Dragic functions now in order, Thio know how toe graph quadratic functions. We still we first want to look at what quadratic functions are care. We have an example of a quadratic function. F of X equals two X squared minus eight X plus. Nine quadratic functions are commonly try No me. ALS, however it can be a binomial is not limited to just to try no mule, and it's consists of three terms. It consists of a term which we is that what we actually call the quadratic and the quadratic is usually the quiet time, which is usually followed by X Square. It's got a B term, which is followed by just the X, or whatever variable is being used, and it has a C term, which is a constant now. The first term the a term for this problem would actually be to R B would be a negative eight, and R C would be positive nine. And just like in linear equations to see also determined the Y intercept. When you're looking at a graph of a quadratic function, you're looking at what we call a parabola, and a parable is a shape. It's usually like you, and it usually looks similar to this right here. It's either going is open upward. RS open downward the first term, the term actually helps determine how it's open. Is that opened up or is it open down? And the best way to know is that if it's a positive term, it's going to be opened up kind of almost like a smiley face. When you smile, you're in a positive mood and your smile is a when it is a when the A term is negative is going downward again? Kind of Think of a smiley face when you're frowning, you're in a negative mood, and so your frown is pointing down. In order to graph these, we're going to create an X Y table. But instead of because we're doing a function instead of an equation, we're not gonna have an X Y table as much as we're going to have and X f of X table. But it's the same concept. So I'm gonna go ahead and I'm gonna write down because your eggs column and in our second column we're gonna have our actual quadratic function in our third column. We're gonna have our Alfa vex. I remember the Alfa Vex is the same as y. So this is how it's similar to an X Y table. And then finally, in our last columnist where we can actually right out our coordinates. So let's start with some coordinates. So for this one, let's do the coordinates. Zero, 12 three and four. Those were gonna be our X coordinates. So we're gonna start off and we're going to put just like we would. It was a linear equation. We're gonna put whatever value for X in place of the X. So I'm gonna put zero in place of my ex. And when I worked that out, I'm going to get nine as my solution. So that means that my first set of coordinates is zero nine. Next I'm going to use to I'm gonna use one. And when I plug that in and work it out, I'm going to get three as my f of x coordinate. So my 1st 2nd set of coordinates should be 13 Next we would have to in place of X, we were sub that in. And once I worked that out, I'm going to get one, so I have to One for three. We have two times three squared, minus eight times three plus nine and we would get three. So I have three. Three and for four would be two times four squared, minus eight times four plus none. And I would get Not now. It is very common to see kind of a pattern in your Y coordinates. If you'll notice we start off with nine and we go to three and we get toe one and the end we come back to Not so it kind of almost starts to repeat. This point right here is actually going to become our Vertex. And you're gonna see this better when we graph. This is gonna be the point that the line is actually gonna change direction and go the opposite way. So let's start. Graph of these and a marauder coordinates down. So our coordinates were 09 13 21 33 and 49 So we're gonna graph thes points, so we're gonna have 09 and we said that would actually end up being our Y intercept, which it is. One, three, 21 33 and four nine. So now that we've graft are points. You can kind of see the way the line goes. It's going to curve right here and change directions. It is a positive parabola. Now, if you'll notice where I said the 21 point, which is this point right here, that's called our Vertex. And that is the point in which our line is going to change its direction. We also have a couple of other things that we can look at The Y intercept we've already said was gonna be 09 and that was gonna be our We got this that from our C term. And we can also find our Vertex from our equation by itself. So our equation waas Alfa Vex equaled two X squared minus eight X plus nine and we can use what we call the axes of symmetry. And the axis of symmetry says that the X coordinates for that vertex and that's gonna be the point at which is gonna be kind of almost the center of the parabola is going to be my B coordinate minus two times a. So for this one, my axis of symmetry would be a this point right here. It would be the point that divides the parabola in half. So let's look at that to see what those actual X coordinates are. So for be, that would be negative. Eight over negative, too. Times my a is to so negative eight over negative four, which would give me positive, too. Well, if I look at my graph that access a symmetry that dotted line I just drew is where X equals two. And so you can plug that in. And we've already said that is the same as my vertex. And as we do some more examples, you're going to see how we're gonna use that to help us find and draw are parabolas out.

