Worked examples: Forms & features of quadratic functions | High School Math | Khan Academy

The function M is given in three equivalent forms, which form most quickly reveals the Y intercept. So let's just remind ourselves, if I have a function, the graph y is equal to M of x. These are all equivalent forms; they tell us that the function M is given in three equivalent forms. I should be able to algebraically manipulate any one of these to get any of the other ones.

And so if I wanted to graph M, y is equal to M of x, and let's say it looks something like this, and I actually know it's a downward opening parabola because I can look at this form right over here and see, "Hey, look! The coefficient on the second degree term here is negative, so it's going to be a downward opening parabola." That's a messy drawing of it.

And so, if we're talking about the Y intercept, we're saying, "Hey, where does it intersect the Y axis?" So what is the value when x is equal to 0? So it boils down to how quickly can we evaluate M of 0? What is M of x when M is when x is equal to zero? So how quickly can you evaluate M of 0?

Well, in this top one, I can substitute 0 for x, and so it'll be -2 * -3 * -9. So it's not too hard to figure out, but there's going to be some calculation in my head. Similarly, in the second choice, for x equals 0, I'd then get -6, which is positive 36 * -2, which is -72.

And then I have to add that to positive 18. I can do that, but it's a little bit of computation. But here for this last one, and this is known as standard form, if I say x equals 0, that term disappears, that term disappears, and I'm just left with M of 0 is equal to -54. So standard form, this is standard form right over here, was by far the easiest.

So we know the Y intercept is (0, -54). Now, one rule of caution: sometimes you might look at what is called vertex form. As we'll see, this is the easiest one where it is to identify the vertex. But when you see this little plus 18 hanging out, it looks a lot like this -54 that was hanging out, and you're like, "Hey, when x equals 0, maybe I can just cross that out."

The same way that I cross these terms out, be very very careful there because if x equals 0, this whole thing does not equal zero. When x is equal to zero, as I just said, you have -6, which is 36 * -2. This is equal to -72, so M of 0 is definitely not 18.

So be very, very, very careful. But we can see that the best choice is this one: standard form, not vertex form or factored form. Factored form, as you can imagine, is very good for figuring out the zeros.

Let's do another example, and actually this is the same M of x, but we're going to ask something else. So it's given in those same three forms: which form most quickly reveals the vertex? Well, I just called this vertex form before, but what's valuable about vertex form is you can really say, "Okay, this is going to achieve its vertex when this thing over here is equal to zero."

How do I know that? Well, once you get used to vertex form, it'll just become a bit of second nature. But if this is a downward opening parabola, the vertex is when you hit that maximum point. And as you can see here, x - 6^2 is always going to be non-negative.

You multiply that times the -2; it's always going to be non-positive. It's either going to be zero or a negative value, so this is always going to take away from this 18. And so if you want to find the vertex, the maximum point here would be the x value that makes this thing equal to zero.

Because for any other x value, this thing is going to be negative. It's going to take away from that 18, and so you can see by inspection: well, what x value will make this equal to zero? Well, if x is equal to 6, 6 - 6 is equal to 0. 0^2 is 0 * -2 is 0, and so M of 6 is equal to 18.

So this lets us know very quickly that the vertex is going to happen at x is equal to 6, and then the y value there, or M of 6, is going to be equal to 18. You can do it with these other ones; the hardest one would be standard form.

Standard form, you could complete the square or do some other techniques, or you could try to get it into factored form. Factored form, you can find the zeros, and then you'd know that the x coordinate of the vertex is halfway between the x coordinate of our two x intercepts, and then you could figure out the y value there.

But this one is definitely the easiest: vertex form. And what is the vertex? Well, it's going to happen at the point (6, 18). Let's do one last example.

So this is a different function. The function f is given in three equivalent forms: which form most quickly reveals the zeros or roots of the function? So once again, when we're talking about zeros or roots, if we have, let's say, that's the x-axis, and if you have a parabola that looks like that, the roots are, or the zeros are the x values that make that function equal to zero.

Or they are the x values of the x intercepts, you could say. And so what x values make— or which one is easy to figure out when this function is equal to zero? Which of these forms, because they're all equivalent, you can expand out these first two, and you should get this last one in standard form? Which one is easy to identify the zeros?

Well, in factored form, I could just say, "Well, what makes either this thing zero or that thing zero?" Because an x that makes this first one zero or the second one zero is going to make this whole expression zero. So you can quickly say, "Well, if x is equal to -1, this is going to be zero," or "if x is equal to 11, this is going to be zero."

So this is a very fast way to find out the zeros. This one here is a lot harder. You would have to solve 3 * x + 6 = -75, do some algebraic manipulation, and you would eventually get to these answers.

So I would rule the vertex form right over here, and this form, standard form, the first step I would do is try to get it into factored form. And then from factored form, I would find the zeros. And so once again, this is definitely more work than if you already have it in factored form.

So factored form is definitely what you want when you're trying to find the zeros. And here it says write one of the zeros. I could write x = 1, or I could have written x = -11.