 In this video, I'm going to talk about reflecting functions. Now, when we talk about reflecting linear functions, again, linear functions are straight lines, linear, straight line. When we talk about reflecting functions, in this case, we're going to talk about reflecting them over either an x-axis or over a y-axis. I'm going to do examples of both. In this case, for this first slide, I'm going to do reflecting functions across a y-axis. So now, if you remember your axes, this is the x-axis, and this is the y-axis here. So what I'm going to do is reflect across the y-axis. So basically, this part here on the right is going to go to the left, and this part here on the left is going to go to the right. And I'm going to reflect this function, and this blue one here, f of x is equal to 2x plus 1 across the y-axis. So what we're going to do is I'm going to do the picture. I'm going to reflect it across the y-axis. And then what I'm going to do is I'm going to rewrite here. I'm going to rewrite the function. And then we're going to compare the two functions to see how we can read this notation down here, to see exactly what's happening. OK, so now to reflect this across, I need some points first. So y intercept is always a good point. Here's another point up here. Slope is 2, so we've got rise to run 1, rise to run 1. There's another point to use. Let's use another point down here. So let's use this point here, which is negative 2. What is that? 1, 2, 3. Negative 2, negative 3. OK, so I'm going to use those four points, and I'm going to reflect them across the y-axis. Now when you reflect points across the axis, the corresponding points are going to be equidistant from the line of reflection. Now it's a very fancy way of saying the points when you reflect them across, they're just going to be the same distance from your line that you reflect them across. So for example, this point up here, if I'm going to reflect across the y-axis, this right here is the distance of 1. When I reflect it across, the new point is still going to have a distance of 1. It's going to be right there. Now what about points that are on that axis? Well, they're just going to stay right there. That one's not going to move at all. This one down here is 1 away from the line of reflection, so that means over here it's going to be just 1 away. Same thing here. This one is 2 away from the line that I'm reflecting across, so over here it's going to be 2 away. It's going to be 2 away. All right, so now I have a couple of dots to go with. My new function is right here. Try to get a nice straight line in there. Try to get a nice straight line as straight as I can get that. I'm not a very good artist here. And that's my new function, G. So what I'm going to do is I'm going to now write the equation for this. So my G of x function, and I'm going to write this in slope-intercept form. y equals mx plus b, so I need the slope in the y-intercept. So as I look at my equation over here, let's do the y-intercept first, which is right here. So actually, the y-intercept doesn't change at all. If you notice, they kind of cross at that same point. So my y-intercept is going to be a plus 1. Plus 1 for the y-intercept, notice I left a gap there for the slope. My slope is rise 2, left 1. Drop 2, right 1. Drop 2, right 1. So I have a slope of a negative 2. Notice this line is going downhill. So I have a slope of a negative 2. Negative 2. Now that I have the equation for this new line, now let's compare that to the old one. So look at these two. The y-intercepts, they didn't change at all. But notice here, 2 and negative 2. The slopes, basically, are just opposites of one another. It's basically an easy way to look at it. OK, I went from 2 to negative 2. So in general, what happened? In general, what happened? If I want to reflect a function across the y-axis, basically what I do is I take that function. I want to reflect it across the y-axis by simply just changing the sign on the x. That's what we did. Right here, the slope on the x, we just changed the sign. That's all we really need to do. So that's what this is. Take your function. If you want to reflect across the y-axis, all you need to do is change the sign on the x. So you just take a negative 2 to the x. And this is reflect across the y-axis. So that's one way of explaining how to reflect across the y-axis. That's just one example of that. So now what I'm going to do is flip to my next slide. Now what I'm going to do is reflecting functions across the x-axis. So this one's just a little bit different from the last one. And we're just going to compare to see how they are different. But we're still taking that same function. f of x is equal to 2x plus 1. We're still taking the same function. And I'm going to reflect it across the x-axis. So here's my y-axis here that I just used. And here's my x-axis. That's the new one that I'm going to use. So basically what's going to happen is that this top part of the line is going to reflect to the bottom. And this bottom part of the line is going to reflect to the top. So just like last time, I need to find some points to reflect. So your y-intercept is always a good point. 1, 3 up here. Negative 1, negative 1. And then negative 2, negative 2, negative 3. There it is, right there. So here are my four points. I'm going to take those. And I'm going to reflect them across the x-axis. And then after I do that, we'll write the equation for it. We'll compare it. And then we'll see what it looks like in general, just like last time. OK, so and again, corresponding points are going to be equidistant from the line of reflection. Again, a very fancy way of saying the points, when I reflect them across, it's just going to be the same distance across. So let's start down here. So 1, 2, 3. There's a distance of 3 here. So I have to go up 3. 1, 2, 3. Remember, my axis that I'm reflecting across is the x-axis. So here's my line of reflection right here. This line is only one away. So when I reflect it, it's going to be one away. There we go. And then this point here is one away from the line of reflection. So again, it's going to be one away when I go down here. Now that's a little different from last time. The y-intercept last time didn't actually change. So that's a little difference from last time. Anyway, this point up here is 3 up. So I'm going to go 3 down. Now 1, 2, 3. Right about there. Right about there. And then I'm going to put a line through these points, put a line through these points, try to make that nice and straight as I can. And then this is my new function, g. We've got to label it so I know which one is which. OK, so now my new equation, g of x. And again, we're going to write this in slope-intercept form. I'm going to write this in slope-intercept form. What I'm going to do is I have to figure out what the y-intercept and what the slope is. Well, the y-intercept has actually changed. It is now negative 1. And my slope, a couple of points, 1, 2, 1, 1, 2, 1, 1, 2, 1. Looks like I have a slope of negative 2. I've got a slope of negative 2. OK, so that's the equation for my new slope. Now let's compare the blue and the black here. Compare these two lines. Notice what happened. Kind of the same thing that happened last time. The sign on this 2 changed here, but then also notice over here, the sign on this number also changed. So basically, we just took the opposite of everything. We just changed the sign of everything, not just the slope, not just the x, but everything got changed. So in general, if I want to reflect across the x-axis, I take my function. And if I want to reflect across the x-axis, I take the opposite of my entire function, opposite of my entire function here. And this is if I want to reflect across the x-axis. So I've got to do a little bit more work when I reflect across the x-axis, because apparently, not only does the slope just change to the opposite, but the y-intercept change. So it's like everything, the whole function, is going to be multiplied by a negative. We're going to change all the signs there. OK, so that's a short video on how to reflect linear functions across the x-axis and also across the y-axis.