Graphing parabola from quadratic in factored form

We're asked to graph the equation ( Y = 12 \cdot x - 6 \cdot x + 2 ) and so, like always, pause this video and take out some graph paper or even try to do it on a regular piece of paper and see if you can graph this equation.

Alright, now let's work through this together. Now, there are many different ways that you could attempt to graph it. Maybe the most basic is to try out a bunch of ( x ) values and a bunch of ( y ) values and try to connect the curve that connects all of those dots. But let's try to see if we can get the essence of this graph without doing that much work.

The key realization here, without even having to do the math, is if I multiplied this out, if I multiplied ( x - 6 \cdot x + 2 ), I'm going to get a quadratic. I'm going to get ( x^2 ) times plus something plus something else. So, this whole thing is going to be a parabola. We are graphing a quadratic equation now.

A parabola, you might remember, can intersect the x-axis multiple times. So, let's see if we can find out where this intersects the x-axis. The way the form that it's in is factored form already. It makes it pretty straightforward for us to recognize when does ( y ) equal ( 0 ), which are going to be the times that we're intersecting the x-axis. Then, from that, we'll actually be able to find the coordinates of the vertex, and we're going to be able to get the general shape of this curve, which is going to be a parabola.

So, let's think about it: when does ( y ) equal ( 0 )? Well, to solve that, we just have to figure out when this expression equals zero. So, let's just solve the equation ( 12 \cdot x - 6 \cdot x + 2 = 0 ).

Now, in previous videos, we've talked about this idea: if I have the product of multiple things and it needs to be equal to zero, the only way that's going to happen is if one or more of these things are going to be equal to zero. Well, one is ( \frac{1}{2} ); it's not going to be equal to zero. But ( x - 6 ) could be equal to ( 0 ).

So, if ( x - 6 = 0 ), then that would make this equation true; or if ( x + 2 = 0 ), that would also make this equation true. The ( x ) values that satisfy either of these would make ( y = 0 ) and those would be places where our curve is intersecting the x-axis.

So, what ( x ) value makes ( x - 6 = 0 )? Well, you can add six to both sides; you probably able to do that in your head and you get ( x = 6 ). Or, you subtract two from both sides here and you get ( x = -2 ). These are the two ( x ) values where ( y ) will be equal to ( 0 ) and you can substitute it back into our original equation.

If ( x = 6 ), then this right over here is going to be equal to ( 0 ) and then ( y ) is going to be equal to ( 0 ). If ( x = -2 ), then this right over here is going to be equal to ( 0 ) and ( y ) would be equal to ( 0 ).

So, we know that our parabola is going to intersect the x-axis at ( x = -2 ) right over there and ( x = 6 ). These are our ( x ) intercepts. So, given this, how do we figure out the vertex?

Well, the key idea here is to recognize that your axis of symmetry for your parabola is going to sit right between your two ( x ) intercepts. So, what is the midpoint, or what is the average of ( 6 ) and ( -2 )? Well, you could do that in your head: ( 6 + -2 = 4 ) divided by ( 2 ) is ( 2 ).

Let me do that, so I'm just trying to find the midpoint between the point ( (-2, 0) ) and ( (6, 0) ). Well, the midpoint is just the average of the coordinates. The average of ( 0 ) and ( 0 ) is just going to be ( 0 ); it's going to sit on the x-axis.

But then, the midpoint of ( -2 ) and ( 6 ), or the average ( (-2 + 6)/2 ), well let's see that's ( 4/2 ); that's just going to be ( 2 ). So, ( (2, 0) ).

You see that there; you might have done that without even doing the math. You're saying, "Okay, if I want to go right in between the two, I want to be two away from each of them." And so just like that, I could draw an axis of symmetry for my parabola.

My vertex is going to sit on that axis of symmetry, and so how do I know what the ( y ) value is? Well, I can figure out, I can substitute back into my original equation to say, "Well, what is ( y ) equal when ( x = 2)?" Because remember, the vertex has a coordinate where ( x = 2 ); it's going to be ( (2, something) ).

So, let's go back and see what ( y ) equals. So, ( y ) will equal ( 12 \cdot \text{when} , x = 2 ). So, ( 2 - 6 \cdot 2 + 2 ). Let's see, this is ( -4 ); this is positive ( 4 ).

So, ( -4 \cdot 4 = -16 ). So, it's equal to ( 12 \cdot -16 ), which is equal to ( 8 ). So, our vertex, when ( x = 2 ), ( y ) is equal to ( 8 ).

Now we can draw the general shape of our actual parabola. It's going to look something like, once again, this is a hand-drawn sketch, so take it with a little bit of a grain of salt, but it's going to look something like this.

It's going to be symmetric around our axis of symmetry; that's why it's called the axis of symmetry. This art program I have has a symmetry tool, but I'll just use this. And there you go, that's a pretty good sketch of what this parabola is or what this graph is going to look like, which is an upward-opening parabola.