 In the previous video, we learned how to graph the standard monomial functions like y equals x squared y equals x cube y also x to the fourth what have you and we saw the very basic pattern that if you have y equals x raised to an even power. Your graph is always going to have the following bucket like shape where you think of these as the points one one negative one one and then zero right here it gets really shallow near the origin and steep otherwise the bigger you make the power. And if you take y equals x to some odd power, you see something like the following where here is your point one one here is your point negative one negative one and the origin is between it. So again as the power gets bigger bigger bigger it gets shallow by the origins deeper everywhere else we have this basic shape when it comes to graphing these monomial functions. Okay, so if we want to graph the function f of x equals one minus x to the fifth. I didn't want to mention, well we can graph this using transformation techniques that we've seen in the past. So we're going to start off with the very basic function y equals x to the fifth, as this is an odd power monomial it's going to have this basic shape it should be steeper than the standard x cube, but not too steep. And so that you see a curve the curve right this dash curve right here in white. I'm going to take those off the screen. So that's going to our standard graph. What have we done to this function? Well there's a negative sign in front of x to the fifth. That tells us that there's going to be some type of reflection reflection here and sticking a negative sign in front of our function is going to be reflection across the x axis. So if you take this white function and reflect it across the x axis you're going to get a picture looks something like the following. So we have a reflection across the x axis. I want to mention that for a moment if you take away the line here if you look at just this graph right here is this a reflection across the x or is this a reflection across the y axis. And as you look at those you might it looks like both right. This is the key thing about being an odd function odd functions are those functions which are symmetric with respect to the origin. One way of describing an odd function is that an odd function. I should say a function is odd if and only if reflection across the x axis is identical to reflection across the y axis. So it turns out odd functions have this property that you can do either reflection and does the exact same thing. So reflect the white curve across the x axis, and you get the green curve right here. And then what about what about the one what is the one doing here. This is going to be a vertical shift up by a factor of one so you take your green curve and you're going to move everything up by one. You're going to move everything up by one everything up by one and personally when it comes to graphing these things. I like to think of what happens to these points. The three standard points as you go through this process like if you take the origin. If you reflect the origin across the x axis it doesn't change. Because reflection across the x axis means you're going to multiply the y corner by negative one. If you then shift everything up by one you're adding one to the white corner you're going to get the point zero one. And so what I'm going to do is I'm actually going to move this over here on the graph it something like this right here. So the vertex of the the vertex of this power function right you went you start off the origin you reflect it which doesn't do anything that you can move it up right to shift it. So we get the point zero comma one that's what happened there. What about some of the other points let's consider the point one one what happens to it. Well when you reflect that across the x axis becomes the point one comma negative one. And then when you shift things up by one that moves it to be one zero. So we start off with this point zero or this point one one you got reflected down here to be one negative one. Then you're going to shift it up by one. And so if we look at just the net change that point moves here and we get the point one comma zero. And then what about the third point. If we take the point negative one negative one by transformation reflection will send that to be negative one positive one and then by shifting it that'll go to negative one comma two. So we started off with the point negative one negative one that reflects up to become negative one one and then you shifted one more upward to give you one two. And if we record just the net change there. You're going to get this point right here. Negative one comma two. And then using these pictures are using these three points we're going to try to connect the dots to give us our curve right here. And so this is how we can draw this thing even by hand. If we keep track of what happens to those three guide post zero zero one one and negative one one. I guess that point should be more like right here. Here's one one and here's negative one one. If we just keep track of what happens those three points we can graph this odd monomial by transformations. Let's look at an example where we have we're going to do some even transformations to this picture. So let's take g of x to be one half x minus one to the fourth power. We have to find a first a basic function. So our basic function is going to be the monomial y equals x to the fourth, which will have the basic shape like this. It'll be obviously no sharp points. It'll be smooth, but it's going to have that shape here. It's going to be steeper than a standard parabola. Now what transformations have we done to our x to the fourth right here? This factor of one half in front. This is doing a scale change. We've rescaled the function. In other words, we've compressed the y coordinate by a factor of two. So if we think about what happens to the origin here, zero zero here, when you vertically compress it, I'm going to move this a little bit higher. When you vertically compress it, you're going to multiply all of the y coordinates by one halves, which of course times something by zero will still give you zero. So you don't see a change if you're on the x axis of vertical compression doesn't do anything. On the other hand, if you take the point one one when you vertically compress it you cut its y coordinate in half. So you get the point one comma one half and same thing for negative one comma one, which is on this graph. You'll end up with negative one comma one half. So everything got vertically compressed. This function gets compressed vertically to give us these points right here, negative one and a half and one and a half, like so. That's what this one half was doing to the picture. What is the x minus one doing? Well, because we have an x minus one inside of the horizontal zone, this represents that we have a horizontal shift to the right by a factor of one. So we're going to take this entire picture and move everyone to the left by one. So the origin moves over by one. And so what that does is it's going to give you the point zero comma one, which we see right here. The point one comma one half is going to move over by one. This is going to give us the point two comma one half. We're just adding one to the x coordinate there. That's all we're doing. And then lastly, what do we do to the point negative one comma one half will move that point over by one. And so that gives us the point zero comma one half. And that's the picture we see right here. Everything got moved over to the right by one. And again, we can use those points to help us graph this line right here as we'll graph the curve. So if we think about what happened to the origin right zero zero zero, it started at zero zero then when it got stretched or compressed vertically didn't do anything. Then we moved it over by one, and that gives us the point zero one comma zero. Then take the point one one. When it got compressed, we move it down by half so it's it's horizontal component got us compressed by a factor of half then we moved it one to the right. And so we end up with the point right here. Two comma one half. Let's take the point negative one one it got vertically compressed so it's why corner got cut in half. Then we moved it one to the right. And so we get this point right here, which would be zero comma one half. And this is going to be x to the fourth so it should be somewhat steep when you're away from the vertex. But it should be somewhat flat, flatter than a standard parabola. So you get a picture looks like this isn't perfectly drawn no, but this really does illustrate here how we can graph power functions particularly monomial functions using the standard transformations that we've learned about in the past.