And I'm really not going to plot a lot more points because I just want to see the idea behind it. The point that used to be at 1, comma, 1, is now going to end up being at 1, comma, 3. So for instance, the point that used to be at the origin is now going to be 2 units up. So everything is going to be shifted upward 2 units. So it's like taking your old y value and then adding 2 to it. So we notice that this is in the format where what I'm doing is applying the function- that is, I'm cubing first- and then I'm adding 2 at the end. What I want to now graph is f of x equals x cubed plus 2. So that's going to be the graph of y equals x cubed. And in the third quadrant, it's going to continue downward like that. We can plot the 1, 1, and we know that in the first quadrant, it's going to continue upward like this. We're going to have- it goes through the origin. Remember that the x cubed function- we can plot a few points if you'd like. So notice that I'm thinking about having an original function, y, which is equal to x cubed, and that should be one of the functions that we're familiar with. So I'm going to look at the function f of x equals x cubed plus 2. So let's look at an example of where we have to do this by hand. You can see that that happens in this diagram. If it turns out that the value of k is negative, so that you'd be subtracting at the end of this function, then we're going to shift the graph down k units. So for instance, this whole graph would move upward. If my value of k is positive- so if I end up writing, for instance, f of x plus 2 or plus 1, then we're going to shift the graph up k units. So in my diagram, you'll see that I have drawn a function that is going to be given as f of x. The graph of y equals f of x plus k is a vertical shift of the graph of y equals f of x. Let's start off with the following diagram. So now that we have some basic graphs that we're familiar with, we're going to use those in order to look at some transformations, and the first transformation we want to consider is a translation, which is going to be a shifting. So again, these are functions that we want to be familiar with, so that we can use them in order to do some problems. And so it looks kind of like a parabola- half of a parabola on its side. The domain will be all values of x greater than or equal 0, and the range is all values of y greater than or equal to 0. Then we look at the square root function, which is going to have the equation f of x equals the square root of x. The range would be all values of y greater than or equal to 0. So again, the distinctive V-shape of the absolute value function. As you substitute in to the absolute value, then your y value ends up being positive, also. The part that's in quadrant II is actually if I take my negative xs and now make them positive. The first part that's in the quadrant is the same as y equals x. It's actually two- you think of it as a linear function that's been separated. This is going to have the distinctive V shape. Let's now look at f of x equals the absolute value of x. The domain is all real numbers, and so is the range. And we can see that it increases as we move from left to right. Notice that we have a section that's in quadrant I and a section that's in quadrant III. Another function that we want to be familiar with is f of x equals x cubed, also knows as the cubing function. We have a range from 0 to infinity, including 0, and we can see where it increases and decreases. The other thing to point out is that we have a domain of all real numbers. Its vertex will be at the origin, and we have this symmetry to the y-axis in this format. Now, this is going to have the shape of a parabola. Let's start off with f of x equals x squared. That is, you should be familiar with them, but it's worth the time at the outset to make sure we're familiar with these different functions that we're going to use in this session. To begin with, let's review some functions that you should already have at your beck and call. We'll look at reflections, compressions and stretches, and then combinations of transformations. We'll look at vertical and horizontal shifts. Now, in particular, we're going to look at some specific types of transformations you can do on functions in order to change their graphs. During this session, we're going to look at a topic called transformations.
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