If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Quadratic systems: graphical solution

CCSS.Math:

## Video transcript

solve the system of equations by graphing check your solution algebraically so let's graph each of these let's start let me find a nice dark color to graph these let's let me graph this top equation in blue this parabola so the first thing to think about is this going to be an upward-opening well one how did I know it's a parabola and that's because it's a quadratic function we have an x squared term we have a second-degree term here and then we have to think about it's going to be an upward opening or downward opening parabola and you see it's a negative coefficient in front of the x squared so it's going to be a downward opening parabola and what is going to be its maximum point let's think about that for a second so this whole term right here this whole term right here is always going to be negative so it's maximum or it's always going to be non positive x squared will be non-negative when you multiply it by a negative it's going to be non positive so the highest value that this thing can take on is when X is going to be equal to zero so the vertex of this parabola is when X is equal to zero and Y is equal to 6 so X is equal to 0 and Y is 1 2 3 4 5 6 so that right there is the highest point of our parabola and then if we want we can graph a couple of other points just to see what happens so let's see what happens when when let's say X is equal to so let me just draw a little table here so when X is equal to 2 what is y so negative x squared plus 6 so when X is 2 what is y you have 2 squared which is 4 but you have negative 2 squared so it's negative 4 plus 6 it is equal to 2 and the same thing when X is negative 2 because you put negative 2 there you square it then you have positive 4 but you have a negative there so it's negative 4 plus 6 is 2 so you have both of those points there so 2 comma 2 so 2 comma 2 and then you have negative 2 comma 2 negative 2 comma 2 and then if I were to graph it I could even do a let's try it with 3 as well so if we put a 3 there 3 squared is 9 you then it becomes a negative 9 plus 3 becomes negative 3 and negative 3 will also become an 8 three right negative three squared is positive nine you have a negative out front becomes negative nine plus six is negative three so you have negative three negative three and then you have three negative three so those are all good points and now we can graph our parabola our parabola will look something oh I was doing well until that second part let's see our parabola will look something like that let me just do the second part that second part is hard to draw something parabola almost let me do it from here almost it looks something like that there well now let me connect this dot right here and then let me connect this so it looks something like that that's what our parabola looks like and obviously it keeps going down in that direction so that's that first graph now let's graph this second one over here y is equal to negative 2x minus 2 this is just going to be a line it's a it's a lot it's a linear equation the highest degree here is one so our y-intercept is negative 2 so 0 1 2 our y-intercept is negative 2 and our slope is negative 2 so if we move one in the X direction were going to go 2 in the Y direction if we move 2 in the X direction we're going to move down 2 down for the Y direction we're going to move down 4 if we move back to we're going to move up 2 in the Y direction and it looks like we found one of our points of intersection and let's just draw that line so that line will look something like it's hard for my hand to draw on that let me let me try as best as I can so it will go something that my hand is having my tablet this is the hardest part it will look something like that right there and so the question is where do they intersect well one point of intersection does immediately pop out as at us they asked us to do it graphically that point right there which is the point negative 2 2 which is the point negative 2 2 seems to pop out of so this is point negative 2 2 and let's see if that makes sense when you have the point negative 2 well at it when you put X is equal to negative two here negative 2 times negative 2 is 4 minus 2 y is equal to 2 when you put negative 2 here Y is also equal to 2 so that makes sense and then there's going to be some other point way out here where they also intersect there's also going to be some other point way out here if we keep making this parabola let's see when Y is equal to positive 4 then you have negative 16 plus 6 you get negative 10 so positive 1 2 3 4 and then you go down 10 1 2 3 4 5 6 7 8 9 10 that looks like that might be our other point of intersection so let me draw let me connect this right there and then our other point of intersection looks to be right there if we if we just follow this red line it looks like we intersect there and let's verify that it works out so 4 negative 10 we know that that's on this blue line let's see if it's on this other line so negative 2 times 4 minus 2 that is negative 8 minus 2 which is equal to negative 10 so the point 4 negative 10 is on both of them when X is equal to 4 y is negative 10 for both equations here so they both definitely they both definitely work out