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Solve the system of equations by graphing.

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Check your solution algebraically.

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Let's graph each of these, and let's start-- let me find a

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nice dark color to graph these with.

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Let me graph this top equation in blue, this parabola.

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The first thing to think about is this going to be an upward

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opening-- one, how did I know it's a parabola?

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That's because it's a quadratic function: we have an

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x squared term, a second degree term, here.

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Then we have to think about: it is going to be upward

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opening, or downward opening parabola?

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You see that it's a negative coefficient in front of the x

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squared, so it's going to be a downward opening parabola.

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What is going to be its maximum point?

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Let's think about that for a second.

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This whole term right here is always going to be negative,

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or it's always going to be non-positive.

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x squared will be non-negative when you multiply it by a

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negative, so it's going to be non-positive.

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So, the highest value that this thing can take on is when

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x is going to be equal to 0-- the vertex of this parabola is

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when x is equal to 0, and y is equal to 6.

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So, x is equal to 0, and y is 1, 2, 3, 4, 5, 6.

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So that right there is the highest point of our parabola.

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Then, if we want, we can a graph a couple of other

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points, just to see what happens.

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So let's see what happens when x is equal to-- let me just

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draw a little table here-- 2, what is y?

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It's negative x squared plus 6.

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So when x is 2, what is y?

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You have 2 squared, which is 4, but you have negative 2

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squared, so it's negative 4 plus 6-- it is equal to 2.

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It's the same thing when x is negative 2.

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You put negative 2 there, you square it, then you have

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positive 4, but you have a negative there, so it's

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negative 4 plus 6 is 2.

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You have both of those points there, so 2 comma 2, and then

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you have a negative 2 comma 2.

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If I were to graph it, Let's try it with 3, as well-- if we

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put a 3 there, 3 squared is 9.

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It then becomes a negative 9 plus 3, it becomes negative 3,

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and negative 3 will also become a negative 3.

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Negative 3 squared is positive 9, you have a negative out

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front, it becomes negative 9 plus 6, which is negative 3.

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You have negative 3, negative 3, and then you have 3,

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negative 3.

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So those are all good points.

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Now we can graph our parabola.

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Our parabola will look something-- I was doing well

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until that second part --like that, and let me just do the

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second part.

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That second part is hard to draw-- let me do it from here.

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It looks something like that.

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We connect to this dot right here, and then

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let me connect this.

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So that it looks something like that.

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That's what our parabola looks like, and obviously it keeps

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going down in that direction.

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So that's that first graph.

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Let's graph this second one over here: y is equal to

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negative 2x minus 2.

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This is just going to be a line.

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It's a linear equation, and the highest degree here is 1.

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Our y-intercept is negative 2, so 0, 1, 2.

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Our y-intercept is negative 2.

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Our slope is negative 2.

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If we move 1 in the x direction, we're going to go 2

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in the y-direction, and if we move 2 in the x direction,

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we're going to move down 4 in the y direction.

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If we move back 2, we're going to move up 2 in the y

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direction, and it looks like we found one of our points of

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intersection.

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Let's just draw that line, so that line will look something

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like-- It's hard for my hand to draw that, but let me try

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as best as I can.

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This is the hardest part.

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It will look something like that right there.

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The question is, where do they intersect?

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One point of intersection does immediately pop out at us,

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because they asked us to do it graphically.

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That point right there, which is the point negative 2, 2.

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It seems to pop out at us, so this is the

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point negative 2, 2.

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Let's see if that makes sense.

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When you have the point negative 2, when you put x is

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equal to negative 2 here, negative 2 times negative 2 is

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4 minus 2, and y is equal to 2.

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When you put negative 2 here, y is also equal to 2, so that

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makes sense.

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There's going to be some other point way out here where they

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also intersect.

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There's also going to be some other point way out here if we

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keep making this parabola.

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When y is equal to positive 4, and you have negative 16 plus

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6, you get negative 10.

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So, positive 1, 2, 3, 4, and then you go down 10.

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1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

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That looks like that might be our other point of

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intersection, so let me connect this right there.

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Our other point of intersection

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looks to be right there.

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If we just follow this red line it looks like we

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intersect there.

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Let's verify that it works out.

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So 4, negative 10.

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We know that that's on this blue line, so let's see if

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it's on this other line.

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So negative 2 times 4 minus 2, that is negative 8 minus 2,

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which is equal to negative 10.

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The point 4, negative 10, is on both of them.

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When x is equal to 4, y is negative 10 for both equations

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here, so they both definitely work out.

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