Graphing quadratics in factored form when a=1

0:00- [Instructor] So, what we have here are two quadratic functions or functions defined by quadratic expressions, and they're in factored form, and we wanna get a little bit of practice actually graphing them straight from factored form. So why don't you pause the video and try to graph these two things, or at least think about how you would try to approach it.

0:19Okay, now let's first try to graph f of x right over here. And so we see that this is of course the x-axis, this is the y-axis, the vertical axis. Now, in factored form, it's very easy to figure out what x values make this function a 0. Essentially, when x - 4 is equal to 0, the whole function's going to be 0, because 0 times anything is 0.

0:42And similarly, when x + 2 is equal to 0, the whole function is going to be equal to 0. Now, what would make x - 4 equal to 0? And I could write this over here. Let me write x - 4 is equal to 0. Well, that happens when x equals 4, we could just add 4 to both sides.

0:58So we know that f of 4 is 0, so we know that this is going to be a point on our curve. And then similarly, we could say, "Well, when x + 2," we can say, "When x plus 2 is equal to 0, our function is 0," so that happens when x is equal to negative -2. Subtract 2 from both sides.

1:19- 2 right there. So that is so far useful, but now how do we think about where the vertex is? Well, the vertex for a quadratic is going to be right halfway in between the two 0s. So we can see that halfway between these two points happens at x equals 1. This is three below the right one and three above the left 0.

1:43Now, what is the value of the function at x equals 1? Well, we could just figure that out. We could say that f of 1, let me do it over here. F of 1 is going to be equal to 1 - 4, which is -3, times 1 + 2 times 3, so this is equal to -9. So right over here, we have this point, that's the vertex.

2:05And now we can sketch what the actual graph looks like. It would look like. It would look like I'm trying to hand draw it, but you can kind of imagine a parabola that looks something like this, and I'm making it go through the vertex and the two 0s. Now, let's use the same idea to figure out the graph of g of x.

2:26Well, by that same notion, the 0s are gonna be when x is equal to -1,

2:31- 1, and when X is equal to -4, - 4. Now, where is the vertex going to be? Well, it's gonna be halfway in between -1 and -4, which is that -2.5, so right over there. So let's figure out what the function is equal to at -2.5. So let me, I will do it right over here.

2:55So g, do the same blue color that I was using before. G, it's a different blue, but -2.5 is equal to -2.5 plus 1 is -1.5, and then - 2.5 + 4 is going to be - 2.5, it's going to be times 1.5. Did I do that right? Yeah, 4 - 2.5 is 1.5. So we have 1.5 or -1.5 times 1.5, which is equal to -2.25.

3:33You might recognize 15 times 15 is 225, that's how I did it. Or I guess you could get a calculator out or something like that. So the point, the vertex is that -2.5 and then -2.25, which is right about there. So the function g of x is going to look, is going to look something, something, something like this, something like that.

3:59And we are done.