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This is part two of graphing parabolas, and on this video we graph the following three parabolas.
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This is part two of parabolas, and at the end of parabolas part one, we noted that a function of the form f of x equals x squared plus c is the parabola opening upward with vertex 0c,
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which also happens to be the y intercept, and then we have something to add on to this.
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The axis of symmetry is the y axis, which has the equation x equals 0.
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If you know what the vertex is, the axis of symmetry of a parabola that's a function will be a vertical line, and all vertical lines are x equals some number.
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It will be whatever the x coordinate of the vertex is, so that's why it will be x equals 0.
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We're going to look at a similar type of function that's also a parabola.
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Let's say we're going to graph the function f of x equals negative x squared.
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Remember, you could also think of that as y equals negative x squared.
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We had already felt in the chart before for y equals x squared. Remember, we just squared each of these numbers.
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Those are the order pairs we put in for the graph of y equals x squared, but we want y equals negative x squared.
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It's going to be the opposite of all these numbers. Negative 9, negative 4, negative 1, 0, negative 1, negative 4, and negative 9.
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The order pairs we want to graph on here are negative 3, negative 9, negative 2, negative 4.
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We're going to have a piece of graph paper and put the video on pause and graph those seven ordered pairs.
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This is what you should get. Here's negative 3, negative 9, negative 2, negative 4, negative 1, negative 1, 0, 0, 1, negative 1,
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2, negative 4, and 3, negative 9. And this time we have a parabola going down.
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In fact, it's a mirror image of the parabola y equals x squared, which was going up.
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If we were going to do y equals x squared or f of x equals x squared, that one opened upward.
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It was the basic look of this parabola going upward.
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So the difference here is that notice when you have x squared, that can't be negative,
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and then you're putting a minus sign in front of it. So all the y values are going to be below the x axis,
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so the y values are going to be negative. So that's what the graph of f of x equals negative x squared is.
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By the way, another way to do it is turn your paper upside down, graph y equals x squared, and then turn it back,
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you know, right side up again and you've got it. So that's another way to graph f of x equals negative x squared.
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Now, how about if we wanted to graph, let's get rid of this one.
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How about graphing in green f of x equals negative x squared plus seven?
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What do you think that might look like? Well, all the y values will be whatever negative x squared is,
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but plus seven more. So all I have to do is add seven to all of these y values,
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negative x squared plus seven. If I add seven to each of those numbers, we get negative two,
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negative three, negative six. Is that right? No.
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Oops. I take that back. We're adding seven, so that'll be positive three. Positive six, seven, six, three, and negative two.
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All right. An easier way to do is imagine just picking up, taking each of these order pairs that you see on the graph of y equals negative x squared and going up seven.
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So that goes up seven spaces, this goes up seven spaces, that goes up seven spaces, et cetera.
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Or I think of this as the origin, right up here. I pretend that's the origin. I go over one and down one, over two and down four.
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So there's lots of ways to think about that over three and see if we're going to have down negative two, downing.
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And we're going to get this, sorry, kind of up a little bit more, that graph.
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We have arrows on them because this parabola also goes down. It's kind of average looking again.
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But the vertex of this parabola is at zero seven. So the vertex here is zero seven, where the vertex for this parabola is zero zero.
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And hopefully you're noticing when something gets added here to the negative x squared, the vertex is going to still be zero c.
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But this minus sign means the whole parabola is going to go down. The axis of symmetry is still the y-axis for these.
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That's going to happen when there's no like x-term. Notice we have an x-squared term.
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We've got a constant, but there's not like a plus five x. There's no x-term. So let's see if we could summarize this.
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So a function of the form f of x equals negative x squared plus c is a parabola opening downward.
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So when you have a minus in front of the x-squared, it's going to go down. When it was positive in front of the x-squared, it went up.
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So it's a parabola opening downward. Still has the vertex zero c, which is also the y-intercept.
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And the axis of symmetry is the y-axis, which is the equation x equals zero. That's a vertical line.
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So let's have you try one.
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Alright, here's one for you to try. Graph f of x equals negative x squared plus four. Accurately is possible on some graph paper.
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State whether the parabola opens up or down. State the vertex. That's an order pair. And state the equation of the axis of symmetry.
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So put the video on pause and try it on your own first. You can either do this by plugging in some ordered pairs or using another method.
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Okay, here's how I do it. The minus sign tells me it goes down. Since there's no x-term, the vertex is the same as the y-intercept. It's just zero four.
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So I know it's going down and I have a vertex here at zero four. And that means the axis of symmetry is this y-axis.
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So this is the vertex. And I'm going to put the axis of symmetry a-o-s. It's x equals zero. It's always of the form x equals some number.
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And then instead of going down, instead of going over one and up one, I'm going over one and down one, over two and down four.
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I remember those basic ordered pairs. One squared, two squared, three squared, see? Over three. Down nine. Because that's three squared.
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Or you could just go on one half and automatically go to the other side and pick the other ones. And so there we go.
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That said, pretty accurate graph of f of x equals negative x squared plus four.
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The last thing you want to note is that often you're asked to label the axis of symmetry. And so we usually do a little dotted line to show where the axis of symmetry is.
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So in this case, it's just the y-axis. So we put a dotted line here and x is the symmetry x equals zero. So if you're asked to label the axis of symmetry, you use a dotted line for the axis of symmetry.
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If you're asked to label the vertex right here at the vertex, you could put v for vertex.
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So now you should be able to graph. Proudly is going up and down that have either just x squared plus c or negative x squared plus c.
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And of course, this could be a, you're adding a positive number or you could be adding a negative number. It could also be negative x squared minus four, for instance, that works as well.
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So those are the ones that we've done on parts 1 and 2.