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CCSS.Math: ,

we're asked to graph the equation y is equal to 1/2 times X minus 6 times X plus 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 try out a bunch of X values as 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 and the key realization here without even having to do the math is if I multiply this out if I multiplied X minus 6 times X plus 2 I'm going to get a quadratic I'm going to get x squared time plus something plus something else and 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 and the way the form that it's in it's in 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 and 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 if we want to know when y equals 0 or if we want it then we have to solve for when is this expression equal 0 so let's just solve the equation 1/2 times X minus 6 times X plus 2 is equal to 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 0 the only way that's going to happen is if one or more of these things are going to be equal to 0 well 1/2 is 1/2 it's not going to be equal to 0 but X minus 6 could be equal to 0 so if X minus 6 is equal to 0 then that would make this equation true or if X plus 2 is equal to 0 that would also make equation true so the X values that satisfy either of these would make y equal zero and those would be places where our curve is intersecting the x axis so what x value makes X minus six equals zero well you can add 6 to both sides you're probably able to do that in your head and you get X is equal to 6 or you subtract 2 to book 4 from both sides here and you get X is equal to these cancel out you get X is equal to negative 2 these are the two X values where Y will be equal to 0 you can substitute it back into our original equation if X is equal to 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 is equal to negative 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 equals negative 2 right over there and X is equal to 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 and so what is the midpoint between or what is the average of 6 and negative 2 well you can do that in your head 6 plus negative 2 is 4 divided by 2 is 2 let me do that so I'm just trying to find the midpoint between the point there's a new color so I'm trying to find the midpoint between the point negative 2 comma 0 and 6 comma 0 well the midpoint those are 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 negative 2 and 6 or the average negative 2 plus 6 over 2 well let's see that's 4 over 2 that's just going to be 2 so 2 comma 0 and you see that there you might you could have done that without even doing the math you think ok if I want to go right in between the two I want to be 2 away from each of them and so just like that I could draw an axis of symmetry for my parabola and so 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 in my original equation say well what is y equal when X is equal to 2 because remember the vertex has a coordinate x equals 2 it's going to be 2 comma something so let's go back let's see what y equals so Y will equal to 1/2 x we're going to see when x equals 2 so 2 minus 6 times 2 plus 2 let's see this is negative 4 this is positive 4 negative 4 times 4 is negative 16 so it's equal to 1/2 times negative 16 which is equal to negative 8 so our vertex when X is equal to 2 y is equal to negative 8 and so our vertex is going to be right over here 2 comma negative 8 and 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 and 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 there's 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 it is and open an upward-opening parabola