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## Video Transcript

Dragic functions now in order, Thio know how toe graph quadratic functions. We still we first want to look at what quadratic functions are care. We have an example of a quadratic function. F of X equals two X squared minus eight X plus. Nine quadratic functions are commonly try No me. ALS, however it can be a binomial is not limited to just to try no mule, and it's consists of three terms. It consists of a term which we is that what we actually call the quadratic and the quadratic is usually the quiet time, which is usually followed by X Square. It's got a B term, which is followed by just the X, or whatever variable is being used, and it has a C term, which is a constant now. The first term the a term for this problem would actually be to R B would be a negative eight, and R C would be positive nine. And just like in linear equations to see also determined the Y intercept. When you're looking at a graph of a quadratic function, you're looking at what we call a parabola, and a parable is a shape. It's usually like you, and it usually looks similar to this right here. It's either going is open upward. RS open downward the first term, the term actually helps determine how it's open. Is that opened up or is it open down? And the best way to know is that if it's a positive term, it's going to be opened up kind of almost like a smiley face. When you smile, you're in a positive mood and your smile is a when it is a when the A term is negative is going downward again? Kind of Think of a smiley face when you're frowning, you're in a negative mood, and so your frown is pointing down. In order to graph these, we're going to create an X Y table. But instead of because we're doing a function instead of an equation, we're not gonna have an X Y table as much as we're going to have and X f of X table. But it's the same concept. So I'm gonna go ahead and I'm gonna write down because your eggs column and in our second column we're gonna have our actual quadratic function in our third column. We're gonna have our Alfa vex. I remember the Alfa Vex is the same as y. So this is how it's similar to an X Y table. And then finally, in our last columnist where we can actually right out our coordinates. So let's start with some coordinates. So for this one, let's do the coordinates. Zero, 12 three and four. Those were gonna be our X coordinates. So we're gonna start off and we're going to put just like we would. It was a linear equation. We're gonna put whatever value for X in place of the X. So I'm gonna put zero in place of my ex. And when I worked that out, I'm going to get nine as my solution. So that means that my first set of coordinates is zero nine. Next I'm going to use to I'm gonna use one. And when I plug that in and work it out, I'm going to get three as my f of x coordinate. So my 1st 2nd set of coordinates should be 13 Next we would have to in place of X, we were sub that in. And once I worked that out, I'm going to get one, so I have to One for three. We have two times three squared, minus eight times three plus nine and we would get three. So I have three. Three and for four would be two times four squared, minus eight times four plus none. And I would get Not now. It is very common to see kind of a pattern in your Y coordinates. If you'll notice we start off with nine and we go to three and we get toe one and the end we come back to Not so it kind of almost starts to repeat. This point right here is actually going to become our Vertex. And you're gonna see this better when we graph. This is gonna be the point that the line is actually gonna change direction and go the opposite way. So let's start. Graph of these and a marauder coordinates down. So our coordinates were 09 13 21 33 and 49 So we're gonna graph thes points, so we're gonna have 09 and we said that would actually end up being our Y intercept, which it is. One, three, 21 33 and four nine. So now that we've graft are points. You can kind of see the way the line goes. It's going to curve right here and change directions. It is a positive parabola. Now, if you'll notice where I said the 21 point, which is this point right here, that's called our Vertex. And that is the point in which our line is going to change its direction. We also have a couple of other things that we can look at The Y intercept we've already said was gonna be 09 and that was gonna be our We got this that from our C term. And we can also find our Vertex from our equation by itself. So our equation waas Alfa Vex equaled two X squared minus eight X plus nine and we can use what we call the axes of symmetry. And the axis of symmetry says that the X coordinates for that vertex and that's gonna be the point at which is gonna be kind of almost the center of the parabola is going to be my B coordinate minus two times a. So for this one, my axis of symmetry would be a this point right here. It would be the point that divides the parabola in half. So let's look at that to see what those actual X coordinates are. So for be, that would be negative. Eight over negative, too. Times my a is to so negative eight over negative four, which would give me positive, too. Well, if I look at my graph that access a symmetry that dotted line I just drew is where X equals two. And so you can plug that in. And we've already said that is the same as my vertex. And as we do some more examples, you're going to see how we're gonna use that to help us find and draw are parabolas out.

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