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What I like to do is show you how to graph a polynomial function.
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We're not going to be using our graph and calculator, what we're going to do is we're going
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to learn how to graph a polynomial function by finding the zeros and by using the leading coefficient test.
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The first thing I want to do is I have my little xy axis here.
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The first thing I want to do is find our zeros. Remember our zeros are going to be an x-intercepts.
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Our zeros are when f and x equals zero. That means our output value, which is f and x, is going to equal zero.
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Remember, that's the same thing as here's your y-axis. It's kind of the same thing as your f of x-axis.
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It's dealing with your output.
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So when that value is zero, we're on the x-axis.
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So I have negative four x cubed plus four x squared plus fifteen x.
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First thing I want to do is see what can I take out of all these.
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Can I factor this in any form?
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I always look and they all share an x. So I'll see if I can take out an x.
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So when I take out an x, I'm also going to notice that's a negative.
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I don't ever want to have, usually I don't like dealing with my negatives in front of my leading term.
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So I'm going to factor the negative x.
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Negative. So just make sure every sign is not going to change.
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All right. Now for the sake of time on the video, the video is not made out of factory.
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If you guys need help on how to factor this.
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I'll be more than happy to show I have all the videos to show, but I'm not going to spend time right now going over the factory.
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This is something you guys can figure out.
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Use the diamond method. Do whatever kind of method you need to.
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But I can factor this further.
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Zero equals a negative x.
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And I previously did it.
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So it ended up being two x minus five.
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Two x minus five times two x plus three.
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So now that I have my factors, we can let this, the next thing I'm going to want to do is
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I need to find exactly what the zeros are.
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So here's the list of, you know, the factors, I need to write them as zeros.
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So I say zero equals negative x.
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Zero equals two x minus five.
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And zero equals two x plus three.
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So when I solve and x, x equals zero and five, divided by two, x equals a five half.
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And here I'm going to subtract by three, bad, all divided by two.
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And what I end up getting is x equals a negative three half.
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So for those of you who are decimal people, what do we have?
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Now I have these three points.
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So to what I want to do is I want to grab these three points.
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So I go back over my table and I just do one, two, three, four, one, two, three, four.
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So I go zero.
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That's one x intercept.
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The next one was two point five right here.
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And third one is a negative one point five.
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Right there.
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Now, another thing you guys should know is these are all the multiplicity of one.
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Each one of these had an exponent of one.
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So they're at odd multiplicity and they're going to be a multiplicity of one.
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One thing we need to remember about that is when they have an odd multiplicity,
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do we know that it's going to cross the abses at the intercepts?
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If they're even, if they're even, then we know they'd actually only touch at these points.
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But fortunately for us, we know that they're going to be odd.
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So they're going to cross at each one of those points.
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The next thing I need to look at is I need to look at our leading coefficient.
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Or the leading coefficient test.
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And what you guys see is that is negative four x cubed.
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And when you're looking at negative four x cubed, the first thing we do is we look at if it's odd or even.
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Since it's odd, we know it's going to fall and it's going to rise.
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Because even functions both fall or both rise.
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Odd functions rise and fall.
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And the next thing I need to determine is if it's going to be, if my A, the number in front of my very term,
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is net positive or negative.
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If it's negative, we rise left, fall right.
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If it's positive, you fall left, rise right.
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So this is all, like I said, all of my videos you guys can see for as far as how to play the leading coefficient test.
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So I know that this is a negative.
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So therefore my final n behavior, my graph is going to go down like that and it's going to fall to the right.
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I don't know what's going to happen in between here, but I know it's going to be fall to the right and rise to left.
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So the last thing we need to do.
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We have, we're going to do a little kind of little table points.
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We have some points, we have zero, zero, we have negative 1.5, zero, and we have 2.5, zero.
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Well, one thing you guys can remember is that the Intermediate Value Zero says that between any two points,
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we're going to be able to find a third point.
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So there always exists another point between these two points. So we know there's a point that's between these two intercepts,
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and we know there's a point between these two intercepts.
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It's always good thinking to always check your work and always do a point outside of your last point and outside of this point,
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which would have been like a negative 2 and for here would have been 1, 2, 3.
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For time purposes, I'm not going to evaluate them.
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Actually, I think I might have already.
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Actually, I did.
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But it's really important to make sure you find the values between your two intercepts.
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So to do that, you need to evaluate for a point and just pick a point.
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Well, it's pretty easy to do negative 1 and 1.
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So you do f of negative 1, which equals negative 4 times negative 1,
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cubed plus 4 times negative 1 squared plus 15 times negative 1.
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And when you do the math, you get negative 1 cubed is a negative 1 times 4 is going to be a positive 4.
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And plus negative 1 squared is a positive 1 plus 4 is 4 and plus negative 15.
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And therefore, you get a negative 7.
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So negative 1 equals a negative 7.
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And for f of 1, you do the exact same thing, but these are all going to be positives.
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So what you actually end up getting is obvious right now.
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And I end up getting what I do in my math.
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I get 1 count of 15.
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So therefore, I go to negative 1 and I go down 7, 1, 2, 3, 4, 5, 6, 7,
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 12, OK.
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It's going to be somewhere nearby, right there.
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So to graph, I now, my graph has to go through these three points.
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And then you can see, and this is at 1, 15, and this is at negative 1, 7.
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OK, my apologies for not really having the most perfect graph.
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But you guys can see all you really need to do to find graph something like that, to graph upon it,
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the function is find the zeros, understand the leading coefficient test, determine the end behavior,
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and then just find your extra points, and you can easily plot the graph.
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And that's pretty much what you guys need to do.
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Go ahead and make the support for the main office.