Worked examples: Forms & features of quadratic functions

- [Instructor] 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. And 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 anyone of these to get any of the other ones. And so, if I wanted to graph y is equal to m of x, and let's say it looks something like this. I actually know it's a downward opening parabola, because I can look at this form right over here and say, "Hey look, the coefficient "on the second degree term here is a 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 y value when x is equal to zero? So it boils down to how quickly can we evaluate m of zero? What is m of x, when x is equal to zero? So how quickly can you evaluate m of zero? Well in this top one, I can substitute zero for x and so it'll be negative two times negative three times negative nine. 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 zero, I then get negative six-squared, which is positive 36 times negative two, which is negative 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 zero that term disappears, that term disappears and I'm just left with m of zero is equal to negative 54. So standard form, this is standard form right over here, was by far the easiest. So we know the y-intercept is zero comma negative 54. Now one rule of caution. Sometimes you might look at what is called, vertex form. And 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 negative 54 that was hanging out. And you say, "Hey, when x equals zero, "maybe I can just cross that out "the same way that I cross these terms out." And be very, very careful there, because if x equals zero, this whole thing does not equal zero. When x is equal to zero, as I just said, you have negative six-squared, which is 36 times negative two. This is equal to negative 72. So m of zero is definitely not 18. So be 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 the vertex form is you can really say, "Okay, this is going to be, "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 minus six-squared is always going to be non-negative. You multiply that times the negative two, 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, it 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 six, six minus six is equal to zero, zero-squared is zero, times negative two is zero. And so m of six is equal to 18. So this lets us know very quickly that the vertex is going to happen at x is equal to six and then the y value there or the m of six 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 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. The vertex form and what is the vertex? Well it's going to happen at the point six comma 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're 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 negative one, "this is going to be zero. "Or if x is equal to negative 11, "this is going to be zero." So this is a very fast way to find out the zeros. This one here, a lot harder. You would have to solve three times x plus six-squared minus 75 is equal to zero. 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. 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 equals negative one or I could have written x equals negative 11.