors too much. So if we say v of five is going to be equal to two times five minus five, trying to keep up with the colors, minus five squared minus 50. Notice this whole thing
becomes zero right over here. So v of five is equal to negative 50, that is when we hit our low point, in terms of the net
value of the restaurant. So t represents months,
so we hit our low point, we rewrote our function
in a form, in vertex form, so it's easy to pick out this value, and we see that this low point
happens at t equals five, which is at time five months. And then what is that lowest net value? Well it's negative 50. And remember, the function
gives us the net value in thousands of dollars,
so it's negative $50,000 is the lowest net value of the restaurant. And you might say how do you have a negative value of something, well imagine if, say the
building is worth $50,000, but the restaurant owes $100,000, then it would have a
negative $50,000 net value.